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Vibroacoustic behavior of a partially immersed cylindrical shell under point-force excitation: Analysis and experiment
Applied Acoustics 161 (2020) 107170
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Vibroacoustic behavior of a partially immersed cylindrical shell under point-force excitation: Analysis and experiment Kaiqi Zhao, Jun Fan ⇑, Bin Wang, Weilin Tang Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 5 July 2019 Received in revised form 11 November 2019 Accepted 26 November 2019
Keywords: Vibroacoustic Sound radiation Cylindrical shell Partially immersed Underwater experiment Elastic wave
a b s t r a c t The vibroacoustic behavior of a partially immersed cylindrical shell under point–force excitation is studied using theoretical and experimental approaches. An analytical form of the vibroacoustic coupling equation is developed using a discretization approach for the circumferential angle along the shell’s wet surface. The radiated sound and radial velocities are analyzed using frequency–depth spectra, and a series of regular oblique bright lines and weak interference fringes can be observed. For a partially immersed cylindrical shell, the subsonic flexural wave a0 can radiate energy into the fluid from airfluid demarcation points on the shell surface, and produce a series of resonant bright lines in the pressure spectra when the shell resonates in the circumferential direction. Interaction between the radiated waves, which propagate on parts of shell above and below the free surface (on the ‘‘dry part” and ‘‘wet part”, respectively), produces interference fringes in the frequency–depth spectra of sound pressure. Furthermore, simple formulas are given to predict the lines and fringes using the phase velocities of a0 wave on the dry and wet parts of the shell. Finally, experimental verification is carried out, and the measured frequency–depth spectra of the surface velocity and the sound pressure are consistent with the theoretical results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The concept of a structure immersed partially in a fluid is prevalent in engineering, such as submarines and ships floating on the sea. Because most of these marine vehicles can be approximated as cylindrical shells, the vibroacoustic characteristics of such shells have been studied extensively. Regarding acoustic–structural coupling effects, previous work has concerned mainly shells that are either fully submerged or semi-submerged in a fluid, including cylindrical shells of either infinite [1–3] or finite [4–5] length and with internal substructures (e.g., ring stiffeners, bulkheads, internal plates, lengthwise stiffeners) [6–11]. However, little attention has been paid to the problem of partial coupling. For a partially immersed shell, the free surface not only affects the external fluid loading but also changes the radiated sound field, which complicates the vibroacoustic characteristics. A semi-submerged cylindrical shell is a particular partial–coupling problem and is relatively straightforward to model by sepa-
⇑ Corresponding author. E-mail address: [email protected] (J. Fan). https://doi.org/10.1016/j.apacoust.2019.107170 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
rating both the structural and acoustic variables in the cylinder’s coordinate system [1–3]. Salaün [1] used a high-frequency approximation to model analytically the sound radiation from both closed and open semi-submerged cylindrical shells, and the free surface was found to have very important effects on the far-field sound pressure. Based on Salaün’s model, Li et al. [2] proposed a diagonal-coupling algorithm to improve the efficiency of computing the sound radiation from a semi-submerged cylindrical shell. Li et al. [3] used the wave–number domain approach to examine the sound radiation from a fluid-filled semi-submerged cylindrical shell, and the sound pressure was found to be influenced by Mach number, damping, and thickness. By contrast, studying partially immersed or filled cylinders is more challenging because the sound pressure is no longer separable in the cylinder’s coordinate system. Instead, the finite-element method/boundary-element method (FEM/BEM) can deal with such problems by considering the half-space Green’s function [12–14]. Ergin and Temarel [15] and Escaler et al. [16] used FEM to study the free vibration of partially immersed or filled cylindrical shells; their results were verified experimentally, but prohibitive computing costs limited the frequency band of the FEM/BEM calculations. To simplify the calculation, Amabili [17,18] approximated the surrounding fluid as either (i) a fan-shaped domain at an angle to the
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Nomenclature a0 flexural wave s0 compressional wave shear wave T0 Oðr; u; zÞ structural coordinate system centered at center of shell O0 ðr 0 ; u0 ; z0 Þ acoustic coordinate system centered on free surface a radius of shell hs thickness of shell qs density of shell Es Young’s modulus of shell rs Poisson’s ratio of shell qf density of fluid cf sound speed in fluid kf acoustic wavenumber in fluid K axial wavenumber L coefficients matrix of Donnell equation ðmÞ um ; v m ; wm Fourier coefficients of axial, circumferential, and radial displacements in wavenumber domain Fourier coefficient of force in wavenumber domain Fm pm Fourier coefficient of pressure in wavenumber domain wm Fourier coefficient of displacement in wavenumber domain p sound pressure in wavenumber domain
free surface or (ii) a ring layer attached to the surface of the shell and confined along the shell length; both approaches render the solutions separable in the polar coordinate system of the shell and solvable using the Galerkin method, but they ignore the reflection effects of the free surface on sound. Later, Guo et al. [19] investigated the vibroacoustic characteristics of a two-dimensional cylindrical shell partially immersed in water; they did this by introducing structural and acoustic coordinate systems to form the coupling equations, which they then solved using the Galerkin method. The work of Guo et al. [19] provides a new way to solve partial-coupling problems, but the sound radiation mechanism of a partially immersed cylindrical shell is yet to be reported. For a partially immersed cylindrical shell, the circumferential orthogonality of the vibroacoustic coupling equation is no longer satisfied on the wet surface. In Section 2, we propose a discretization approach for the circumferential angle along the wet surface to (i) solve the coupling equations directly and (ii) derive the analytical expressions for the far-field pressure. To describe the results for the sound pressure and radial velocity, we introduce the frequency–depth spectrum (FDS), which is a two-dimensional color plot of either the sound pressure or radial velocity in which frequency is plotted on the horizontal axis, immersion depth is plotted on the vertical axis, and color represents the pressure/velocity amplitude at a given frequency and immersion depth. Theoretical results are presented in Section 3, including the discovery of a series of regular bright spots and weak interference fringes in the FDSs. The subsonic flexural wave a0 cannot be radiated into the fluid via phase matching, but such waves can radiate energy to the far-field from discontinuous points on the shell provided by the free surface, similar to the sound radiation and scattering from submerged cylindrical shells with internal substructures [6– 8,20,21]. The mechanism behind the bright spots and weak interference fringes in the FDSs of sound pressure is discussed and evaluated in Section 4, and simple predictive formulas are derived. The experiment is presented in Section 5. Experimental and theoretical FDSs of the radial velocity and radiated sound pressure are compared, and the experimental and theoretical results agree well. Finally, in Section 6, we present a brief summary and draw some conclusions.
pn S
c
nth component of sound pressure wet surface of shell immersion angle 0 ¼ u u, circumferential angle between measurement point in acoustic coordinate system and measurement point in structural coordinate system immersion depth (distance from center of shell to free surface) ¼ H=a, normalized immersion depth elevation angle in spherical coordinate system excitation location qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Es = 1 r2s qs , compressional wave velocity of plate of same thickness as shell phase velocity wavenumber of a0 wave wavenumber of s0 wave inscribed angle of the wet surface
cph
phase velocity of a0 wave on a shell in vacuum
a
b
H Hr h ðu0 ; z0 Þ cs cph ka0 ks0 ð0Þ ðwÞ
cph
phase velocity of a0 wave on a submerged shell
d
distance between the demarcation points ‘‘1” and ‘‘2”
2. Theory Consider the sound radiation of an evacuated infinitely long cylindrical shell immersed partially in a fluid and driven by a point force F. The shell has a thickness hs , radius a, density qs , Young’s modulus Es , and Poisson’s ratio rs . Based on Ref.19, two cylindrical coordinate systems are employed, as shown in Fig. 1: one is the 0 0 acoustic coordinate system r ; u ; z0 centered on the free surface 0 O to describe the sound field; the other is the structural coordinate system ðr; u; zÞ centered at the shell center O to describe the vibration of the shell. The external fluid is inviscid with a sound speed cf and density qf . 2.1. Shell vibration in an infinite fluid In the structural coordinate system, the shell vibration is governed by the Donnell equation [6]:
0
um ðK Þ
1
0
B C B C LðmÞB @ v m ðK Þ A ¼ J @
wm ðK Þ
1
0
0
C A
ð1Þ
F m ðK Þ pm ðK Þ
in which the time-harmonic factor eixt is omitted. Here, m is the circumferential order, K is the axial wavenumber, and J ¼ a2 1 r2s =Es hs . The functions um , v m , wm , F m , pm and LðmÞ are given in the Appendix as Eqs. (A.1)–(A.3). Using Cramer’s rule, the radial velocity can be solved by _ ZM m wm ðK Þ ¼ F m ðK Þ pm ðK Þ
ð2Þ
_ where wm ðK Þ ¼ ixwm ðK Þ is the radial velocity in the wavenumber
domain, and Z M m ¼ jLðmÞj=ðixJ ðL11 L22 L12 L21 ÞÞ represents the mechanical impedance. 2.2. Vibroacoustic partial coupling equation In the acoustic coordinate system, the sound pressure is expressed in the wavenumber domain as
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 1. Coordinate systems for infinitely long cylindrical shell immersed partially in fluid: (a) front view; (b) side view.
(P
p ðr ; u ; K Þ ¼ 0
0
0 0 u0 ÞAn ðK ÞHð1Þ 0 u p; n ðjr Þ;
1 n¼1 sinðn
0;
others:
ð3Þ
where kf ¼ x=cf is the exterior acoustic wavenumber, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ k2f K 2 is the radial wavenumber, An ðK Þ is the sound pressure coefficient, and Hð1Þ n is the cylindrical Hankel function of order n. The boundary condition on the wet surface S is
0 0 @ p r ; u ; K @r
_ ¼ ixqf wðu; K Þ
ð4Þ
S;r¼a
but a coordinate transformation is needed because the left- and right-hand sides of Eq. (4) are expressed in different coordinate systems. As shown in Fig. 1, the two coordinate systems are related geometrically by
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 > > < r ¼ r 2 þ H 2Hrcosðu þ p=2Þ ¼ r2 þ H þ 2HrsinðuÞ;
Fig. 2. Discretization of wet surface in circumferential direction.
cosu ¼ rcosu=r ; > > : 0 z¼z; 0
0
ð5Þ 0
where H is the distance from O to O . The wet surface of the shell is in the range ½a; p þ a, and a is calculated via sina ¼ H=a. We have 0 H > 0; r > r; a > 0 when the shell’s center is beneath the free sur0 face, H < 0; r < r; a < 0 when the shell’s center is above the free 0 surface, and H ¼ 0; r ¼ r; a ¼ 0 when the shell’s center is on the free 0 surface. We define another useful angle b ¼ u u which satisfies
cosb ¼ ðr þ HsinuÞ=r0 ;
sinb ¼ Hcosu=r0 :
ð6Þ
By substituting Eqs. (5) and (6) into Eq. (4) and using P _ ilu wðu; K Þ ¼ 1 l¼1 wl ðK Þe , the partial-coupling equation of the shell can be derived as _
1 X n¼1
1 X 0 _ wl ðK Þeilu ; An ðK ÞBn u ; a ¼ ixqf
a u p þ a
ð7Þ
l¼1
where the partial derivative of the pressure is given in the Appendix as Eq. (A.4), and
0 0 0 sinb Bn u ; a ¼ Hð1Þ jr n cos nu n r0 0 ð1Þ 0 0 þ sin nu jHn jr cosb
As shown in Fig. 2, the wet surface is divided into 2N þ 1 strips along u with the interval Du ¼ ðp þ 2aÞ=2N; the ith strip is ui ¼ a þ Duði 1Þ, where i ¼ 1; 2; ; 2N þ 1: The matrix form of the partial-coupling equation is
2
38 9 B11 B2Nþ1;1 A > < 1 > = 6 . 7 .. .. .. 6 . 7 . . . 4 . 5> > : ; A2Nþ1 B1;2Nþ1 B2Nþ1;2Nþ1 8 9 3 2 _ > > C 11 C 2Nþ1;1 > > w 1 > > 7< . = 6 . . . 7 . . . ¼ iqf x6 . . . . 5> . > 4 > > > _ > ; C 1;2Nþ1 C 2Nþ1;2Nþ1 : w 2Nþ1
where B and C are matrices of size ð2N þ 1Þ ð2N þ 1Þ and ele 0 0 0 ments Bn;i ¼ Bn ui ; a , C l;i ¼ eilui , where cosui ¼ a=ri and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ri ¼ a2 þ H2 þ 2Hasinui . Thus, the sound pressure coefficient is _ A ¼ iqf xB1 C w and can be calculated using quadrature rectangle (QR) decomposition [22]; B1 is the inverse matrix of B. Applying a modal expansion to the acoustic load on the wet surface, we have
ð8Þ
In Eq. (7), the circumferential orthogonality of u is no longer satisfied, thus a discretization method for u on the wet surface is proposed. Eq. (7) can be transformed into a matrix equation and solved directly by discretizing u along the wet surface and truncating the infinite series as n ¼ 1; 2; ; 2N þ 1 and l ¼ N; ; 0; N.
ð9Þ
pm ðK Þ ¼
1 X
An
n¼1
1 2p
Z pþa a
sinðnu ÞHðn1Þ ðjr Þeimu du 0
0
ð10Þ
We let
Dn;m
jr ¼ 0
1 2p
Z pþa a
sinðnu ÞHðn1Þ ðjr Þeimu du 0
0
and Eq. (10) can be rewritten in matrix form as
ð11Þ
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
_ p ¼ DA ¼ Z a w
ð12Þ
where D is the matrix of Dn;m , which can be integrated numerically using Gaussian quadrature [22]. Z a ¼ DB1 C is the acoustic radiation impedance of size m l; the elements of m ¼ l and m–l are referred to as the self-impedance and mutual impedance, respectively. The mutual impedance is produced by the partial coupling of structural vibration and sound, and the governing equation for the free vibration of a partially immersed cylindrical shell is _ _ ZM w þ Za w ¼ 0
ð13Þ
where the diagonal matrix Z M impedance.
represents the mechanical
2.3. Far-field sound pressure under point-force excitation Consider a harmonic point ðr ¼ a; u ¼ u0 ; z0 ¼ 0Þ and satisfying
force
F m ðK Þ ¼ F 0 eimu0 =2pa
applied
at
ð14Þ
in the wavenumber domain, where F 0 is the magnitude of the force. Substituting Eqs. (12) and (14) into Eq. (4) gives the spectral radial velocity, namely
1 _ w ¼ ZM þ Za F
ð15Þ
The far-field sound pressure can be derived via the stationary phase method, namely
X 0 eikf R 2Nþ1 0 nþ1 p R; h; u ¼ sin nu An kf cos h ðiÞ pR n¼1 0
ð16Þ
0
where we assume R H and R R in the far-field, with z ¼ Rcosh and r ¼ Rsinh. The far-field pressure can also be written in modal series form as 2Nþ1 X 0 0 p R; h; u ¼ pn ðR; h; u Þ
ð17Þ
n¼1
Using the convergence condition of modal series, the convergence of the far-field pressure can be ensured by setting
pn1 < e and pn < e
ð18Þ
in the following calculations of the far-field sound pressure and radial velocity, we set e ¼ 1 1026 . The theoretical results are given in the form of FDSs of both the sound pressure and radial velocity with a 1-N force excitation, and the sound pressure is corrected to a distance of 1 m. 3. Numerical simulation results and discussion The shell is made of 304 stainless steel, and its geometrical and material parameters are as follows: radius a ¼ 0:1365m; ratio of thickness to radius hs =a ¼ 2:93% or hs =a ¼ 1:46%; density
qs ¼ 7930kg=m3 ;Young’s modulus Es ¼ 1:94201 1011 Pa; Poisson’s ratio rs ¼ 0:25. The sound speed and density of the external fluid are cf ¼ 1500m=s and qf ¼ 1000kg=m3 , respectively. The excitation location is at ðr ¼ a; u0 ¼ p=2; z0 ¼ 0Þ with force
F 0 ¼ 1N. The observation location is assumed to be at the broadside ðh ¼ p=2orK ¼ 0Þ and 1 m from the center of the shell. To explain how the immersion depth influences the sound radiation of the partially immersed cylindrical shell, we conduct an FDS analysis. Fig. 3 shows the spectra of the far-field sound pres0 sure, with the observations located at (a) u ¼ p=2 and (b) 0 u ¼ 3p=4. The thickness-to-radius ratio is 2.93%. The color repre-
sents the magnitude of the sound pressure [Pa], the horizontal coordinate is the frequency [Hz], and the vertical coordinate is the normalized immersion depth Hr ¼ H=a ranging from 1 (i.e., the shell is almost fully in the air, with only the bottom of its surface in contact with the free surface) to 1 (i.e., the shell is almost fully submerged, with only the top of its surface in contact with the free surface). A series of regular oblique bright lines (spots) and two groups of weak interference fringes can be observed in both Fig. 3(a) and (b). One group of dark fringes bends upward and the other downward, characteristics that are related to both the frequency and immersion depth. The observation location affects only the sound pressure amplitude. Studying the radial velocity helps us to understand the mechanism of the oblique bright lines in the pressure FDSs. The measuring point on the outer surface of the cylindrical shell is located at ðr ¼ a; u ¼ p=2; z ¼ 0Þ. For comparison, Fig. 4(a) plots the radial velocity for a submerged, semi-submerged, and airloaded cylindrical shell with a thickness-to-radius ratio of 2.93%. As shown in Fig. 4(a), for the same modal order (e.g. m ¼ 6; where m is the circumferential mode order), the resonant frequency of the air-loaded shell is the highest, while that of the submerged shell is the lowest. Fig. 4(b) plots the m ¼ 6 mode shapes of the (I) submerged, (II) semi-submerged, and (III) airloaded cylindrical shell, each labeled with the corresponding resonant frequency. Fig. 4(c) shows the FSDs of the radial velocity of a partially immersed cylindrical shell, the color representing the radial velocity in [m/s]. The oblique bright lines in Fig. 4(c) indicate the resonant frequency, which decreases with the immersion depth. There is a one-to-one correspondence between the bright spots in Fig. 3 and the bright lines in Fig. 4(c). Thus, the bright spots in the sound pressure spectra are generated by resonance of the shell.
4. Analysis of radiation mechanisms The dominant radiation characteristics of the partially immersed cylindrical shell in the FDSs are resonance and interference. 1) A harmonic force exerted on a smoothly curved elastic shell generates elastic waves on the shell surface, and those waves spread over the shell surface as a series of damped elastic waves [23]. For K ¼ 0, those waves include the flexural wave a0 and the compressional wave s0 , which propagate on the shell surface circumferentially. Shell resonance occurs if the elastic wave on the shell forms a circumferential standing wave [7], and some of the resonant elastic waves radiate into the fluid via phase matching if their phase velocity exceeds the sound speed in the fluid. The other elastic waves whose phase velocity is less than the sound speed in the fluid cannot radiate by phase matching, and at very low frequency only a few waves can radiate because of the curvature of the shell. 2) Because of the impedance discontinuity on the shell surface, the free surface forms two air-fluid demarcation points from where the subsonic flexural waves radiate into the fluid and form a series of bright resonance lines. This is similar to a cylindrical shell with an internal plate or lengthwise ribs [6,7]. Consequently, the air-fluid demarcation points are the sources of the acoustic radiation. 4.1. Resonance phenomenon As mentioned in Section3, the bright lines in the FDSs of the sound pressure are generated by shell resonance. To understand clearly the resonant characteristics of the shell, we explain the relationship between elastic waves and normal modes. Because the theoretical model is an infinite cylindrical shell, we consider
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 3. Frequency–depth spectra (FDSs) of sound pressure radiated from partially immersed cylindrical shell with observation location at (a) u0 ¼ p=2 and (b) u0 ¼ 3p=4. The thickness-to-radius ratio is 2.93%.
Fig. 4. Radial velocity at ðr ¼ a; u ¼ p=2; z ¼ 0Þ on shell: (a) radial velocity of the shell when it is (I) submerged, (II) semi-submerged, and (III) air-loaded; (b) m ¼ 6 modal shapes of (I) submerged, (II) semi-submerged, and (III) air-loaded shell; (c) FDSs of radial velocity of partially immersed cylindrical shell with a thickness-to-radius ratio of 2.93%.
only the circumferential modes. As presented in Ref.23, the resonant characteristics can be described in the modal form expðið mu ixt ÞÞ, where m is the mode order and represents the mth normal mode. Each mode is a multipole, with m ¼ 0 being the ‘‘breathing mode” (where the pressure contours move radially in and out), m ¼ 1 being the ‘‘dipole” (where the pressure contours move radially back and forth), etc. The motions of the mth normal mode are all standing waves in the circumferential direction and traveling with phase velocity cph ðxÞ ¼ ax=m. This is easily understood because, for the mth mode, exactly m wavelengths fit around the shell circumference. Obviously, the pressure contours of the mth normal mode are coincident with the mth circumferential vibration mode shape of the shell. Because the free surface divides the shell into a dry part and a wet part, the resonant characteristics can be analyzed by using the dispersion of the elastic waves on the dry and wet parts of the evacuated shell. From Eq. (1), the dispersion equation for an infinitely long cylindrical shell in vacuum is
jLðmÞj ¼ 0
ð19Þ
which has three types of root corresponding to three types of elastic wave, namely s0 (compressional), a0 (flexural), and T 0 (torsional). We take the observation point at the broadside (h ¼ p=2 or K ¼ 0), thereby allowing axial displacement to be neglected. The dominant elastic waves are s0 and a0 in the circumferential direction, and the dispersion equation for a fluid-loaded cylindrical shell with K ¼ 0 can be simplified as [7]
8 9 2 4 x2 qf Hðð1xÞ a=c Þ ðkf aÞ= < xa ph 2 xa 1 þ bs þJ ð1Þ0 : cs cph kf H xa=c ðkf aÞ ; ð ph Þ ( 2 ) 2 2 xa xa xa ¼0 cph cs cph
where bs ¼
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 hs =12a2 , c2s ¼ Es =ð1 r2s Þqs , and cph is the phase
velocity. Fig. 5 shows the dispersion curves of the s0 and a0 waves in a fluid-loaded cylindrical shell with a thickness-to-radius ratio of 2.94% or 1.46%. The phase velocities of the s0 and a0 waves are real when the shell is in vacuum but become complex when the shell is submerged. The phase velocity of the s0 wave is related weakly to the shell thickness, but that of the a0 wave is related to the shell thickness whether the shell is in vacuum or submerged. In the calculated frequency range, the a0 wave is subsonic (or ka0 > kf ) and cannot radiate energy via phase matching, while the s0 wave is supersonic (or ks0 < kf ) and can radiate into the fluid via phase matching when u > uc , where uc ¼ sin ðcf =cs0 Þ is the critical angle of the s0 wave. For comparison, Fig. 6 plots the far-field sound pressure at u0 ¼ p=2 for submerged and semi-submerged (Hr ¼ 0) cylindrical shells with a thickness-to-radius ratio of 2.93%. The small peaks in Fig. 6(I) are produced by the subsonic a0 wave because of the curvature of the shell [24], that being because phase-matching is 1
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 5. Phase-velocity dispersion curves of a0 and s0 waves on submerged cylindrical shell with two different thicknesses: (a) real part of a0 and s0 waves; (b) real and imaginary parts of s0 wave. The solid and dashed curves are for thickness-to-radius ratios of 1.46% and 2.93%, respectively.
Fig. 7(a) and (b) are produced by resonance of the subsonic a0 waves that are radiated from the air-fluid demarcation points on the shell. As shown in Fig. 8, the wet surface is represented by c ¼ p þ 2a, where c ¼ 0 (Hr ¼ 1) and c ¼ 2p (Hr ¼ 1) correspond to the shell nearly in air and in fluid, respectively. From the propagation velocities of the elastic waves on the dry and wet parts, the following simple formula is derived for predicting the resonant frequency of a partially immersed cylindrical shell:
f ¼m
ð2p cÞa ð0 Þ
cph ð 0Þ
þ
ca ðw Þ
cph
!1 ;
ð21Þ
ðwÞ
where cph and cph are the phase velocities of the elastic waves on the shell when it is in vacuum and submerged, respectively. Also, f ¼ mcph =2pa with c ! 0 and f ¼ mcph =2pa with c ! 2p. Thus, ð0Þ
Fig. 6. Far-field sound pressure at u0 ¼ p=2 for (I) submerged and (II) semisubmerged cylindrical shells with a thickness-to-radius ratio of 2.93%.
not satisfied. However, Fig. 6(II) contains a series of strong resonant peaks that could be due to the resonance of the a0 wave. Fig. 7 shows the FDSs of the far-field sound pressure for a partially immersed cylindrical shell with a thickness-to-radius ratio of (a) 2.93% and (b) 1.46%. The observation location is at u0 ¼ p=2. Clearly, the horizontal interval of the bright spots decreases with decreasing shell thickness. Therefore, the oblique bright spots in
ðwÞ
the resonant frequency of a partially immersed cylindrical shell decreases with immersion depth to form a series of oblique bright spots in the pressure spectra in Fig. 7. The solid yellow lines overlaid on the spectra show the resonant frequency of the a0 wave as predicted by Eq. (21). As shown, the predicted results are almost consistent with the resonant bright lines. In addition to the oblique bright spots produced by the a0 waves, there is a strong bright stripe at higher frequencies, labeled by the dashed red wireframes in Fig. 7(a) and (b). The interference stripes are produced by resonance of the zeroth-order s0 wave (the
Fig. 7. FDSs of sound pressure radiated from partially immersed cylindrical shell with thickness-to-radius ratio of (a) 2.93% and (b) 1.46%. The solid yellow curves show the resonant bright lines predicted by Eq. (21). The dashed red wireframes show the resonance of the zeroth-order s0 wave.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
only within the immersion depth range of c=2 > uc , when the observation point is at u0 ¼ p=2. Thus, the bright stripe in Fig. 7 is stronger in the depth range of Hr ¼ 0:8 to 1. Furthermore, the second path is ineffective when the observation point falls in the ranges of u < a þ uc and u > a þ p uc , so the bright stripe shrinks as the observation point approaches the free surface, and those results with the observation point near the free surface are omitted here. 4.2. Interference phenomenon
Fig. 8. Schematic of circumferential resonance formed by elastic waves. ‘‘1” and ‘‘2” represent the air–fluid demarcation points on the shell.
resonant frequency is referred to as the ring frequency), which is different from the interference stripes generated by the a0 wave and is difficult to predict using Eq. (21) because Eq. (21) fails at m ¼ 0. The ring frequency is defined such that the length of the compressional wave at this frequency is equal to the circumference of the shell. However, we can estimate the ring frequency of a partially immersed cylindrical shell based on the dry shell and wet shell. For a shell in vacuum, the ring frequency can be calculated by letting m ¼ 0 in Eq. (19), which leads to f ¼ cs =2pa, where cs is the longitudinal wave velocity on a plate of the same thickness as the shell. The ring frequency of a submerged cylindrical shell is calculated by letting xa=cph ðxÞ ¼ 0 in Eq. (20) to satisfy
ð1Þ xa 2 a2 1 r2 x2 qf H0 kf a ¼ 0: 1 þ ð1Þ0 E s hs cs kf H0 kf a
ð22Þ
Two types of weak interference fringes correlated with the shell thickness, as shown in Figs. 3 and 7, are therefore caused by interference among the a0 waves radiated along different paths. The a0 waves are generated by the force and then propagate along the circumference in clockwise and counterclockwise directions, to finally radiate through the two air–fluid demarcation points. Figs. 10 and 11 illustrate the two types of paths related to the dry part and wet part. 4.2.1. Interference fringes related to dry part From the geometric relationship illustrated in Fig. 10, the acoustic path difference formed by the dry part is dup ¼ ð2p cÞa. The corresponding paths are as follows: (i) the a0 wave propagates along the dry part in the clockwise direction and radiates first from point 1 and then from point 2; (ii) the a0 wave propagates along the dry part in the counterclockwise direction and radiates first from point 2 and then from point 1. Thus, simple formulas for predicting the constructive and destructive interference fringes, respectively, are
f ¼ mcph =ð2p cÞa; ð0Þ
m ¼ 1; 2; ;
f ¼ ðm þ 0:5Þcph =ð2p cÞa; ð0Þ
ð23Þ
m ¼ 1; 2; ;
ð24Þ
ð0Þ
where cph is the phase velocity of the a0 wave on the shell in vacuum. The resonant frequency is f ¼ mcph =2pa at c ! 0 while ð0Þ
Eq. (22) can be solved using a root-finding algorithm. The ring frequencies of a cylindrical shell in vacuum and submerged are listed in Table1, and these two frequencies are the upper and lower limits, respectively, of the ring frequency of a partially immersed cylindrical shell. Therefore, the ring frequency of a partially immersed cylindrical shell can be estimated based on Table1, and the estimated results represented by the dashed red wireframes agree with the location of the bright stripe shown in Fig. 7. The imaginary part of the phase velocity of the elastic waves represents the radiation efficiency of a submerged cylindrical shell [24]. As shown in Fig. 5(b), the peak of the imaginary part of the s0 wave covers a certain frequency range, which results in a wider resonant region of the zeroth-order s0 wave in the pressure spectra in Fig. 7. As shown in Fig. 9, the s0 wave can radiate into the fluid through two paths, namely (i) from the air–fluid demarcation points on the shell and (ii) from the critical angle due to phase matching; the latter possesses a higher radiation efficiency. Therefore, the bright stripe produced by the s0 wave is stronger than that produced by the a0 wave. However, the second path is effective
f ! 1 at c ! 2p. Thus, the resonant frequency of the interference fringes related to the dry part is correlated positively with the immersion depth to form the upward-bending dark fringes in Fig. 7.
4.2.2. Interference fringes related to wet part As illustrated in Fig. 11, the acoustic path difference formed by the wet part is ddown ¼ ca. The corresponding paths are as follows: (i) the a0 wave propagates along the wet part in the counterclockwise direction and radiates first from point 1 and then from point 2; (ii) the a0 wave propagates along the wet part in the clockwise direction and radiates first from point 2 and then from point 1. As in Section 4.2.1, simple formulas for predicting the constructive and destructive interference fringes, respectively, are
f ¼ mcph =ca; ðwÞ
m ¼ 1; 2; ;
f ¼ ðm þ 0:5Þcph =ca; ðwÞ
m ¼ 1; 2; ;
ð25Þ ð26Þ
ðwÞ
where cph is the phase velocity of the a0 wave on a submerged cylindrical shell. Obviously, the resonant frequency is f ! 1 at
Table 1 Resonant frequency of the zeroth-order s0 wave (ring frequency) of shell in vacuum or submerged. Order
m¼0 m¼0
Thickness-to-radius ratio
1.46% 2.93%
Frequency [Hz] In vacuum
Submerged
5956.9 5956.9
1195.2 2619.7
ð0Þ c ! 0 and f ¼ mcph =2pa at c ! 2p. Thus, the resonant frequency
of the interference fringes related to the wet part is correlated negatively with immersion depth, forming the downward-bending dark fringes in Fig. 7. Fig. 12 shows the FDSs of the sound pressure of a partially immersed cylindrical shell of varying thickness. The data in Fig. 12(a)–(d) are the same as those of Fig. 7(a) and (b), the difference being that the dashed green and solid blue curves show the
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 9. Propagation paths of s0 wave with observation point at (a) u0 ¼ p=2 and (b) obliquely below, e.g., u0 ¼ 3p=4.
Fig. 10. Propagation paths of a0 wave along dry part to produce the upward-bending interference fringes: (a) clockwise direction; (b) counterclockwise direction.
Fig. 11. Propagation paths of a0 wave along wet part to produce the downward-bending interference fringes: (a) counterclockwise direction; (b) clockwise direction.
two types of interference fringes, which agree well with the theoretical results. The constructive interference fringes represented by the dashed green curves in Fig. 12(a) and (b) are predicted by Eq. (23), and those in Fig. 12(c) and (d) are predicted by Eq. (25). The destructive interference fringes represented by the solid blue curves in Fig. 12(a) and (b) are predicted by Eq. (24), and those in Fig. 12(c) and (d) are predicted by Eq. (26). Because the phase velocity of the a0 wave increases with frequency, the intervals of
both types of fringes narrow, as can be understood from Eqs. (23)–(26). In addition to the interference related to the dry part and wet part, sound waves radiate from points 1 and 2, producing geometric interference. The path difference between points 1 and 2 is dsina, and dsina < 2a, where d is the distance between points 1 to 2. Because a< kmin ¼ 0:3m in the calculated frequency range, the geometric interference between the points can be ignored. In
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
9
Fig. 12. FDSs of sound pressure radiated from partially immersed cylindrical shell with thickness-to-radius ratios (a, c) 2.93% and (b, d) 1.46%. The dashed green and solid blue curves represent the predicted results for the constructive and destructive interference fringes, respectively.
fact, these discontinuous points are the sound radiation sources for a partially immersed cylindrical shell. Therefore, in the lowfrequency range (no s0 -wave component), the virtual source interference related to the free surface is nonexistent, and the directivity pattern can be approximated as a superposition field of two equivalent strength dipoles. Fig. 13 shows the directivity patterns of the far-field sound pressure level (20lgðp=p0 Þ withp0 = 1lPa) of a partially immersed cylindrical shell with a thickness-toradius ratio of 2.93%; the maximum value at u0 ¼ p=2 can be observed. As is known, the directivity function is jsinu0 j for an acoustical dipole and is cos kf d=2 cosu0 for two in-phase sources [25]. Therefore, the directivity function for a partially
immersed cylindrical shell at low frequency can be derived approximately as Dðu0 Þ ¼ cosðkf d=2 cosu0 Þsinu0 .
5. Experiment The air-filled cylindrical shell is made of seamless stainless steel, with a radius of 0.1365 m, a length of 0.5 m, a thicknessto-radius ratio of 2.93%, and material parameters as listed in Section 3. Both ends of the shell are closed by two 0.01-m-thick flat end-caps of the same material as the shell. The end-caps also serve as the counterweights. Photographs of the experimental model and
Fig. 13. Directivity patterns of far-field sound pressure levels of a partially immersed cylindrical shell at different immersion depths Hr ¼ 0:5; 0; 0:5 at frequencies (a) 500 Hz and (b) 2500 Hz.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 14. Photograph of experimental model.
a sketch of the experimental setup are shown in Figs. 14 and 15, respectively. The experimental model is immersed horizontally in a 5 m 5 m 5 m water-filled tank with a 4.5-m-deepth water layer and a 0.2-m-deep layer of fine sand at the bottom. The shell is connected to the lifting platform located at the central axis of the tank by elastic ropes, and the lifting motor is controlled by a computer at a speed of 0.2 m/min. To counteract the buoyancy of the shell during descent, two wooden planks (with high damping capability) are installed between the end-caps and the lifting platform. The exciter is hung elastically at the top of the shell and is connected to it by a PCB-208C02 force sensor. The receiver is a B&K8103 hydrophone mounted on a 1-m-radius quarter-arc support and fixed under the shell 1 m from the free surface. During the measurement, the hydrophone is static when the shell is descending. In this process, we obtain the radiated sound pressure at a fixed measuring point as the shell is lowered continuously. A B&K4534-B accelerometer is mounted at the bottom of the shell to measure the radial velocity, and a high-capacity uninterruptible power supply is used during the entire measurement. Two
synchronizations are used in the measurement as shown in Fig. 15. The first one is the synchronization between the signal generator and the data acquisition system. This is a multiple-trigger mode, and the trigger signal is the output from the signal generator. The other one is the synchronization between the data acquisition system and the lifting platform. This is a single-trigger mode, and the trigger signal is the output from the data acquisition system. These two synchronizations ensure that the recorded data corresponds accurately to the shell’s immersion depth. Limited by the measurement system, the frequency range of the experiment is 0.5–2.5 kHz. To achieve a higher resolution for the immersion depth, a chirp pulse (40 ms, 0.5–2.5 kHz) is used as the excitation signal, with its time plot shown in Fig. 16. The excitation signal is generated repeatedly with a time period of 450 ms. The sample frequency satisfies the Nyquist sampling theorem. The acquisition time for a single pulse is 100 ms, and totally 184 pulse counts are collected within a complete measurement as the shell is lowered continuously from Hr ¼ 1 to Hr ¼ 1. The normalized immersion depth of the shell is calculated by Hr ¼ 0:2 count=a, where count is the count number of the collected data. The time signal is transformed into the frequency domain by a fast Fourier transform, and this process results in the FDSs for the corresponding measurement. As illustrated in Fig. 15, because the path difference between the direct and reflected waves from the sidewalls
Fig. 15. Sketch of experimental setup.
Fig. 16. Time plot of chirp signal.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 17. Comparison of analytical and experimental results for (a) radial velocity at ðr ¼ a; u ¼ p=2; z ¼ 0Þ and (b) sound pressure amplitude at u0 ¼ p=2 of a shell immersed at Hr ¼ 0.
Table 2 Measured resonant frequencies of experimental model at Hr ¼ 0, where q and m represent the axial and circumferential mode orders, respectively. Orderðq; mÞ
(1,4)
(1,5)
(1,6)
(3,5)
(1,7)
Frequency [Hz]
701.6
1093.1
1615.2
2065.4
2230.2
and bottom exceeds 5 m, by considering the reflection loss, the reflected waves are attenuated by more than 20lg5 14dB and can be ignored. For comparison, the measured results are corrected to those for an excitation force of 1 N. To assess the accuracy of the theoretical results, the analytical and experimental frequency spectra of the radial velocity at
ða; p=2; 0Þ and the far-field sound pressure amplitude at u ¼ p=2 are compared in Fig. 17(a) and (b), respectively. The analytical results agree well with the experimental results except for several small discrepancies in the resonant frequencies. As can be seen, the experimental resonant peaks are slightly larger than the analytical ones, especially for m ¼ 4. This may be due to the differences in materials and geometry between the theoretical and experimental models, given that the experimental model is finite rather than an infinite shell. The peaks at 2065.4 Hz in Fig. 17(a) and (b) of the experimental results are due to the axial resonance of the shell, as is shown by the mode test results in Table2. The measured resonant frequencies of the model at Hr ¼ 0 are listed in Table2, where q and m represent the axial and circumferential mode orders, respectively. 0
Fig. 18. FDSs of radial velocity of partially immersed cylindrical shell: (a) experimental and (b) calculated results with measurement point at ðr ¼ a; u ¼ p=2; z ¼ 0Þ; (c) experimental and (d) calculated results with measurement point at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ. The thickness-to-radius ratio is 2.93%.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 19. FDSs of sound pressure radiated from partially immersed cylindrical shell: (a) experimental and (b) calculated results with measurement point at u ¼ p=2; (c) 0 experimental and (d) calculated results with measurement point at u ¼ 3p=4. The thickness-to-radius ratio is 2.93%, and the solid yellow curves show the resonant bright lines predicted by Eq. (22). 0
Fig. 20. Interference fringes predicted for experimental results in Fig. 19(a): (a) bending upward; (b) bending downward. The constructive interference fringes represented by the dashed green curves in (a) are predicted from Eqs. (23) and (25), and those in (b) from Eqs. (24) and (26).
Fig. 18 illustrates the measured and calculated FDSs of the radial velocity on the shell. The left and right parts are the measured and calculated results, respectively, with the measuring point at ðr ¼ a; u ¼ p=2; z ¼ 0Þ for Fig. 18(a) and (b) and at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ for Fig. 18(c) and (d). Fig. 19 shows the measured and calculated FDSs of the sound pressure of the shell. The left and right parts are the measured and calculated results, respectively, with the observation point at u0 ¼ p=2 for Fig. 19(a) 0 and (b) and at u ¼ 3p=4 for Fig. 19(c) and (d). The yellow curves overlaid on Fig. 19(a) and (c) show the resonant frequencies predicted by Eq. (21). The oblique resonant bright spots and interference fringes generated by the a0 wave in the pressure spectra can be observed clearly in the experimental results. By contrast, the resonant bright spots produced by the s0 wave in the pressure spectra cannot be observed because the maximum experimental
frequency is lower than the zeroth-order s0 wave resonant frequency (or ring frequency of 2619.7 Hz) of a submerged shell, which is the lowest resonant frequency of an s0 wave on a partially immersed cylindrical shell. The experimental and calculated results agree well in Figs. 18 and 19, except for small deviations at low frequency, e.g., the fourth-order resonant frequency. Differences in radial velocity can also be observed for the sixth-order resonant peaks at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ in Fig. 18(c) and (d). These deviations may be caused by (i) the experimental model being finite and the end-caps and supporting wooden planks producing prestress on the shell, which leads to the lower-order modal frequencies in the experiment being larger than those in the calculated results, or (ii) the phase velocity of the elastic waves of the experimental model being not completely the same as that of the theoretical model. Consequently, the fourth-order modal
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
frequency of the a0 wave in the experimental results is not predicted accurately by the formula. Two types of interference fringes can be observed in the experimental results shown in Fig. 19(a) and (c), which agree well with the calculated results. Because the same interface fringes are observed in Fig. 19(a) and (c), only the predicted results of u0 ¼ p=2 are presented. The predicted interference fringes for upward bending and downward bending are shown in Fig. 20(a) and (b), respectively, and agree well with those in the experimental results. The data in Fig. 20(a) and (b) are the same as those in Fig. 19(a), the difference being that the dashed green and solid blue curves show the predicted interference fringes. The dashed green and solid blue curves in Fig. 20(a) are calculated form Eqs. (23) and (24), and those in Fig. 20(b) are calculated from Eqs. (25) and (26). Additionally, some vertical fringes are observed in the experimental results as shown in Fig. 19(a) and (c). These fringes may be caused by the following two aspects: (i) the experimental model is finite while the theoretical model is infinite, so the axial resonant fringes cannot be predicted by the present theoretical method; (ii) sound in air generated by the shaker will propagate into the water and produce some of the fringes. Therefore, the far-field sound pressure of a partially immersed cylindrical shell is quite different from that of a fully submerged one. The main differences are as follows. 1) The resonant lines that appear in the sound pressure spectra of a partially immersed shell almost disappear for a submerged shell, as shown in Fig. 6. Because of the curvature of the shell, only a small amount of the subsonic a0 wave can radiate sound to the far field to form a few resonant peaks at very low frequency. 2) The resonant frequencies of a partially immersed cylindrical shell decrease with the immersion depth, and the horizontal intervals of the resonant frequencies increase with the frequency. 3) Interference between the radiated waves that propagate on the dry part and the waves on the wet part produces the interference fringes in the FDSs of sound pressure. This interference means that the resonant peaks of the pressure spectra will diminish or even disappear at a certain immersion depth of the shell. This physical phenomenon does not occur in the sound pressure spectra of a submerged cylindrical shell.
bright lines in the pressure spectra, while the supersonic wave s0 can radiate out not only from the demarcation points but also through phase matching to form an enhanced bright strip in the pressure spectra. Because the phase velocity of a0 wave is different on the dry and wet parts of the shell, the resonant frequencies of partially immersed cylindrical shell decrease with the immersion depth. 2. Interaction between a0 waves radiated from different air-fluid demarcation points produce interference fringes in the pressure spectra. The acoustic path differences of the upward and downward bending interference fringes are related to the dry and wet parts of the shell, respectively. 3. The predicted results from simple formulas agree well with the experimental and numerical results.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Contract No. 11774229). We would like to thank Dr. Yunzhe Tong and Bo Liu for illuminating discussions. Appendix A The axial, circumferential, and radial displacements of the shell are
9 uðu; zÞ > =
v ðu; zÞ > ¼
wðu; zÞ
;
1 2p
1 X
eimu
m¼1
8 9 > > > < u m ðK Þ > = eiKz dK: v m ðK Þ > 1 > > > : ; wm ðK Þ
Z
1
ðA1Þ
The acoustic and force loads are: 6. Conclusions
pðu; zÞ
In this paper, the radiation mechanisms of a partially immersed cylindrical shell undergoing point-force excitation are analyzed using theoretical and experimental methods. The spectra for both vibration and sound are measured and calculated with the shell in continuous immersion. The major characteristics are a series of regular oblique bright lines and weak interference fringes in the FDSs of radiated sound pressure, which can be observed by experiment. Simple formulas for predicting the bright lines and interference fringes are derived. The radiation mechanisms of the partially immersed cylindrical shell can be explained as follows: 1. The a0 and s0 waves are excited by the force. Shell resonance occurs when these elastic waves form a standing wave circum-
0 0 @ p r ; u ; K @r
(P ¼ S;r¼a
1 ð1 Þ n¼1 An ðK ÞðHn
jr0 ncos nu0
sinb r
0
Fðu; zÞ
8 9 Z 1< 1 = P ð K Þ 1 X m eiKz dK: eimu ¼ ; 2p m¼1 1 : F m ðK Þ
The elements of matrix L ðmÞ are 2 6 L ðmÞ ¼ 6 4
X2s þ v2 þ r m2
rþ mv 2 Xs þ r v2 þ m2
rþ mv irs v
im
3
irs v
7 7 im 5 2 2 2 2 2 1 Xs þ b v þ m ðA3Þ
2
where Xs ¼ xa=cs , b2s ¼ hs =12a2 , c2s ¼ Es =qs 1 rs , v ¼ Ka, and K is the axial wavenumber. For economy of space, we introduce the symbols rþ ¼ ð1 þ rs Þ=2 and r ¼ ð1 rs Þ=2. The partial derivative of the pressure is 2
0 0 0 þ sin nu jHnð1Þ jr cosbÞ;
0u p 0
ðA4Þ
p u0 2p
0;
ferentially. The resonance of the subsonic wave a0 can radiate from the air-fluid demarcation points to form a series of oblique
ðA2Þ
( where @p=@r ¼
0
@p @r 0 @r @r
þ
0
@p @ u 0 @ u @r
and
@r =@r ¼ ðr þ HsinuÞ=r ¼ cosb; 0 uÞ @ u =@r ¼ Hcosru0 3ðHþrsin ¼ sinb 0 0 : sinu r 0
0
14
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[13] Chen PT, Lin CS, Yang T. Responses of partially immersed elastic structures using a symmetric formulation for coupled boundary element and finite element methods. J Acoust Soc Am 2002;112:866–75. https://doi.org/10.1121/ 1.1494448. [14] Wu H, Chen P. Application of coupled FEM/BEM on the analysis of underwater radiated noise of a surface ship induced by hull vibrations. J. Marine Sci. Technol 2017;25:196–204. https://doi.org/10.6119/JMST-016-1118-2. [15] Ergin A, Temarel P. Free vibration of a partially liquid-filled and submerged, horizontal cylindrical shell. J Sound Vib 2002;254:951–65. https://doi.org/ 10.1006/jsvi.2001.4139. [16] Escaler X, De La Torre O, Goggins J. Experimental and numerical analysis of directional added mass effects in partially liquid-filled horizontal pipes. J Fluid Struct 2017;69:252–64. https://doi.org/10.1016/j.jfluidstructs.2017.01.001. [17] Amabili M. Free vibration of partially filled, horizontal cylindrical shells. J Sound Vib 1996;191:757–80. https://doi.org/10.1006/jsvi.1996.0154. [18] Amabili M. Flexural vibration of cylindrical shells partially coupled with external and internal fluids. ASME J Vib Acoust 1997;119:476–84. https://doi. org/10.1115/1.2889748. [19] Guo W, Li T, Zhu X, Qu K. Semi-analytical research on acoustic-structure coupling calculation of partially submerged cylindrical shell. Acta Phys Sin 2018;67. 084302, (in Chinese). [20] Klauson A, Maze G, Metsaveer J. Acoustic scattering by submerged cylindrical shell stiffened by an internal lengthwise rib. J Acoust Soc Am 1994;96:1575–81. https://doi.org/10.1121/1.410236. [21] Guo YP. Sound scattering from cylindrical shells with internal elastic plates. J Acoust Soc Am 1993;93:1936–46. https://doi.org/10.1121/1.406708. [22] Golub GH, Van Loan CF. Matrix computations Chap. 5. JHU Press; 2012. p. 206–74. [23] Überall H. Relation between creeping waves and normal modes of vibration of a curved body. J Acoust Soc Am 1977;61:711–5. https://doi.org/10.1121/ 1.381347. [24] Marston PL, Sun NH. Backscattering near the coincidence frequency of a thin cylindrical shell: Surface wave properties from elasticity theory and an approximate ray synthesis. J Acoust Soc Am 1995;97:777–83. https://doi.org/ 10.1121/1.412124. [25] Morse PM, Ingard KU. Theoretical acoustics. Princeton University Press; 1986. p. 312.
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Vibroacoustic behavior of a partially immersed cylindrical shell under point-force excitation: Analysis and experiment Kaiqi Zhao, Jun Fan ⇑, Bin Wang, Weilin Tang Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 5 July 2019 Received in revised form 11 November 2019 Accepted 26 November 2019
Keywords: Vibroacoustic Sound radiation Cylindrical shell Partially immersed Underwater experiment Elastic wave
a b s t r a c t The vibroacoustic behavior of a partially immersed cylindrical shell under point–force excitation is studied using theoretical and experimental approaches. An analytical form of the vibroacoustic coupling equation is developed using a discretization approach for the circumferential angle along the shell’s wet surface. The radiated sound and radial velocities are analyzed using frequency–depth spectra, and a series of regular oblique bright lines and weak interference fringes can be observed. For a partially immersed cylindrical shell, the subsonic flexural wave a0 can radiate energy into the fluid from airfluid demarcation points on the shell surface, and produce a series of resonant bright lines in the pressure spectra when the shell resonates in the circumferential direction. Interaction between the radiated waves, which propagate on parts of shell above and below the free surface (on the ‘‘dry part” and ‘‘wet part”, respectively), produces interference fringes in the frequency–depth spectra of sound pressure. Furthermore, simple formulas are given to predict the lines and fringes using the phase velocities of a0 wave on the dry and wet parts of the shell. Finally, experimental verification is carried out, and the measured frequency–depth spectra of the surface velocity and the sound pressure are consistent with the theoretical results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The concept of a structure immersed partially in a fluid is prevalent in engineering, such as submarines and ships floating on the sea. Because most of these marine vehicles can be approximated as cylindrical shells, the vibroacoustic characteristics of such shells have been studied extensively. Regarding acoustic–structural coupling effects, previous work has concerned mainly shells that are either fully submerged or semi-submerged in a fluid, including cylindrical shells of either infinite [1–3] or finite [4–5] length and with internal substructures (e.g., ring stiffeners, bulkheads, internal plates, lengthwise stiffeners) [6–11]. However, little attention has been paid to the problem of partial coupling. For a partially immersed shell, the free surface not only affects the external fluid loading but also changes the radiated sound field, which complicates the vibroacoustic characteristics. A semi-submerged cylindrical shell is a particular partial–coupling problem and is relatively straightforward to model by sepa-
⇑ Corresponding author. E-mail address: [email protected] (J. Fan). https://doi.org/10.1016/j.apacoust.2019.107170 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
rating both the structural and acoustic variables in the cylinder’s coordinate system [1–3]. Salaün [1] used a high-frequency approximation to model analytically the sound radiation from both closed and open semi-submerged cylindrical shells, and the free surface was found to have very important effects on the far-field sound pressure. Based on Salaün’s model, Li et al. [2] proposed a diagonal-coupling algorithm to improve the efficiency of computing the sound radiation from a semi-submerged cylindrical shell. Li et al. [3] used the wave–number domain approach to examine the sound radiation from a fluid-filled semi-submerged cylindrical shell, and the sound pressure was found to be influenced by Mach number, damping, and thickness. By contrast, studying partially immersed or filled cylinders is more challenging because the sound pressure is no longer separable in the cylinder’s coordinate system. Instead, the finite-element method/boundary-element method (FEM/BEM) can deal with such problems by considering the half-space Green’s function [12–14]. Ergin and Temarel [15] and Escaler et al. [16] used FEM to study the free vibration of partially immersed or filled cylindrical shells; their results were verified experimentally, but prohibitive computing costs limited the frequency band of the FEM/BEM calculations. To simplify the calculation, Amabili [17,18] approximated the surrounding fluid as either (i) a fan-shaped domain at an angle to the
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Nomenclature a0 flexural wave s0 compressional wave shear wave T0 Oðr; u; zÞ structural coordinate system centered at center of shell O0 ðr 0 ; u0 ; z0 Þ acoustic coordinate system centered on free surface a radius of shell hs thickness of shell qs density of shell Es Young’s modulus of shell rs Poisson’s ratio of shell qf density of fluid cf sound speed in fluid kf acoustic wavenumber in fluid K axial wavenumber L coefficients matrix of Donnell equation ðmÞ um ; v m ; wm Fourier coefficients of axial, circumferential, and radial displacements in wavenumber domain Fourier coefficient of force in wavenumber domain Fm pm Fourier coefficient of pressure in wavenumber domain wm Fourier coefficient of displacement in wavenumber domain p sound pressure in wavenumber domain
free surface or (ii) a ring layer attached to the surface of the shell and confined along the shell length; both approaches render the solutions separable in the polar coordinate system of the shell and solvable using the Galerkin method, but they ignore the reflection effects of the free surface on sound. Later, Guo et al. [19] investigated the vibroacoustic characteristics of a two-dimensional cylindrical shell partially immersed in water; they did this by introducing structural and acoustic coordinate systems to form the coupling equations, which they then solved using the Galerkin method. The work of Guo et al. [19] provides a new way to solve partial-coupling problems, but the sound radiation mechanism of a partially immersed cylindrical shell is yet to be reported. For a partially immersed cylindrical shell, the circumferential orthogonality of the vibroacoustic coupling equation is no longer satisfied on the wet surface. In Section 2, we propose a discretization approach for the circumferential angle along the wet surface to (i) solve the coupling equations directly and (ii) derive the analytical expressions for the far-field pressure. To describe the results for the sound pressure and radial velocity, we introduce the frequency–depth spectrum (FDS), which is a two-dimensional color plot of either the sound pressure or radial velocity in which frequency is plotted on the horizontal axis, immersion depth is plotted on the vertical axis, and color represents the pressure/velocity amplitude at a given frequency and immersion depth. Theoretical results are presented in Section 3, including the discovery of a series of regular bright spots and weak interference fringes in the FDSs. The subsonic flexural wave a0 cannot be radiated into the fluid via phase matching, but such waves can radiate energy to the far-field from discontinuous points on the shell provided by the free surface, similar to the sound radiation and scattering from submerged cylindrical shells with internal substructures [6– 8,20,21]. The mechanism behind the bright spots and weak interference fringes in the FDSs of sound pressure is discussed and evaluated in Section 4, and simple predictive formulas are derived. The experiment is presented in Section 5. Experimental and theoretical FDSs of the radial velocity and radiated sound pressure are compared, and the experimental and theoretical results agree well. Finally, in Section 6, we present a brief summary and draw some conclusions.
pn S
c
nth component of sound pressure wet surface of shell immersion angle 0 ¼ u u, circumferential angle between measurement point in acoustic coordinate system and measurement point in structural coordinate system immersion depth (distance from center of shell to free surface) ¼ H=a, normalized immersion depth elevation angle in spherical coordinate system excitation location qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Es = 1 r2s qs , compressional wave velocity of plate of same thickness as shell phase velocity wavenumber of a0 wave wavenumber of s0 wave inscribed angle of the wet surface
cph
phase velocity of a0 wave on a shell in vacuum
a
b
H Hr h ðu0 ; z0 Þ cs cph ka0 ks0 ð0Þ ðwÞ
cph
phase velocity of a0 wave on a submerged shell
d
distance between the demarcation points ‘‘1” and ‘‘2”
2. Theory Consider the sound radiation of an evacuated infinitely long cylindrical shell immersed partially in a fluid and driven by a point force F. The shell has a thickness hs , radius a, density qs , Young’s modulus Es , and Poisson’s ratio rs . Based on Ref.19, two cylindrical coordinate systems are employed, as shown in Fig. 1: one is the 0 0 acoustic coordinate system r ; u ; z0 centered on the free surface 0 O to describe the sound field; the other is the structural coordinate system ðr; u; zÞ centered at the shell center O to describe the vibration of the shell. The external fluid is inviscid with a sound speed cf and density qf . 2.1. Shell vibration in an infinite fluid In the structural coordinate system, the shell vibration is governed by the Donnell equation [6]:
0
um ðK Þ
1
0
B C B C LðmÞB @ v m ðK Þ A ¼ J @
wm ðK Þ
1
0
0
C A
ð1Þ
F m ðK Þ pm ðK Þ
in which the time-harmonic factor eixt is omitted. Here, m is the circumferential order, K is the axial wavenumber, and J ¼ a2 1 r2s =Es hs . The functions um , v m , wm , F m , pm and LðmÞ are given in the Appendix as Eqs. (A.1)–(A.3). Using Cramer’s rule, the radial velocity can be solved by _ ZM m wm ðK Þ ¼ F m ðK Þ pm ðK Þ
ð2Þ
_ where wm ðK Þ ¼ ixwm ðK Þ is the radial velocity in the wavenumber
domain, and Z M m ¼ jLðmÞj=ðixJ ðL11 L22 L12 L21 ÞÞ represents the mechanical impedance. 2.2. Vibroacoustic partial coupling equation In the acoustic coordinate system, the sound pressure is expressed in the wavenumber domain as
3
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 1. Coordinate systems for infinitely long cylindrical shell immersed partially in fluid: (a) front view; (b) side view.
(P
p ðr ; u ; K Þ ¼ 0
0
0 0 u0 ÞAn ðK ÞHð1Þ 0 u p; n ðjr Þ;
1 n¼1 sinðn
0;
others:
ð3Þ
where kf ¼ x=cf is the exterior acoustic wavenumber, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ k2f K 2 is the radial wavenumber, An ðK Þ is the sound pressure coefficient, and Hð1Þ n is the cylindrical Hankel function of order n. The boundary condition on the wet surface S is
0 0 @ p r ; u ; K @r
_ ¼ ixqf wðu; K Þ
ð4Þ
S;r¼a
but a coordinate transformation is needed because the left- and right-hand sides of Eq. (4) are expressed in different coordinate systems. As shown in Fig. 1, the two coordinate systems are related geometrically by
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 > > < r ¼ r 2 þ H 2Hrcosðu þ p=2Þ ¼ r2 þ H þ 2HrsinðuÞ;
Fig. 2. Discretization of wet surface in circumferential direction.
cosu ¼ rcosu=r ; > > : 0 z¼z; 0
0
ð5Þ 0
where H is the distance from O to O . The wet surface of the shell is in the range ½a; p þ a, and a is calculated via sina ¼ H=a. We have 0 H > 0; r > r; a > 0 when the shell’s center is beneath the free sur0 face, H < 0; r < r; a < 0 when the shell’s center is above the free 0 surface, and H ¼ 0; r ¼ r; a ¼ 0 when the shell’s center is on the free 0 surface. We define another useful angle b ¼ u u which satisfies
cosb ¼ ðr þ HsinuÞ=r0 ;
sinb ¼ Hcosu=r0 :
ð6Þ
By substituting Eqs. (5) and (6) into Eq. (4) and using P _ ilu wðu; K Þ ¼ 1 l¼1 wl ðK Þe , the partial-coupling equation of the shell can be derived as _
1 X n¼1
1 X 0 _ wl ðK Þeilu ; An ðK ÞBn u ; a ¼ ixqf
a u p þ a
ð7Þ
l¼1
where the partial derivative of the pressure is given in the Appendix as Eq. (A.4), and
0 0 0 sinb Bn u ; a ¼ Hð1Þ jr n cos nu n r0 0 ð1Þ 0 0 þ sin nu jHn jr cosb
As shown in Fig. 2, the wet surface is divided into 2N þ 1 strips along u with the interval Du ¼ ðp þ 2aÞ=2N; the ith strip is ui ¼ a þ Duði 1Þ, where i ¼ 1; 2; ; 2N þ 1: The matrix form of the partial-coupling equation is
2
38 9 B11 B2Nþ1;1 A > < 1 > = 6 . 7 .. .. .. 6 . 7 . . . 4 . 5> > : ; A2Nþ1 B1;2Nþ1 B2Nþ1;2Nþ1 8 9 3 2 _ > > C 11 C 2Nþ1;1 > > w 1 > > 7< . = 6 . . . 7 . . . ¼ iqf x6 . . . . 5> . > 4 > > > _ > ; C 1;2Nþ1 C 2Nþ1;2Nþ1 : w 2Nþ1
where B and C are matrices of size ð2N þ 1Þ ð2N þ 1Þ and ele 0 0 0 ments Bn;i ¼ Bn ui ; a , C l;i ¼ eilui , where cosui ¼ a=ri and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ri ¼ a2 þ H2 þ 2Hasinui . Thus, the sound pressure coefficient is _ A ¼ iqf xB1 C w and can be calculated using quadrature rectangle (QR) decomposition [22]; B1 is the inverse matrix of B. Applying a modal expansion to the acoustic load on the wet surface, we have
ð8Þ
In Eq. (7), the circumferential orthogonality of u is no longer satisfied, thus a discretization method for u on the wet surface is proposed. Eq. (7) can be transformed into a matrix equation and solved directly by discretizing u along the wet surface and truncating the infinite series as n ¼ 1; 2; ; 2N þ 1 and l ¼ N; ; 0; N.
ð9Þ
pm ðK Þ ¼
1 X
An
n¼1
1 2p
Z pþa a
sinðnu ÞHðn1Þ ðjr Þeimu du 0
0
ð10Þ
We let
Dn;m
jr ¼ 0
1 2p
Z pþa a
sinðnu ÞHðn1Þ ðjr Þeimu du 0
0
and Eq. (10) can be rewritten in matrix form as
ð11Þ
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
_ p ¼ DA ¼ Z a w
ð12Þ
where D is the matrix of Dn;m , which can be integrated numerically using Gaussian quadrature [22]. Z a ¼ DB1 C is the acoustic radiation impedance of size m l; the elements of m ¼ l and m–l are referred to as the self-impedance and mutual impedance, respectively. The mutual impedance is produced by the partial coupling of structural vibration and sound, and the governing equation for the free vibration of a partially immersed cylindrical shell is _ _ ZM w þ Za w ¼ 0
ð13Þ
where the diagonal matrix Z M impedance.
represents the mechanical
2.3. Far-field sound pressure under point-force excitation Consider a harmonic point ðr ¼ a; u ¼ u0 ; z0 ¼ 0Þ and satisfying
force
F m ðK Þ ¼ F 0 eimu0 =2pa
applied
at
ð14Þ
in the wavenumber domain, where F 0 is the magnitude of the force. Substituting Eqs. (12) and (14) into Eq. (4) gives the spectral radial velocity, namely
1 _ w ¼ ZM þ Za F
ð15Þ
The far-field sound pressure can be derived via the stationary phase method, namely
X 0 eikf R 2Nþ1 0 nþ1 p R; h; u ¼ sin nu An kf cos h ðiÞ pR n¼1 0
ð16Þ
0
where we assume R H and R R in the far-field, with z ¼ Rcosh and r ¼ Rsinh. The far-field pressure can also be written in modal series form as 2Nþ1 X 0 0 p R; h; u ¼ pn ðR; h; u Þ
ð17Þ
n¼1
Using the convergence condition of modal series, the convergence of the far-field pressure can be ensured by setting
pn1 < e and pn < e
ð18Þ
in the following calculations of the far-field sound pressure and radial velocity, we set e ¼ 1 1026 . The theoretical results are given in the form of FDSs of both the sound pressure and radial velocity with a 1-N force excitation, and the sound pressure is corrected to a distance of 1 m. 3. Numerical simulation results and discussion The shell is made of 304 stainless steel, and its geometrical and material parameters are as follows: radius a ¼ 0:1365m; ratio of thickness to radius hs =a ¼ 2:93% or hs =a ¼ 1:46%; density
qs ¼ 7930kg=m3 ;Young’s modulus Es ¼ 1:94201 1011 Pa; Poisson’s ratio rs ¼ 0:25. The sound speed and density of the external fluid are cf ¼ 1500m=s and qf ¼ 1000kg=m3 , respectively. The excitation location is at ðr ¼ a; u0 ¼ p=2; z0 ¼ 0Þ with force
F 0 ¼ 1N. The observation location is assumed to be at the broadside ðh ¼ p=2orK ¼ 0Þ and 1 m from the center of the shell. To explain how the immersion depth influences the sound radiation of the partially immersed cylindrical shell, we conduct an FDS analysis. Fig. 3 shows the spectra of the far-field sound pres0 sure, with the observations located at (a) u ¼ p=2 and (b) 0 u ¼ 3p=4. The thickness-to-radius ratio is 2.93%. The color repre-
sents the magnitude of the sound pressure [Pa], the horizontal coordinate is the frequency [Hz], and the vertical coordinate is the normalized immersion depth Hr ¼ H=a ranging from 1 (i.e., the shell is almost fully in the air, with only the bottom of its surface in contact with the free surface) to 1 (i.e., the shell is almost fully submerged, with only the top of its surface in contact with the free surface). A series of regular oblique bright lines (spots) and two groups of weak interference fringes can be observed in both Fig. 3(a) and (b). One group of dark fringes bends upward and the other downward, characteristics that are related to both the frequency and immersion depth. The observation location affects only the sound pressure amplitude. Studying the radial velocity helps us to understand the mechanism of the oblique bright lines in the pressure FDSs. The measuring point on the outer surface of the cylindrical shell is located at ðr ¼ a; u ¼ p=2; z ¼ 0Þ. For comparison, Fig. 4(a) plots the radial velocity for a submerged, semi-submerged, and airloaded cylindrical shell with a thickness-to-radius ratio of 2.93%. As shown in Fig. 4(a), for the same modal order (e.g. m ¼ 6; where m is the circumferential mode order), the resonant frequency of the air-loaded shell is the highest, while that of the submerged shell is the lowest. Fig. 4(b) plots the m ¼ 6 mode shapes of the (I) submerged, (II) semi-submerged, and (III) airloaded cylindrical shell, each labeled with the corresponding resonant frequency. Fig. 4(c) shows the FSDs of the radial velocity of a partially immersed cylindrical shell, the color representing the radial velocity in [m/s]. The oblique bright lines in Fig. 4(c) indicate the resonant frequency, which decreases with the immersion depth. There is a one-to-one correspondence between the bright spots in Fig. 3 and the bright lines in Fig. 4(c). Thus, the bright spots in the sound pressure spectra are generated by resonance of the shell.
4. Analysis of radiation mechanisms The dominant radiation characteristics of the partially immersed cylindrical shell in the FDSs are resonance and interference. 1) A harmonic force exerted on a smoothly curved elastic shell generates elastic waves on the shell surface, and those waves spread over the shell surface as a series of damped elastic waves [23]. For K ¼ 0, those waves include the flexural wave a0 and the compressional wave s0 , which propagate on the shell surface circumferentially. Shell resonance occurs if the elastic wave on the shell forms a circumferential standing wave [7], and some of the resonant elastic waves radiate into the fluid via phase matching if their phase velocity exceeds the sound speed in the fluid. The other elastic waves whose phase velocity is less than the sound speed in the fluid cannot radiate by phase matching, and at very low frequency only a few waves can radiate because of the curvature of the shell. 2) Because of the impedance discontinuity on the shell surface, the free surface forms two air-fluid demarcation points from where the subsonic flexural waves radiate into the fluid and form a series of bright resonance lines. This is similar to a cylindrical shell with an internal plate or lengthwise ribs [6,7]. Consequently, the air-fluid demarcation points are the sources of the acoustic radiation. 4.1. Resonance phenomenon As mentioned in Section3, the bright lines in the FDSs of the sound pressure are generated by shell resonance. To understand clearly the resonant characteristics of the shell, we explain the relationship between elastic waves and normal modes. Because the theoretical model is an infinite cylindrical shell, we consider
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 3. Frequency–depth spectra (FDSs) of sound pressure radiated from partially immersed cylindrical shell with observation location at (a) u0 ¼ p=2 and (b) u0 ¼ 3p=4. The thickness-to-radius ratio is 2.93%.
Fig. 4. Radial velocity at ðr ¼ a; u ¼ p=2; z ¼ 0Þ on shell: (a) radial velocity of the shell when it is (I) submerged, (II) semi-submerged, and (III) air-loaded; (b) m ¼ 6 modal shapes of (I) submerged, (II) semi-submerged, and (III) air-loaded shell; (c) FDSs of radial velocity of partially immersed cylindrical shell with a thickness-to-radius ratio of 2.93%.
only the circumferential modes. As presented in Ref.23, the resonant characteristics can be described in the modal form expðið mu ixt ÞÞ, where m is the mode order and represents the mth normal mode. Each mode is a multipole, with m ¼ 0 being the ‘‘breathing mode” (where the pressure contours move radially in and out), m ¼ 1 being the ‘‘dipole” (where the pressure contours move radially back and forth), etc. The motions of the mth normal mode are all standing waves in the circumferential direction and traveling with phase velocity cph ðxÞ ¼ ax=m. This is easily understood because, for the mth mode, exactly m wavelengths fit around the shell circumference. Obviously, the pressure contours of the mth normal mode are coincident with the mth circumferential vibration mode shape of the shell. Because the free surface divides the shell into a dry part and a wet part, the resonant characteristics can be analyzed by using the dispersion of the elastic waves on the dry and wet parts of the evacuated shell. From Eq. (1), the dispersion equation for an infinitely long cylindrical shell in vacuum is
jLðmÞj ¼ 0
ð19Þ
which has three types of root corresponding to three types of elastic wave, namely s0 (compressional), a0 (flexural), and T 0 (torsional). We take the observation point at the broadside (h ¼ p=2 or K ¼ 0), thereby allowing axial displacement to be neglected. The dominant elastic waves are s0 and a0 in the circumferential direction, and the dispersion equation for a fluid-loaded cylindrical shell with K ¼ 0 can be simplified as [7]
8 9 2 4 x2 qf Hðð1xÞ a=c Þ ðkf aÞ= < xa ph 2 xa 1 þ bs þJ ð1Þ0 : cs cph kf H xa=c ðkf aÞ ; ð ph Þ ( 2 ) 2 2 xa xa xa ¼0 cph cs cph
where bs ¼
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 hs =12a2 , c2s ¼ Es =ð1 r2s Þqs , and cph is the phase
velocity. Fig. 5 shows the dispersion curves of the s0 and a0 waves in a fluid-loaded cylindrical shell with a thickness-to-radius ratio of 2.94% or 1.46%. The phase velocities of the s0 and a0 waves are real when the shell is in vacuum but become complex when the shell is submerged. The phase velocity of the s0 wave is related weakly to the shell thickness, but that of the a0 wave is related to the shell thickness whether the shell is in vacuum or submerged. In the calculated frequency range, the a0 wave is subsonic (or ka0 > kf ) and cannot radiate energy via phase matching, while the s0 wave is supersonic (or ks0 < kf ) and can radiate into the fluid via phase matching when u > uc , where uc ¼ sin ðcf =cs0 Þ is the critical angle of the s0 wave. For comparison, Fig. 6 plots the far-field sound pressure at u0 ¼ p=2 for submerged and semi-submerged (Hr ¼ 0) cylindrical shells with a thickness-to-radius ratio of 2.93%. The small peaks in Fig. 6(I) are produced by the subsonic a0 wave because of the curvature of the shell [24], that being because phase-matching is 1
6
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 5. Phase-velocity dispersion curves of a0 and s0 waves on submerged cylindrical shell with two different thicknesses: (a) real part of a0 and s0 waves; (b) real and imaginary parts of s0 wave. The solid and dashed curves are for thickness-to-radius ratios of 1.46% and 2.93%, respectively.
Fig. 7(a) and (b) are produced by resonance of the subsonic a0 waves that are radiated from the air-fluid demarcation points on the shell. As shown in Fig. 8, the wet surface is represented by c ¼ p þ 2a, where c ¼ 0 (Hr ¼ 1) and c ¼ 2p (Hr ¼ 1) correspond to the shell nearly in air and in fluid, respectively. From the propagation velocities of the elastic waves on the dry and wet parts, the following simple formula is derived for predicting the resonant frequency of a partially immersed cylindrical shell:
f ¼m
ð2p cÞa ð0 Þ
cph ð 0Þ
þ
ca ðw Þ
cph
!1 ;
ð21Þ
ðwÞ
where cph and cph are the phase velocities of the elastic waves on the shell when it is in vacuum and submerged, respectively. Also, f ¼ mcph =2pa with c ! 0 and f ¼ mcph =2pa with c ! 2p. Thus, ð0Þ
Fig. 6. Far-field sound pressure at u0 ¼ p=2 for (I) submerged and (II) semisubmerged cylindrical shells with a thickness-to-radius ratio of 2.93%.
not satisfied. However, Fig. 6(II) contains a series of strong resonant peaks that could be due to the resonance of the a0 wave. Fig. 7 shows the FDSs of the far-field sound pressure for a partially immersed cylindrical shell with a thickness-to-radius ratio of (a) 2.93% and (b) 1.46%. The observation location is at u0 ¼ p=2. Clearly, the horizontal interval of the bright spots decreases with decreasing shell thickness. Therefore, the oblique bright spots in
ðwÞ
the resonant frequency of a partially immersed cylindrical shell decreases with immersion depth to form a series of oblique bright spots in the pressure spectra in Fig. 7. The solid yellow lines overlaid on the spectra show the resonant frequency of the a0 wave as predicted by Eq. (21). As shown, the predicted results are almost consistent with the resonant bright lines. In addition to the oblique bright spots produced by the a0 waves, there is a strong bright stripe at higher frequencies, labeled by the dashed red wireframes in Fig. 7(a) and (b). The interference stripes are produced by resonance of the zeroth-order s0 wave (the
Fig. 7. FDSs of sound pressure radiated from partially immersed cylindrical shell with thickness-to-radius ratio of (a) 2.93% and (b) 1.46%. The solid yellow curves show the resonant bright lines predicted by Eq. (21). The dashed red wireframes show the resonance of the zeroth-order s0 wave.
7
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
only within the immersion depth range of c=2 > uc , when the observation point is at u0 ¼ p=2. Thus, the bright stripe in Fig. 7 is stronger in the depth range of Hr ¼ 0:8 to 1. Furthermore, the second path is ineffective when the observation point falls in the ranges of u < a þ uc and u > a þ p uc , so the bright stripe shrinks as the observation point approaches the free surface, and those results with the observation point near the free surface are omitted here. 4.2. Interference phenomenon
Fig. 8. Schematic of circumferential resonance formed by elastic waves. ‘‘1” and ‘‘2” represent the air–fluid demarcation points on the shell.
resonant frequency is referred to as the ring frequency), which is different from the interference stripes generated by the a0 wave and is difficult to predict using Eq. (21) because Eq. (21) fails at m ¼ 0. The ring frequency is defined such that the length of the compressional wave at this frequency is equal to the circumference of the shell. However, we can estimate the ring frequency of a partially immersed cylindrical shell based on the dry shell and wet shell. For a shell in vacuum, the ring frequency can be calculated by letting m ¼ 0 in Eq. (19), which leads to f ¼ cs =2pa, where cs is the longitudinal wave velocity on a plate of the same thickness as the shell. The ring frequency of a submerged cylindrical shell is calculated by letting xa=cph ðxÞ ¼ 0 in Eq. (20) to satisfy
ð1Þ xa 2 a2 1 r2 x2 qf H0 kf a ¼ 0: 1 þ ð1Þ0 E s hs cs kf H0 kf a
ð22Þ
Two types of weak interference fringes correlated with the shell thickness, as shown in Figs. 3 and 7, are therefore caused by interference among the a0 waves radiated along different paths. The a0 waves are generated by the force and then propagate along the circumference in clockwise and counterclockwise directions, to finally radiate through the two air–fluid demarcation points. Figs. 10 and 11 illustrate the two types of paths related to the dry part and wet part. 4.2.1. Interference fringes related to dry part From the geometric relationship illustrated in Fig. 10, the acoustic path difference formed by the dry part is dup ¼ ð2p cÞa. The corresponding paths are as follows: (i) the a0 wave propagates along the dry part in the clockwise direction and radiates first from point 1 and then from point 2; (ii) the a0 wave propagates along the dry part in the counterclockwise direction and radiates first from point 2 and then from point 1. Thus, simple formulas for predicting the constructive and destructive interference fringes, respectively, are
f ¼ mcph =ð2p cÞa; ð0Þ
m ¼ 1; 2; ;
f ¼ ðm þ 0:5Þcph =ð2p cÞa; ð0Þ
ð23Þ
m ¼ 1; 2; ;
ð24Þ
ð0Þ
where cph is the phase velocity of the a0 wave on the shell in vacuum. The resonant frequency is f ¼ mcph =2pa at c ! 0 while ð0Þ
Eq. (22) can be solved using a root-finding algorithm. The ring frequencies of a cylindrical shell in vacuum and submerged are listed in Table1, and these two frequencies are the upper and lower limits, respectively, of the ring frequency of a partially immersed cylindrical shell. Therefore, the ring frequency of a partially immersed cylindrical shell can be estimated based on Table1, and the estimated results represented by the dashed red wireframes agree with the location of the bright stripe shown in Fig. 7. The imaginary part of the phase velocity of the elastic waves represents the radiation efficiency of a submerged cylindrical shell [24]. As shown in Fig. 5(b), the peak of the imaginary part of the s0 wave covers a certain frequency range, which results in a wider resonant region of the zeroth-order s0 wave in the pressure spectra in Fig. 7. As shown in Fig. 9, the s0 wave can radiate into the fluid through two paths, namely (i) from the air–fluid demarcation points on the shell and (ii) from the critical angle due to phase matching; the latter possesses a higher radiation efficiency. Therefore, the bright stripe produced by the s0 wave is stronger than that produced by the a0 wave. However, the second path is effective
f ! 1 at c ! 2p. Thus, the resonant frequency of the interference fringes related to the dry part is correlated positively with the immersion depth to form the upward-bending dark fringes in Fig. 7.
4.2.2. Interference fringes related to wet part As illustrated in Fig. 11, the acoustic path difference formed by the wet part is ddown ¼ ca. The corresponding paths are as follows: (i) the a0 wave propagates along the wet part in the counterclockwise direction and radiates first from point 1 and then from point 2; (ii) the a0 wave propagates along the wet part in the clockwise direction and radiates first from point 2 and then from point 1. As in Section 4.2.1, simple formulas for predicting the constructive and destructive interference fringes, respectively, are
f ¼ mcph =ca; ðwÞ
m ¼ 1; 2; ;
f ¼ ðm þ 0:5Þcph =ca; ðwÞ
m ¼ 1; 2; ;
ð25Þ ð26Þ
ðwÞ
where cph is the phase velocity of the a0 wave on a submerged cylindrical shell. Obviously, the resonant frequency is f ! 1 at
Table 1 Resonant frequency of the zeroth-order s0 wave (ring frequency) of shell in vacuum or submerged. Order
m¼0 m¼0
Thickness-to-radius ratio
1.46% 2.93%
Frequency [Hz] In vacuum
Submerged
5956.9 5956.9
1195.2 2619.7
ð0Þ c ! 0 and f ¼ mcph =2pa at c ! 2p. Thus, the resonant frequency
of the interference fringes related to the wet part is correlated negatively with immersion depth, forming the downward-bending dark fringes in Fig. 7. Fig. 12 shows the FDSs of the sound pressure of a partially immersed cylindrical shell of varying thickness. The data in Fig. 12(a)–(d) are the same as those of Fig. 7(a) and (b), the difference being that the dashed green and solid blue curves show the
8
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 9. Propagation paths of s0 wave with observation point at (a) u0 ¼ p=2 and (b) obliquely below, e.g., u0 ¼ 3p=4.
Fig. 10. Propagation paths of a0 wave along dry part to produce the upward-bending interference fringes: (a) clockwise direction; (b) counterclockwise direction.
Fig. 11. Propagation paths of a0 wave along wet part to produce the downward-bending interference fringes: (a) counterclockwise direction; (b) clockwise direction.
two types of interference fringes, which agree well with the theoretical results. The constructive interference fringes represented by the dashed green curves in Fig. 12(a) and (b) are predicted by Eq. (23), and those in Fig. 12(c) and (d) are predicted by Eq. (25). The destructive interference fringes represented by the solid blue curves in Fig. 12(a) and (b) are predicted by Eq. (24), and those in Fig. 12(c) and (d) are predicted by Eq. (26). Because the phase velocity of the a0 wave increases with frequency, the intervals of
both types of fringes narrow, as can be understood from Eqs. (23)–(26). In addition to the interference related to the dry part and wet part, sound waves radiate from points 1 and 2, producing geometric interference. The path difference between points 1 and 2 is dsina, and dsina < 2a, where d is the distance between points 1 to 2. Because a< kmin ¼ 0:3m in the calculated frequency range, the geometric interference between the points can be ignored. In
K. Zhao et al. / Applied Acoustics 161 (2020) 107170
9
Fig. 12. FDSs of sound pressure radiated from partially immersed cylindrical shell with thickness-to-radius ratios (a, c) 2.93% and (b, d) 1.46%. The dashed green and solid blue curves represent the predicted results for the constructive and destructive interference fringes, respectively.
fact, these discontinuous points are the sound radiation sources for a partially immersed cylindrical shell. Therefore, in the lowfrequency range (no s0 -wave component), the virtual source interference related to the free surface is nonexistent, and the directivity pattern can be approximated as a superposition field of two equivalent strength dipoles. Fig. 13 shows the directivity patterns of the far-field sound pressure level (20lgðp=p0 Þ withp0 = 1lPa) of a partially immersed cylindrical shell with a thickness-toradius ratio of 2.93%; the maximum value at u0 ¼ p=2 can be observed. As is known, the directivity function is jsinu0 j for an acoustical dipole and is cos kf d=2 cosu0 for two in-phase sources [25]. Therefore, the directivity function for a partially
immersed cylindrical shell at low frequency can be derived approximately as Dðu0 Þ ¼ cosðkf d=2 cosu0 Þsinu0 .
5. Experiment The air-filled cylindrical shell is made of seamless stainless steel, with a radius of 0.1365 m, a length of 0.5 m, a thicknessto-radius ratio of 2.93%, and material parameters as listed in Section 3. Both ends of the shell are closed by two 0.01-m-thick flat end-caps of the same material as the shell. The end-caps also serve as the counterweights. Photographs of the experimental model and
Fig. 13. Directivity patterns of far-field sound pressure levels of a partially immersed cylindrical shell at different immersion depths Hr ¼ 0:5; 0; 0:5 at frequencies (a) 500 Hz and (b) 2500 Hz.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 14. Photograph of experimental model.
a sketch of the experimental setup are shown in Figs. 14 and 15, respectively. The experimental model is immersed horizontally in a 5 m 5 m 5 m water-filled tank with a 4.5-m-deepth water layer and a 0.2-m-deep layer of fine sand at the bottom. The shell is connected to the lifting platform located at the central axis of the tank by elastic ropes, and the lifting motor is controlled by a computer at a speed of 0.2 m/min. To counteract the buoyancy of the shell during descent, two wooden planks (with high damping capability) are installed between the end-caps and the lifting platform. The exciter is hung elastically at the top of the shell and is connected to it by a PCB-208C02 force sensor. The receiver is a B&K8103 hydrophone mounted on a 1-m-radius quarter-arc support and fixed under the shell 1 m from the free surface. During the measurement, the hydrophone is static when the shell is descending. In this process, we obtain the radiated sound pressure at a fixed measuring point as the shell is lowered continuously. A B&K4534-B accelerometer is mounted at the bottom of the shell to measure the radial velocity, and a high-capacity uninterruptible power supply is used during the entire measurement. Two
synchronizations are used in the measurement as shown in Fig. 15. The first one is the synchronization between the signal generator and the data acquisition system. This is a multiple-trigger mode, and the trigger signal is the output from the signal generator. The other one is the synchronization between the data acquisition system and the lifting platform. This is a single-trigger mode, and the trigger signal is the output from the data acquisition system. These two synchronizations ensure that the recorded data corresponds accurately to the shell’s immersion depth. Limited by the measurement system, the frequency range of the experiment is 0.5–2.5 kHz. To achieve a higher resolution for the immersion depth, a chirp pulse (40 ms, 0.5–2.5 kHz) is used as the excitation signal, with its time plot shown in Fig. 16. The excitation signal is generated repeatedly with a time period of 450 ms. The sample frequency satisfies the Nyquist sampling theorem. The acquisition time for a single pulse is 100 ms, and totally 184 pulse counts are collected within a complete measurement as the shell is lowered continuously from Hr ¼ 1 to Hr ¼ 1. The normalized immersion depth of the shell is calculated by Hr ¼ 0:2 count=a, where count is the count number of the collected data. The time signal is transformed into the frequency domain by a fast Fourier transform, and this process results in the FDSs for the corresponding measurement. As illustrated in Fig. 15, because the path difference between the direct and reflected waves from the sidewalls
Fig. 15. Sketch of experimental setup.
Fig. 16. Time plot of chirp signal.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 17. Comparison of analytical and experimental results for (a) radial velocity at ðr ¼ a; u ¼ p=2; z ¼ 0Þ and (b) sound pressure amplitude at u0 ¼ p=2 of a shell immersed at Hr ¼ 0.
Table 2 Measured resonant frequencies of experimental model at Hr ¼ 0, where q and m represent the axial and circumferential mode orders, respectively. Orderðq; mÞ
(1,4)
(1,5)
(1,6)
(3,5)
(1,7)
Frequency [Hz]
701.6
1093.1
1615.2
2065.4
2230.2
and bottom exceeds 5 m, by considering the reflection loss, the reflected waves are attenuated by more than 20lg5 14dB and can be ignored. For comparison, the measured results are corrected to those for an excitation force of 1 N. To assess the accuracy of the theoretical results, the analytical and experimental frequency spectra of the radial velocity at
ða; p=2; 0Þ and the far-field sound pressure amplitude at u ¼ p=2 are compared in Fig. 17(a) and (b), respectively. The analytical results agree well with the experimental results except for several small discrepancies in the resonant frequencies. As can be seen, the experimental resonant peaks are slightly larger than the analytical ones, especially for m ¼ 4. This may be due to the differences in materials and geometry between the theoretical and experimental models, given that the experimental model is finite rather than an infinite shell. The peaks at 2065.4 Hz in Fig. 17(a) and (b) of the experimental results are due to the axial resonance of the shell, as is shown by the mode test results in Table2. The measured resonant frequencies of the model at Hr ¼ 0 are listed in Table2, where q and m represent the axial and circumferential mode orders, respectively. 0
Fig. 18. FDSs of radial velocity of partially immersed cylindrical shell: (a) experimental and (b) calculated results with measurement point at ðr ¼ a; u ¼ p=2; z ¼ 0Þ; (c) experimental and (d) calculated results with measurement point at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ. The thickness-to-radius ratio is 2.93%.
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
Fig. 19. FDSs of sound pressure radiated from partially immersed cylindrical shell: (a) experimental and (b) calculated results with measurement point at u ¼ p=2; (c) 0 experimental and (d) calculated results with measurement point at u ¼ 3p=4. The thickness-to-radius ratio is 2.93%, and the solid yellow curves show the resonant bright lines predicted by Eq. (22). 0
Fig. 20. Interference fringes predicted for experimental results in Fig. 19(a): (a) bending upward; (b) bending downward. The constructive interference fringes represented by the dashed green curves in (a) are predicted from Eqs. (23) and (25), and those in (b) from Eqs. (24) and (26).
Fig. 18 illustrates the measured and calculated FDSs of the radial velocity on the shell. The left and right parts are the measured and calculated results, respectively, with the measuring point at ðr ¼ a; u ¼ p=2; z ¼ 0Þ for Fig. 18(a) and (b) and at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ for Fig. 18(c) and (d). Fig. 19 shows the measured and calculated FDSs of the sound pressure of the shell. The left and right parts are the measured and calculated results, respectively, with the observation point at u0 ¼ p=2 for Fig. 19(a) 0 and (b) and at u ¼ 3p=4 for Fig. 19(c) and (d). The yellow curves overlaid on Fig. 19(a) and (c) show the resonant frequencies predicted by Eq. (21). The oblique resonant bright spots and interference fringes generated by the a0 wave in the pressure spectra can be observed clearly in the experimental results. By contrast, the resonant bright spots produced by the s0 wave in the pressure spectra cannot be observed because the maximum experimental
frequency is lower than the zeroth-order s0 wave resonant frequency (or ring frequency of 2619.7 Hz) of a submerged shell, which is the lowest resonant frequency of an s0 wave on a partially immersed cylindrical shell. The experimental and calculated results agree well in Figs. 18 and 19, except for small deviations at low frequency, e.g., the fourth-order resonant frequency. Differences in radial velocity can also be observed for the sixth-order resonant peaks at ðr ¼ a; u ¼ 3p=4; z ¼ 0Þ in Fig. 18(c) and (d). These deviations may be caused by (i) the experimental model being finite and the end-caps and supporting wooden planks producing prestress on the shell, which leads to the lower-order modal frequencies in the experiment being larger than those in the calculated results, or (ii) the phase velocity of the elastic waves of the experimental model being not completely the same as that of the theoretical model. Consequently, the fourth-order modal
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K. Zhao et al. / Applied Acoustics 161 (2020) 107170
frequency of the a0 wave in the experimental results is not predicted accurately by the formula. Two types of interference fringes can be observed in the experimental results shown in Fig. 19(a) and (c), which agree well with the calculated results. Because the same interface fringes are observed in Fig. 19(a) and (c), only the predicted results of u0 ¼ p=2 are presented. The predicted interference fringes for upward bending and downward bending are shown in Fig. 20(a) and (b), respectively, and agree well with those in the experimental results. The data in Fig. 20(a) and (b) are the same as those in Fig. 19(a), the difference being that the dashed green and solid blue curves show the predicted interference fringes. The dashed green and solid blue curves in Fig. 20(a) are calculated form Eqs. (23) and (24), and those in Fig. 20(b) are calculated from Eqs. (25) and (26). Additionally, some vertical fringes are observed in the experimental results as shown in Fig. 19(a) and (c). These fringes may be caused by the following two aspects: (i) the experimental model is finite while the theoretical model is infinite, so the axial resonant fringes cannot be predicted by the present theoretical method; (ii) sound in air generated by the shaker will propagate into the water and produce some of the fringes. Therefore, the far-field sound pressure of a partially immersed cylindrical shell is quite different from that of a fully submerged one. The main differences are as follows. 1) The resonant lines that appear in the sound pressure spectra of a partially immersed shell almost disappear for a submerged shell, as shown in Fig. 6. Because of the curvature of the shell, only a small amount of the subsonic a0 wave can radiate sound to the far field to form a few resonant peaks at very low frequency. 2) The resonant frequencies of a partially immersed cylindrical shell decrease with the immersion depth, and the horizontal intervals of the resonant frequencies increase with the frequency. 3) Interference between the radiated waves that propagate on the dry part and the waves on the wet part produces the interference fringes in the FDSs of sound pressure. This interference means that the resonant peaks of the pressure spectra will diminish or even disappear at a certain immersion depth of the shell. This physical phenomenon does not occur in the sound pressure spectra of a submerged cylindrical shell.
bright lines in the pressure spectra, while the supersonic wave s0 can radiate out not only from the demarcation points but also through phase matching to form an enhanced bright strip in the pressure spectra. Because the phase velocity of a0 wave is different on the dry and wet parts of the shell, the resonant frequencies of partially immersed cylindrical shell decrease with the immersion depth. 2. Interaction between a0 waves radiated from different air-fluid demarcation points produce interference fringes in the pressure spectra. The acoustic path differences of the upward and downward bending interference fringes are related to the dry and wet parts of the shell, respectively. 3. The predicted results from simple formulas agree well with the experimental and numerical results.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Contract No. 11774229). We would like to thank Dr. Yunzhe Tong and Bo Liu for illuminating discussions. Appendix A The axial, circumferential, and radial displacements of the shell are
9 uðu; zÞ > =
v ðu; zÞ > ¼
wðu; zÞ
;
1 2p
1 X
eimu
m¼1
8 9 > > > < u m ðK Þ > = eiKz dK: v m ðK Þ > 1 > > > : ; wm ðK Þ
Z
1
ðA1Þ
The acoustic and force loads are: 6. Conclusions
pðu; zÞ
In this paper, the radiation mechanisms of a partially immersed cylindrical shell undergoing point-force excitation are analyzed using theoretical and experimental methods. The spectra for both vibration and sound are measured and calculated with the shell in continuous immersion. The major characteristics are a series of regular oblique bright lines and weak interference fringes in the FDSs of radiated sound pressure, which can be observed by experiment. Simple formulas for predicting the bright lines and interference fringes are derived. The radiation mechanisms of the partially immersed cylindrical shell can be explained as follows: 1. The a0 and s0 waves are excited by the force. Shell resonance occurs when these elastic waves form a standing wave circum-
0 0 @ p r ; u ; K @r
(P ¼ S;r¼a
1 ð1 Þ n¼1 An ðK ÞðHn
jr0 ncos nu0
sinb r
0
Fðu; zÞ
8 9 Z 1< 1 = P ð K Þ 1 X m eiKz dK: eimu ¼ ; 2p m¼1 1 : F m ðK Þ
The elements of matrix L ðmÞ are 2 6 L ðmÞ ¼ 6 4
X2s þ v2 þ r m2
rþ mv 2 Xs þ r v2 þ m2
rþ mv irs v
im
3
irs v
7 7 im 5 2 2 2 2 2 1 Xs þ b v þ m ðA3Þ
2
where Xs ¼ xa=cs , b2s ¼ hs =12a2 , c2s ¼ Es =qs 1 rs , v ¼ Ka, and K is the axial wavenumber. For economy of space, we introduce the symbols rþ ¼ ð1 þ rs Þ=2 and r ¼ ð1 rs Þ=2. The partial derivative of the pressure is 2
0 0 0 þ sin nu jHnð1Þ jr cosbÞ;
0u p 0
ðA4Þ
p u0 2p
0;
ferentially. The resonance of the subsonic wave a0 can radiate from the air-fluid demarcation points to form a series of oblique
ðA2Þ
( where @p=@r ¼
0
@p @r 0 @r @r
þ
0
@p @ u 0 @ u @r
and
@r =@r ¼ ðr þ HsinuÞ=r ¼ cosb; 0 uÞ @ u =@r ¼ Hcosru0 3ðHþrsin ¼ sinb 0 0 : sinu r 0
0
14
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