Response Green's function and acoustic scattering from a fluid-loaded cylindrical shell with discontinuities

Compurers& Brucrures Vol. 65, No. 3, pp. 337-384. 1997 0 1997Elxvier ScienceLtd. All rightsreserved

Pergamon PII: SOO45-7949(%)00254-4

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RESPONSE GREEN’S FUNCTION AND ACOUSTIC SCATTERING FROM A FLUID-LOADED CYLINDRICAL SHELL WITH DISCONTINUITIES J. M. Cuschierit and D. Feit$ tCenter for Acoustics and Vibration, Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431, U.S.A. SCarderock Division, NSWC, David Taylor Research Center, Bethesda, MD 20084-5000, U.S.A. (Received 7 August 1995) Abstract-A classical problem in structural acoustics is the response and acoustic scattering from a fluid-loaded cylindrical shell with ring discontinuities, to excitation by incident acoustic waves. The solution for l.hese type of problems is generally formulated using integral transforms to the wavenumber domain. The solution in the spatial domain is obtained by inverse transforming the wavenumber solution. The inverse transform is typically obtained using a contour integral approach which can handle the singularities of the inverse transform. A contour solution is required for every spatial location of interest. Structural damping can be introduced to remove the singularities from the integrand of the inverse transform, and the spatial solution obtained by performing the inverse transform numerically. However, damping can mask the influence of the fluid-loading. An alternative to these approaches to obtain the spatial solution is a hybrid numerical analytical method which does not require the introduction of structural damping, and therefore retains only the influence of the fluid-loading on the response and the scattering. Csing this hybrid numerical/analytical approach, the response Green’s function and the nearfield scattered pressure are obtained for a fluid-loaded shell with a wall thickness to radius ratio of l%, excited by a circumferential ring load, which is representative of scattering by a ring impedance, excluding the shell to impedance interaction. Apart from the response Green’s function and the nearfield scattered pressure, of interest in structural acoustics is the far field scattered pressure. A solution for the bistatic and monostatic scattering from a fluid-loaded cylindrical shell with one or two internal plate bulkheads is presented. The solution is based on knowledge of the junction forces between the shell and the internal plate bulkheads which are obtained using a mobility (or impedance) approach. The mobility functions are obtained using the hybrid method. Results for the response Green’s function and the nearfield and farlield scattered pressures are presented for the circumferential modes n equals 0 and 2. The results for the n equals 2 case are representative of high order circumferential modes which include the presence of compressional and shear helical waves. 0 1997 Elsevier Science Ltd.

1. INTRODUCTION

The first part of this paper deals with the solution for the velocity response Green’s function and the nearfield scattered pressure for a fluid-loaded, thin, elastic shell with a circumferential ring discontinuity. The solution is derived using a hybrid numerical analytical approach. The cylindrical shell is fluid loaded on the outside with an unbounded acoustic fluid. The inside of the shell is considered to be in-vacua. The solution is obtained for the case when the cylindrical shell is excited by an obliquely incident acoustic wave, ang,le of incidence f3 to the radial direction of the shell. The ring discontinuity is modeled by four impedance functions along the coordinates of the shell. The loads induced by the ring discontinuity are given by the product of the impedance functions and the shell velocities at the location of the ring discontinuity. Since a shell has three types of displacements, radial, circumferential and axial, the ring discontinuity may induce a radial

force and moment, a circumferential load and an axial load. The relative magnitude, phase and coupling between these loads are dependent on the characteristics of the ring impedance discontinuity and the interaction with the fluid-loaded shell. While the solution is for a general ring impedance discontinuity, the response Green’s function and nearfield scattered pressure results are for normalized induced loads. The results for a specific ring discontinuity, which would include coupling and the relative magnitude and phase of the forces requires a complete definition of the ring impedance discontinuity. This paper presents the method by which the response Green’s function can be obtained for an arbitrary set of forces, but the method also applies to coupled forces. The solution for the response or the scattered pressure from a fluid-loaded cylindrical shell is generally obtained by first transforming the differential equation of motion of the shell from the spatial domain to the wavenumber domain. This approach 337

338

J. M. Cuschieri and D. Feit

is a convenient way to properly take into account the influence of the fluid loading. The response Green’s function or the nearfield scattered pressure are obtained by inverse transforming the resulting wavenumber equations of motion. However, these wavenumber equations of motion have singularities which are on or near the axis of integration (real axis). These singularities can cause numerical problems if one attempts a direct, inverse Fourier transform and the classical solution is to use a contour integral approach. However, in this paper the method of solution used is based on a hybrid approach [1], which generates a solution for the response Green’s function of the shell from the evaluation of only one integral. The scattered pressure along planes parallel to the surface of the shell are also obtained by evaluation of this single integral. This single integral is performed numerically using an inverse discrete Fourier transform (IDFT). To avoid the problems that the singularities will pose when evaluating the IDFT, these singularities are removed and their contribution evaluated analytically. The combined result from the numerical part and the analytical part forms the hybrid solution. The second part of this paper deals with the scattering of acoustic waves when the interaction between the shell and the discontinuity or discontinuities is taken into account. The farfield scattered pressure and the response Green’s function are considered for the case of the fluid-loaded shell when internally loaded by a single or double plate bulkhead (Fig. 1). The two plate bulkheads are separated by a distance equal to a shell diameter. Interaction, both structurally and acoustically, takes place between the two bulkheads, creating a complex scattering pressure pattern. The structural interaction between the plate bulkhead discontinuities and the fluid-loaded shell is described using the mobility power flow (MPF) approach [2] used in previous work to evaluate the power input and flow between coupled substructures.

The MPF approach describes the junction forces and moments, created by the interaction between the substructures, in terms of the junction mobility functions of the uncoupled substructures, together with an input of the shell free response under the influence of obliquely incident acoustic waves. Through compatibility of the forces, moments and velocities at the junctions between the shell and the internal plate bulkheads, a mobility matrix is defined, the solution of which gives the junctions’ forces and moments. For the two junction case, the mobility matrix not only includes mobility functions with the response and loading defined at the same location, but also transfer mobility functions, with the response and the loading defined at different locations. These transfer mobility functions are the basis for the structural interaction between the two bulkhead discontinuities and thus the overall scattered pressure. The advantage of using this approach over other methods is that, through the evaluation of the power flow from the shell to the internal plate, a check is performed on the consistency of the analysis. The net flow of power, that is, the power flow from the shell to the plate, must be greater than zero since the shell is the excited substructure. This check will ensure that the proper configuration is taken into account when evaluating the junction forces. It has been found from previous work [2] that unless the proper mobility functions with the proper signs are considered for the interaction between the uncoupled motions of the substructures, the resulting net power flow from the substructure with the excitation (source substructure) to the coupled substructure (receiver substructure) may be negative. This is physically impossible, since the receiver substructure does not have any external input power. In implementing the MPF approach, junction and input mobility functions must be u-priori known. The junction mobility functions are evaluated for the uncoupled substructures. In solving for the mobility functions, the boundary conditions for both the fluid-loaded shell

Fig. 1. Cylindrical shell with single or double internal plate bulkheads, excited by an obliquely incident acoustic wave with angle of incidence 0.

Response Green’s function and acoustic scattering and the plate at the location of the junction are taken to be free, or with externally applied loads. In some instances, it may be possible to impose other boundary conditions that would simplify the problem by reducing the number of mobility expressions and degrees of freedom that must be considered. However, a free boundary condition would provide the most general solution without any imposed assumptions. The junction mobility functions for the internal plate bulkhead are evaluated using classical thick (Mindlin) plate equations [3). For the case of the cylindrical shell, lthe input and transfer mobility functions are evaluated from the response Green’s functions solution developed in the first part of this paper. Having evaluated the mobility functions, the solution for the junction forces and moments is obtained from a matrix solution of the equations describing the consistency of forces and velocities at the junction between the shell and the plate. From knowledge of the junction forces and moments, the power flow, the re:sponse and the scattered pressure components associated with each of these forces and the total far field pressure can be evaluated.

339

from the source is obtained. Also, the convention will influence the choice of the sign of the square root term in the contribution due to the scattered pressure. The equations of motion that describe the behavior of the shell are the Donnell-Mushtari shell equations as used in Junger and Feit [S], with some minor differences, for a thin shell with a large radius to thickness ratio. Because of the curvature of the shell, there are three coupled equations of motion that describe the shell motion in the axial (u), circumferential (0) and radial directions (w), (Fig. 2). The equations of motion are,

+va$+fYu= -yF”

(qqa$

+a(q azv

a2V

+B’)@

x(1 +P2),,2+(1 2. SHELL EQUATIONS

OF MOTION

The cylindrical structure with the definition of the coordinate system is shown in Fig. 2. In the analysis a harmonic time dependency of the form e-j”’ is assumed. That is, waves travelling in the positive x-direction will have an exponential function of the form e”r”. If a solution to the fluid-loaded shell problem was to be obtained using a classical contour integral approach [4], the selected convention would influence the shape of the contour and thus, which roots (singularities) are included within that contour. The selection of the contour is such that the Sommerfeld radiation condition of a decreasing far-field acoustic pressure with increased distance

+

aw

(

j$-

a3w

-a2P2ax2 B@ d3W

2

a&

+R% = -yF,

au

(

au

-vaXi- a+

_ a2p2

\

w c; x

\

fl

4

u

ring disconti uity

Y

Y shell radius a wall thickness h

6

Fig. 2. Cylindrical shell with ring discontinuity and excited by an obliquely incident acoustic wave with angle of incidence 0 I:Othe shell radial direction.

> (2)

&-B2@)

aB

incident plane acoustic waves

\

(1)

-Fw+z+@i+ps) 1

(3)

where a and h are, respectively, the radius and wall thickness of the shell, E and v are the shell material modulus of elasticity and Poisson’s ratio, respectively, b = h/(a&), y = a’(1 - v2)/(Eh), F,, F, and F, are, respectively, the axial, circumferential and radial external forces per unit area, B is the external bending moment per unit area, pi and pS are, respectively, the incident and scattered pressure acting on the exterior of the shell, and n is the frequency parameter normalized with respect to the ring frequency, R = w/o~, or = q/a and c, = ,/E/ [~(l - v’)], Q is the shell material density, and pi and ps are, respectively, the incident and scattered acoustic pressure components. The solution to these coupled equations of motion [eqns (l)-(3)] in the case of an infinite shell is readily

J. M. Cuschieri and D. Feit

340

obtained by using an axial spatial Fourier transform to the axial wavenumber (k) domain. In the circumferential direction, a modal decomposition on the circumferential modes of the shell is implemented, and the solutions are obtained for specific values of the circumferential mode number n. For the displacements, the following modal decomposition is used, [S]

kind [6]. That is the nth component of the scattered pressure on the surface of the shell is given by, using the normalized notation,

n jn

I+ $?!

[ (>

u(x, 4) =

U(X, 4)

c Wx)cosW)

= C vn(x)sW4)

4% 4) = 1 w”(x)cos(n~). ”

(4)

Spatial Fourier transforming eqns (lH3) and introducing the modal decomposition of eqn (4), the resulting set of equations written in matrix form are given by,

~,,=f2~-(RM7i)2-

=

2

n f2M

PI

c0ge)

.Q+ ‘&ClMz - l2M sin(@) H;(QM cos(8))

’

where 6. = 1 if n = 0 and E”= 2 for n > 0. P, is the magnitude of the incident pressure wave. The delta function, d(QM7i - RM sin(@), is obtained since the incident wave has only one wavenumber component, which is equivalent to the trace wavenumber of the incident acoustic wave on the surface of the shell. The external forces and moment in the cylindrical shell equations of motion represent the influence of the ring discontinuity. These loads are a function of the characteristics of the discontinuity, and can be described by the discontinuity impedance functions and the motion of the shell at the location of the discontinuity. That is, for a ring discontinuity at x0, F”,,= &(XO)ZU”

( > F

(7)

where x = ClMm. The incident acoustic wave is impinging on the cylinder from a direction of cp = 180” and at angle 8 to the shell radial direction (Fig. 2). The incident and reflected acoustic waves can be expressed in terms of cylindrical coordinates [6]. The normalized, axial, trace wavenumber of the incident acoustic wave on the surface of the shell is thus koa sin(e) = RM sin(e). The resulting expression, spatial Fourier transformed, and with the nondimensionalized terms introduced is given by, p&q

where S(k, n) is a 3 x 3 matrix and the terms with subscript n denote the nth modal component of that term. The tilde (-) above the displacements, forces and pressure terms in eqn (5) indicates that these are spatial transformed quantities, that is functions of k, the spatial wavenumber. Introducing the following normalizing terms, E= k/k,, and M = c&co, where k. is the acoustic wavenumber and co the acoustic wave speed in the external acoustic medium, the relationships ka = LIME and koa = RM are obtained. The elements of matrix S are then given by,

1 >

n2

R;, = ~“(XO)Z,; R:,,;= IQ”(XO)Z,,,, & = IQ;(xo)Z,, ,

~33 =

[l

+

/92{(52M7i)2+ n’}‘] - Cl2

where Z,,, (i = u, v, w, or B) are the discontinuity impedance functions in the directions u, V, w and the radial bending, respectively. ir.(x,), V”(xO), and &(x0) represent the velocity response of the shell for the circumferential mode n at the discontinuity location x0.

( >

1+v srr = srI = j 2 s13 =

sjl

=

82; =

~32 =

nRME

jvf&fli

n[l + p’{(nMli)’

+ n’}].

(9)

(6)

The radial component of the scattered pressure can be expressed in terms of Hankel functions of the first

3. RESPONSE GREEN’S FUNCTION AND NEARFIELD SCA’ITERED PRESSURE

The solution to eqn (5) for the shell velocity components, with the radial rotation separately

Response Green’s function and acoustic scattering

considered, in the wavenumber

341

domain, for mode order n is given by,

X

where AS, is the value of the determinant of the 3 by 3 [S(7i, n)] matrix, AUare the co-factors of the same matrix

and 3” is the derivative of the radial velocity with respect to the nondimensional axial variable X, this represents the axial bending of the shell. From eq_n(lo), the solution for the nth mode radial response l@“(E) in the axial wavenumber domain is given by,

(1 l).The overall response can be written in terms of two components,

one component associated directly with the excitation pressure, I,,,, and is made up of the inverse Fourier transform of the first term in the second bracketed term of eqn (1 1), and a second component Ii’,&%‘) associated with the forces or moment introduced by the presence of the ring discontinuity. In eqn (12), X is the axial distance, normalized with respect to the shell radius and is defined by X = [(x - xo)/a]. Evaluating first the component directly associated with the excitation pressure,

jR2M

Wdx, = F

-A,&,,(~) +A,&($

s0 m_

yo,

_-oc Pi,, y-

- Au iYrnMX d7;. ( 13) ’ AS. + A,,FLn . (11)

The radial response in the spatial domain is obtained by inverse Fourier transforming eqn

x

Substituting for pin from eqn (8), eqn (13) can be analytically evaluated because of the presence of the delta function. The solution of the inverse Fourier transform is thus given by,

Aap(RM sin(e) AS,@M sin(O)) + A&L%4 sin(Q)FL,(nM

C

cos(@).

M. Cuschieri and D. Feit

342

The contributions to the radial displacement from the forces and moment generated by the ring impedance are obtained from a solution of the equations,

The roots can be classified into four categories: purely real roots; complex roots with a very small imaginary part, which will be referred to as “near real” roots; complex roots with the real and imaginary components of the same order of magnitude; and purely imaginary roots. There are no purely imaginary roots for n equals 0, these only exist for values of n greater than 0, and represent nonpropagating waves below the cut-on frequency. The purely real roots represent quasi-flexural waves which are always subsonic. Since the shell has no structural damping, these subsonic waves propagate along the shell unattenuated. The nearly real, but complex, roots represent torsional and axial compression waves. These waves are supersonic and therefore lose some of their energy to acoustic radiation. This loss of energy gives these roots the small imaginary component. Since the energy lost to radiation is very small, the imaginary part is also very small, making the roots “nearly real”. Even though these roots do not lie on the real axis, because of their proximity to the real axis, these roots have contributions similar to those from the singularities of the purely real roots when evaluating the IDFT. The complex roots occur in sets of complex conjugate pairs which are the negative of each other. That is, there are four of these roots and these are placed symmetrically about both the real and the imaginary axes. When considering these roots in pairs, formed by combining the roots with equal imaginary parts, these represent resonant fast decaying waves. The resonant behavior is created by the inertial loading of the shell due to the presence of the fluid loading. These type of evanescent waves exist near the boundaries and discontinuities of the cylindrical shell.

where FI = -G,(ywJa), F = FP,,(yw,/a), ft = F,,,,(yo,/a), Fd = B,(ro,/a2), and Aj4 = jmMAj1. The loads induced by the ring discontinuity are dependent on the response of the shell at the location of the discontinuity. 3.1. Response Green’s function The inverse Fourier transform of eqn (15) is obtained using a hybrid numerical analytical solution [l]. Before implementing the IFT (inverse Fourier transform), the singularities of the integrand of eqn (15) which are on or close to the real axis, are removed. After removing the singularities, the inverse Fourier transform is performed numerically using an IDFT. The contributions from the singularities are analytically evaluated and combined with the IDFT solution. The singularities are located at the roots of the denominator of the integrand of eqn (15) which is the characteristic equation-wavenumber as a function of frequency-for a fluid-loaded cylindrical shell. The loci of the roots of the denominator of the integrand of eqn (15), as a function of frequency, give the dispersion curves for the fluid-loaded shell. Using a numerical approach [7, 81, the roots of (AS, + AJX,) are evaluated for n equals 0 and 2. The number of roots that have to be determined is based on the order of the characteristic equation and this is compared to the number of roots suggested by the approach of Scott [9]. For a shell in UCICUO, the number of roots is eight [lo]. The fluid-loading introduces two additional roots, hence the total number of roots to be found is 10. These two additional roots represent combined structural acoustic waves. Five of the 10 roots represent positive going waves, while the other five represent negative going waves. The wave representation of these roots is discussed in the following paragraph. The results from the root finding algorithm are shown in Figs 3 and 4 for n equals 0 and 2, respectively. __ [Wn,(x)lm _ jR2M Fill

-xi-

h&l g(E,)(E2- E:)

3.2. Hybrid solution For the n equals 0 case, only four singularities must be removed from the integrand of eqn (15), while for n greater than 0, there are six singularities. Let the denominator in eqn (15) be represented by the function f(E) = AS, + Aj,FL,,, and the numerator, excluding the exponential term, by h(E) = A,,,,. The function h(K) is either an even or odd function of 8, depending on the value of m. For m = 1 and 4, h(x) is an odd function and for m = 2 or 3, h(E) is an even function. The function f(E) is always an even function. Considering then = 0 case, let the real roots of f(E) representing the quasi-flexural waves be given by E = +g, and the near real roots, representing the axial compressional waves be given by E = f x2. When h(x) is an even function, (m = 2 or 3),

s OLI

h(7i2)

@Mx dii+ _-ILIg(7i*)(E2-

eJEnMx dK

7;:)

Wd g(&)(P

h(x2)

1 3

- iq - g(E*)(P - E) ePMY d7i

(16)

Response Green’s function and acoustic scattering 50.0

I

I

343

I

I

25.0 -

f S

! 6

-25.0

-

- (4 -50.0 0.0

50.0 r

8

I

I

I 1.0

I

2.0 3.0 Norme&ed Fmqwney n I

I

I

4.0

5.0

I

1

_---_/--25.0 -7,9

,/

f Z .B 1

0.0 - 12,3,45,5

P

--\ a.110

P

'.

-25.0

-

-1 ------

- @I -50.0

I--

0.0

’

I

I

I

I

1.0

2.0

3.0

4.0

I 5.0

Fig. 3. Dispersion characteristics of a fluid-loaded cylindrical shell for n = 0. (a) Real part of the normalized wavenumber roots; (b) imaginary part of the normalized wavenumber roots.

and when h&) is an odd function, (m = 1 or 4),

-

Eh(7id WE,) E,g(E,>(F - 7i:)- i&g(&)(E2 - E:)1

1

eJxnMx dE’ U7)

J. M. Cuschieri

344 50.0

,

-25.0

-

-50.0

I

I

- 64 ’ 0.0

and D. Feit

I

I

I

I

I

I

I

1.0

2.0

3.0

4.0

5.0

Nonnaked Fmq~~~ey n

50.0

,

I

I

I

I

-

25.0

-

L

.’

_

7,9 --I

t

-50.0

-

(b)

0.0

1

I

I

I

1.0

2.0

3.0

4.0

5.0

NornmlizedFrequencyR Fig. 4. Dispersion normalized

characteristics of a fluid-loaded wavenumber roots; (b) imaginary

where in eqns (16) and (17), (18)

cylindrical shell for n = 2. (a) Real part of the part of the normalized wavenumber roots.

The first two terms in eqns (16) and (17) represent the contributions from the singularities of the integrand of eqn (15), and these are evaluated analytically using a contour integral approach. The

Response Green’s function and acoustic scattering term in eqns (16) or (17), is the original integrand with the singularities removed. This term is a continuous functiomn of 7i and therefore an IDFT can be applied to this last term, to evaluate the inverse spatial Fourier transform. The two sets of results generated from the analytical part of the solution, correspond to the contributions from the real roots and near real, but complex roots. When the integrand is an even function [eqn (16)], the analytic contributions are given by,

With the sign convention used in this paper, the &lnMx terms represents waves propagating in the positive X-direction, while the terms e-jrl.lnwx represent waves propagating in the negative X-direction. For the response ‘Green’s function the appropriate term is selected for positive or negative propagation. At X = 0, the solution consists of the average contribution from both negative and positive propagating components, that is,

when the term on lthe left hand side is evaluated for x= 0. When the integrand is an odd function of x, [eqn (17)], the contribution from the real and near real roots is of the form,

In this case as well, with the sign convention used in this paper, the first term in the squiggly bracket represent waves propagating in the positive Xdirection, while th.e second term represent waves propagating in the negative X-direction. At X = 0, taking the average of the contribution from the positive going and negative going waves, the contribution from these two components to the overall response at X = 0 is zero, that is

with the left hand iride evaluated at X = 0.

345

Using this approach, the analytical contributions would appear to create a discontinuity in the response, near X = 0. However, when combining the analytical contributions with the numerical contribution (result of the IDFT), the numerical contribution compensates for the apparent discontinuity in the analytical part. The cancellation of the discontinuity may not be complete, but this is only a consequence of the resolution of the IDFT. The distortion in the IDFT result very close to X= 0 is due to the truncation in the inverse Fourier transform. If the length of the inverse Fourier transform is made infinite (or very large), then the result would show a step discontinuity which is of opposite sign to the step discontinuity, in the analytical part of the solution. Combining the numerical results (IDFT) with the analytical results gives the overall response Green’s function which is associated with the presence of the ring impedance. For axial (m = I) and radial (m = 3) impedances and for the circumferential mode n equals 0, the radial response Green’s functions for the two normalized frequencies of 0.2 and 2.0 are shown in Fig. 5. For R = 0.2, for excitation by both the axial and radial discontinuities, the response is dominated by the long wavelength membrane waves (Fig. 5a, b). In the case of the radial discontinuity excitation (Fig. 5a), a high response localized at the excitation location can be observed, which is characteristic of a membrane type behavior. For G = 2.0, the response from the radial discontinuity excitation is predominantly subsonic quasiflexural waves. The magnitude of the response settles to a constant amplitude away from the discontinuity location as the quasi-flexural waves propagate unattenuated away from the discontinuity (Fig. SC). The response due to excitation by the axial discontinuity has two predominant wavelengths, one associated with the short wavelength of the quasi-flexural waves and one associated with the long wavelength of the axial longitudinal waves. While the axial longitudinal waves are present for excitation from any type of discontinuity, their amplitude is generally lower than that of the quasi-flexural waves. The axial longitudinal waves are harder to excite, due to their high structural impedance. However, these waves become more pronounced (larger amplitude) when the excitation is from an axial load or discontinuity. For all conditions considered the response near the discontinuity fluctuates in amplitude which is indicative of the interchange of energy between the fluid and the cylindrical shell. Also, for the excitation from the axial ring discontinuity, the response at the discontinuity is zero, due to the asymmetric behavior of the shell under this type of loading. Figure 6 shows the radial response Green’s function for excitation from an axial (m = l), torsional (m = 2) or radial (m = 3) impedances for the circumferential mode n equals 2, and for the two

346

J. M. Cuschieri and D. Feit

Response Green’s function and acoustic scattering

I

347

-3

E -4

t

-0

d

_Q

01

I 0

I 0

,8

348

J. M. Cuschieri and D. Feit

normalized frequencies of 0.2 and 2.0. For R = 0.2, similar behavior to the n equals 0 case is observed, mainly that the radial response is dominated by membrane, long wavelength waves. In this case as well, for the excitation by the radial discontinuity a large amplitude response is observed at the discontinuity, typical of the membrane behavior. Furthermore, this frequency is below the cut-on frequency of the axial longitudinal and tangential shear waves and thus the only contribution to the response away from the discontinuity comes from the membrane waves (Fig. 6aac). For G = 2.0, which is above the cut-on frequency of the axial longitudinal and tangential shear waves, the contribution from these waves is apparent in all the radial responses. Observed in the radial responses are the contributions from the quasi-flexural short wavelength waves and the longer wavelength tangential shear and axial longitudinal waves, which are, respectively, more predominant when the excitation is by a tangential discontinuity (Fig. 6e) and an axial discontinuity (Fig. 6f).

presence of the l/x term in the numerator. This singularity represents slowly decaying (l/J[distance from ring load]) waves which can lead to problems in numerically evaluating the inverse Fourier transform of eqn (23). Evaluating the limit values of the terms in eqn (24) which include x, that is as K-+1 or x-+0,

(25)

therefore, when E = 1, the term in the square bracket in eqn (24) reduces to

1%,,(7i= LR,Q)=

A,m+$(f)( -A) AS, + A&$)($(

-;)

= Am

(26) 3.3. Scattered pressure From eqn (7), the mode n component of the scattered pressure, at a normal distance R from the surface of the shell, due to a radial, axial, circumferential or moment ring load (respectively m = 1, 2, 3, or 4) excluding the specular component, is given by

with Ajm, AS, and Aa evaluated at 3; = 1. For the of n = 0, A, reduces to (Ajm/A,,), A,.4 and evaluated at 7i = 1. Using the same approach as used for the fluid-loaded plate [ 11,the expression the scattered pressure in the wavenumber domain be written as,

case A13 was for can

(23) Rearranging,

The exponential term dX(RmI) divided by the 3 term in the above equation is introduced instead e+-I~/fi is the of the H&R)/H,(x) term. asymptotic value of H&R)/H&) for large argument. That is. x $$-$ n

(24)

(28) The scattered pressure in the spatial domain, as a function of X and R, is obtained by inverse Fourier transforming eqn (24). In evaluating the IFT, apart from having to deal with the singularities as in the evaluation of the response Green’s function, a new singularity has been introduced at x = 1 due to the

The reason for this substitution is to facilitate the evaluation of the inverse spatial Fourier transform to obtain the scattered pressure in the axial spatial (x) domain. Thus, the component of the scattered

Response Green’s function and acoustic scattering

0

1

2

3

4

5

0

1

Normalized AxkJ Distanm

0

i

349

2

3

N-

AxiaImce

NC-~

Axial 3e

4

i

Fig. 7. Near field scattered pressure for n equals 0. Excitation by (a) axial discontinuity, fi = 0.2; (b) radial discontinuity, R = 0.2; (c) axial discontinuity, R = 2.0; (d) radial discontinuity, R = 2.0.

5

350

J. M. Cuschieri

-=ma

Plp=ll PWI-‘ON

and D. Feit

Response Green’s function and acoustic scattering pressure from an axial, tangential, radial or bending moment ring load (m = 1, 2, 3, or 4), is given by PI,,W,

R, fi)

00

AS. + A&Y ff h

e” -H.(X) e xHXX)

(29)

where, as before, it = -F”,,(ywJa), F1 = F?“(yo,/a), fi = f’,,,,(yo,/a), I? .= &(yo,/a2), and Aj4 = jib2MA33. The first integral in eqn (29) is fast decaying and can be evaluated numerically, after taking care of the real and near real roots of the denominator of the first term. The last integral represents the slowly decaying waves which would create “aliasing” if evaluated numerically over some truncated range of the wavenumber [ll]. However, this term can be evaluated analytically,

x Hj”[QM,/X

+

(R - l)*],

(30)

where fl’)[{] is a Hankel function of the first kind of order 1. In implementing a solution, the exponential term e’~(~-‘)in eqn (29), subtracted from the portion of the equation which is inverse Fourier transformed numerically, imposes some requirements on the resolution or size of the FFT that needs to be considered. As the distance from the surface of the shell (R) increases, and for E between 0 and 1, where x is real, the function e’xcR-I) is a rapidly oscillating function of x, and the resolution of the wavenumber spectrum must be selected such that the rapid oscillations are properly resolved. Increasing

351

the resolution while maintaining the same range of the wavenumber spectrum requires a larger number of data points, resulting in a large size FFT in the inverse spatial Fourier transform. Furthermore, the range of 7i controls the resolution in the spatial domain. For a normalized frequency (a) of 5.0 and a maximum normal range of 5.0 from the surface of the shell, the size of the FFT which had to be used in this analysis is 32 K samples giving a maximum value for x included in the inverse transform of approximately 75. Figure 7 shows the nearfield scattered pressure for circumferential mode n equals 0 for the two normalized frequencies R of 0.2 and 2.0. For 0 = 0.2, the nearfield scattered pressure is dominated by the direct scattering from the discontinuity for both excitation by the radial and axial discontinuities. The membrane waves are subsonic and therefore do not radiate acoustically. For R = 2.0, the nearfield scattering from the radial discontinuity excitation is mainly due to the discontinuity, with dipole like radiation characteristics. The pattern for the nearfield scattering due to the influence of axial discontinuity is, however, different. In this case, the axial longitudinal waves, which are supersonic, are more dominant, and thus the predominant scattering is along the coincidence angle of these waves. Very close to the shell surface, one can still observe the interaction that occurs between the shell and the fluid due to the presence of the subsonic quasi-flexural waves. The nearfield scattered pressure results for n equals 2 are shown in Fig. 8. In this case, three excitation conditions may exist: excitation by an axial ring discontinuity; excitation by a tangential ring discontinuity; and excitation by a radial ring discontinuity. The results for C?= 0.2 show that the predominant behavior of the shell comes from the membrane waves and the scattering mainly comes from the discontinuity. This frequency is below the cut-on frequency of the axial longitudinal and tangential shear waves, and the subsonic membrane waves are the only waves that can propagate. A feature in the results is the null in the scattering, which is similar to that obtained in the case of a fluid-loaded plate below the critical frequency [l]. R = 2.0 is above the cut-on frequency of the axial longitudinal and tangential shear waves, and the scattering clearly exhibits the contribution from these waves, especially for the excitation by the axial or tangential discontinuities. Both of these waves are supersonic, hence the scattering from these waves occurs along their coincidence angles. The scattering when the shell is excited by the radial discontinuity has contributions from both these waves and that directly from the discontinuity. The generated scattering patterns are more complex than for any of that other discontinuities. The influence of the subsonic quasi-flexural waves can be observed very close to the shell surface.

352

J. M. Cuschieri and D. Feit 4. SHELL AND BULKHEAD INTERACTION

In the first part of this paper, while a solution for an arbitrary impedance discontinuity was considered, the result for the nearfield scattered pressure and the response Green’s function are obtained for normalized forces created by the discontinuity. That is, the interaction between the fluid-loaded shell and the discontinuity is not included in the formulation. In the following sections, a solution for the farfield scattered pressure is presented which includes the interaction between a fluid-loaded cylindrical shell and the internal impedance discontinuity or discontinuities, which in this case consist of one or two thick plate bulkheads. This section starts with a development of the mobility approach that is used to represent the interaction between the internal plate bulkheads and the shell. 4.1. Single plate bulkhead The junction model of the cylindrical shell and the single internal plate bulkhead is shown in Fig. 1.

= I MI,

-t

MM

+

0

0

A’33

0

A433

+

NII

0

N43 M23

+

N12

The plate bulkhead creating a ring discontinuity on the shell is located at the axial coordinate X = 0.0. The internal plate bulkhead is assumed to have the same thickness to radius ratio, and the

same material characteristics as the shell. The forces and moments acting at the junction are as shown in Fig. 9. Not shown in this figure are the tangential motion and forces. These have been omitted from the figure for the sake of clarity. The forces as shown in Fig. 9 would represent the complete set of loading and motion for the first circumferential mode, the n equals 0 or the breathing mode. In this case, the tangential motion in both the shell and the plate would be zero and the solution is independent of the circumferential variable 4 (Fig. 2). For higher order circumferential modes, n greater than 0, the tangential force and motion are not zero. From consistency of forces or moments and response at the junction between the internal plate and the fluid-loaded shell, a matrix equation for the junction forces can be derived. This matrix equation consists of mobility functions of the fluid-loaded shell and the plate. For any general value of the circumferential mode order n, the matrix equation is given by,

MM + NM 0 MufNu 0

0 M23

+

N12

(31)

0 M22

+

N22

where the same convention as before is used, that is 1, 2, 3 and 4, respectively, represent the axial direction, the tangential direction, the radial direction and the radial bending.

Shell

shell axial direction x

II Plate Fig. 9. Coupling between fluid-loaded shell and internal plate bulkhead.

Response Green’s function and acoustic scattering Equation (30) is not written in the order of this notation. The tangential contributions are placed in the last column and row, since for n equals 0, these components would not be included in the solution. All tangential mobility functions are zero for n equals 0. The M terms in eqn (30) represent mobility functions of the fluid-loaded shell, while the N terms represent mobility functions of the internal plate. M,i and N, represent input mobility functions where the response and the load are applied at the same location and in the same direction, and M, and NV, with i #j represent transfer mobility functions where the load and the response are in different directions. Since the system is symmetric, i and j can be interchanged. The same convention for representing the directions is used for the shell and the plate. Thus, as an example, the axial motion of the shell (1) would couple with the transverse motion of the plate (3). Fun, F,,“, B, and F,.” represent, respectively, the junction ring loads in the axial, radial, bending and tangential directions. The superscript fin the terms on the left hand side vector

indicates that these terms represent the shell free velocity response at the junction location when the shell is decoupled from the plate bulkhead. 4.2. Double plate bulkhead The model of the forces and moments at the junctions for the double plate bulkhead case is shown in Fig. 10. The bulkheads are located at the normalized axial locations X = + 1.O. In this case as well, omitted from this figure are the tangential components of the forces and the velocities. These tangential forces and velocities are however included in the MPF formulation. At each of the junctions there is an associated set of forces and velocities corresponding to the axial, tangential radial and radial bending directions. The junction dependency for the loads and the velocities is, respectively, indicated by the superscripts and subscripts 1 and 2. Using the continuity condition, a mobility matrix equation is obtained that relates the six junction forces and two moments, to the free shell velocities at the locations of the junctions, that is,

M;: + NM 0 Mi+N, M:: Ml’34 44: 0 MZ

Ml

MI: Ml:

M:: + Nn

M::

Ml: M:

0

M2’12

0

M:: + N II

M2’ 32

M:: + N12

0

442’ 42

0

Ml:

M:: + NE

X

M::

+

N21

353

M:

M;; + N22

J. M. Cuschieri and D. Feit

shell axial direction x

Fig. IO. Coupling between fluid-loaded shell and internal plate bulkheads.

where M; represents mobility functions of the shell and N, represent the mobility functions of the internal plates. The subscripts 0 would indicate motion in direction i due to load in directionj, where i, j = 1, 2, 3 or 4, corresponding, respectively, to the axial, tangential, radial and radial bending directions. The superscripts rs indicate position. Hence Mr would represent motion in direction i at position r due to load in directionj at position s. When r equals S, and i equals j, the mobility function is an input mobility, with the motion and force applied in the same direction and at the same location. The transfer mobility functions can have various forms. w; would be a transfer mobility with different directions for the velocity and force, but applied at the same location. These type of mobility functions are also used for the single bulkhead case. w; is a transfer mobility with the velocity and the load in the same direction, but defined at different locations, and My would represent a transfer mobility with the velocity and the load in different directions and for different locations. The last two types of transfer mobility functions are introduced because of the separation between the internal bulkheads. FL,?,F,‘,,, &:;, B,!, e,,, e,,, Ft. and Bz are the junction ring loads at locations 1 and 2, respectively (Fig. 10). rit;,, I?,, @,,, a@‘L,Px, rifl,, I?*, I@‘?!& and a%@~, are the shell free velocity response, also at locations 1 and 2, when the shell is subjected to the incident acoustic wave. The junction loads are evaluated by inverting the mobility matrix and multiplying the matrix inverse by the free response vector. In eqn (31) the last two rows and columns are for the tangential forces contributions. For n equals 0, there are no tangential force components, since the circumferential mode is the symmetric breathing mode. To solve for this mode, the last two rows and two columns, together with the corresponding c and F,.,,terms are removed from the matrix equation. In solving these matrix equations for the junction loads, the mobility functions are required. The

evaluation of the mobility functions for the plate and the fluid-loaded shell is discussed in the next section.

5. MOBILITY 5.1.

FUNCTIONS

Internal plate bulkhead

The mobility functions for the thick plate can be obtained from a solution of the Mindlin thick plate equations [3] written in cylindrical coordinates. Using the Mindlin equations, the in-plane and out-of-plane motions of the plate are uncoupled. For the out-of-plane motion, the mobility functions are obtained from a solution of the following matrix equation: r

(33)

when an out-of-plane (transverse) force, F,,;(yo,/a), or bending moment, B.(yw,/a2), is applied along the plate edge boundary. In eqn (33), y has the same definition as for the shell and a and h are, respectively, the radius and thickness of the plate. The other terms are as defined for the shell. The ring frequency has been introduced to make these equations compatible with the equations for the fluid-loaded shell. The matrix in eqn (33) represents the solution of the equations of motions when expressed in terms of potential functions [3]. The potential functions are used to decouple the plate velocity components. d,, 9& and V. are the coefficients of the solution of the equations of motion for the potential functions. These coefficients are evaluated from knowledge of the boundary conditions.

Response Green’s function and acoustic scattering

In this equation., the twisting moment along the edge of the-plate that could be induced by the shell is assumed to always be negligible. In eqn (33) the PO terms are defined as follows: PI, = (a, -

1)(&C(&)

+ vS,h(b,)

P,2 = (62 -

l){s;J;(s,)

+ v&J;(&) -

355

Repeating the procedure for an applied edge moment, FWfl(yo,/a) = 0, and &(yw,/a*) = 1, and solving for the coefficients and inserting into the equations for the plate velocity, the following mobility functions are obtained,

- V??J,(&)} vn2J,(62)}

PI, = n(l - v){&J;(83) - J”(h)} P2, = ad,J;(&) p22 = a2625.‘(62)

(34) = [(1 ~~v)&AJn(W

p23 = n.Jn(&>

+ %Jn(&)j

(36)

PM = 2(u, - l)n{J,(&) - S,J;(&)} p32 = 2(‘J2 -

l)n{Jn(&)

-

S,J;(S,)}

NA~=%=,(~ p33 = { -n2Jn(&)

~~&][dd,(a,-

1)J.(6,)

+ S,J;(&) - S;Ji’(&)}, ” a2

where J,(h) are Bessel function of order n and a prime indicates differentiation with respect to the argument. Additionally,

K = n712, ai.2 = (&%{a

- /)’

+ 4/6,4]“2},

[S&6,4 - l/f], 6:= -(12 v)

A unique set of coefficients, d,, 9. and W”exists for each applied load. That is, to obtain the mobility functions associated with a transverse edge force, eqn (33) is solved for F,,(yo,/a) = 1, and B,,(yw,/a2) = 0. The values of the coefficients d,, 1, and V, thus obtained are inserted into the equations that relate the potential functions to the velocity components of the plate to obtain the mobility functions. The mobility for the transverse edge velocity per unit applied edge transverse load is given by

Nn =

2jn (1 +h)K(l

-

v)

1

{dnJn(&) + .%Jn(&)}.

(37)

For the in-plane mobility functions of the plate, the same procedure can be used. In this case, to keep the complexity of the solution to a minimum, only the first order in-plane deformation is retained. This is sometimes referred to as the thin plate approximation to in-plane motion [12]. Using the potential function approach to decouple the in-plane velocity components of the plate, the solution to the equations of motion of the two potential functions, expressed in matrix form is given by,

21,” = Q’/{P’(l +jq)j -t- % AZ [(a

+ %~,(a2 - l)JL(82) + nWnJn(&)].

where LZZ’; and SK,are the coefficients of the potential functions solutions, which are unique for a given set of boundary conditions. In eqn (38) the P,; terms are defined by ph = e:p(e,)

+ ve,J,(e,)

P;~ = (1 - v)n{e2r(e2)

- n2vJ,(&)

- J.(e,))

p;, = 2n{~.(e,) - e,r(e,)) pi2 = -n2Jn(e2)

(35)

where 6: = a’/(1 v)}.

+ e,r(e,)

+

- e2J;(e2)

(39)

jq) and (3: = 2n2/{(l + jq)(l -

J. M. Cuschieri and D. Feit

356

-9 In

-9

*

t----

-. - _-_- - - - _----\

1-___.+_ t

--====+ ___{:-___

-9

\ _ -=-

-.

mC

E

_

-0;

4

-9

>--t-

---_

2---

-\ zl_ 2-e

\

_---

-\

1

\

_---

L=_--

_

_

-

I

I-4

_

_-L-

-\

_-‘_-.

Response Green’s function and acoustic scattering With an applied edge radial in-plane load, F,(yw,/a) = 1 and F,.(rwJa) = 0, the mobility function for the in-plane radial motion per unit applied edge radial load is given by,

ri,

NII =

=

& [

1

{Xz;e,J,‘(e,) + O;nJ”(e2)}.

(40)

Similarly, to evaluate the transfer mobility functions and the tangential mobility, setting Fu~(:,(ror/a) = 0 and J’“,(ro,/a) = 1, the following set of mobility functions are obtained:

351

obtained using the hybrid numerical/analytical approach. In this case, the normalized separation distance is X = 2.0, and the equations for the velocity response are solved for this normalized distance. When evaluating the shell transfer mobility functions, m;, the combined separation distance can be considered provided that the proper sign to indicate a phase lead or lag is included when combining the functions in eqn (32). Therefore, for the fluid-loaded shell, apart from the mobility functions shown in Figs 13 and 14 for, respectively, n equals 0 and 2, there are additional mobility functions. The additional mobility functions are shown in Figs 15 and 16, respectively, for n equals 0 and 2. The most significant difference between the mobility functions shown in Figs 15 and 16, and the mobility functions shown in Figs 13 and 14 is the oscillating nature of these functions. These oscillating characteristics are due to the phase shift introduced by the separation distance between the two locations on the shell. As the two locations move farther apart, the oscillations get closer together. 6. FREE SHELL VELOCITY

=

& [

N22

=

1

{-d;n~,(e,)

r’, -

- a;e,J;(e,)J

2jQ (1

+h)U

x { -d:nJ,(er)

-

v)

(41)

1

- 99:e,J;(e,)).

The remaining input required in eqns (31) and (32) to solve for the junction forces, is the free (uncoupled) velocity of the shell at the locations of the junctions with the internal plate bulkheads. The free velocity is the response of the uncoupled shell when subjected to the external oblique incidence acoustic excitation. From eqn (14), the free velocity components of the shell, per unit incident pressure Pi, obliquely incident on the shell at an angle of incidence 0, are given by,

(42)

The mobility functions expressed by eqns (35x37) and eqns (40)-(42) for n equals 0 and 2 are shown in Figs 11 and 12, respectively. For the out-of-plane components a number of resonance frequencies can be observed, while for the in-plane components very few resonances (one or two) are within the frequency range of interest. Both the in-plane and out-of-plane resonance frequencieis play a significant role in the far field scattered pressure. 5.2. Shell mobility fkctions The fluid-loaded shell mobility functions are derived using equations of the form of eqn (15). The nonzero mobility functions required for the solution of the single bulkhead problem for the circumferential mode orders n equals 0 and 2 are shown in Figs 13 and 14. In this case, there are no apparent resonance frequencies, since the shell is assumed to be infinite. Also, these mobility functions are independent of the location on the shell, as long as the velocity and the force are at the same location. The shell transfer mobility functions, mainly required for the double bulkhead case, with the load and the velocity at different locations, are also CAS 65/3-D

@xx> _1

Q

@MX

sin(*,*lM

x2 !&I4 cos(e)H.‘(nM cos(8))

p, yw, ( U >

J. M. Cuschieri and D. Feit

358

1o:

_!__

. . . . . . . . .[

,J_

___._. .,

:;

.’

22

t

-

Real

...---.

lmag I

-

(4 Real lmag

i

(4

-

Real ... .. lmag

33 .1003.0~ 0.0

’ 1.0

’ ’ 2.0 3.0 Frequencyn

’

’ 4.0

’

I

1.o

,

I

I

I

3.0 2.0 FrequencyR

I

I

4.0

Fig. 12. Plate mobility functions for n = 2. (a) In-plane radial input mobility; (b) in-plane tangential input mobility; (c) transfer mobility between in-plane radial and tangential; (d) transfer mobility between transverse and rotational; (e) input mobility for the transverse direction; (f) inut mobility for the rotational direction.

a*(x) ax

-

p,

=

i 7 RM

(> yu, a

cJn

@W*sinp3)**~

cos(B)H,’ (cm cos(B))

Each of the terms in the square bracket on the right hand side of the above equations are evaluated for values of the normalized wavenumber CM4E of RM sin(e). For the single bulkhead case, X = 0, while for the two bulkheads, since these are located at plus or minus one radius from the origin of the coordinate axis, the free response is calculated for X = f 1.0. The reason why the free shell response needs to be

Response Green’s function and acoustic scattering

_P

_

-9

_

Q

OC

L-

_ok

4

e

_

359

J. M. Cuschieri and D. Feit

360

1 i\\

--__db)j _-------_-_-

TX___/

__-______---0.0 -

I

\

11 ,.-~---~-:--~_~_l;;-l

\ _

.s.o 0.0

\

! 1.0

., , 2.0

33 I 3.0

Frequencyfl

/

I, 4.0

5.0

0.0

1.0

2.0

3.0

4.0

Frequencyfl

Fig. 14. Shell mobility functions for n = 2. (a) Axial input mobility; (b) axial to rotational transfer mobility; (c) input tangential mobility; (d) tangential and radial transfer mobility; (e) radial input mobility; (f) radial rotational input mobility.

5.0

361

Response Green’s function and acoustic scattering

I

--

..

lmag

” ‘I

44 ““0.0 F-Y

n

t

*

l

1.0

*

I

#

2.0 Fmqtmcy

I

3.0 n

I

I

I

4.0

Fig. 15. Shell I:ransfer mobility functions between the two bulkhead locations for n = 0. The numbers in each graph indicate the directions of the velocity and the load.

1

5.0

J. M. Cuschieri

and D. Feit

4om.o

moo.0

0.0

~20411)

I

I

.o.o

’ ,

3

,

’

’

’

tooa,

.fmo

,

0..

020,

,

0.2

0.10

0.0

0.00

’

’ ,

’

,

,

’

’

’

.

,

,

,

,

,

’ ,

’

’

*

,

,

,

(

(

Q.,O

9.2

-0.4 0.0

(

’ ,

,

1.0

Fig. 16. Shell transfer

20 F-n

3.0

mobility

functions

4.0

5.0

1.0

0.0

between the two bulkheads

20 F-n

3.0

for n = 2. The numbers

4.0

5.0

in each graph

indicate the directions of the force and the velocity.

evaluated at X = + 1.0 is to include the relative phase difference (lag or lead in time) in the response of the shell at the two bulkhead locations with respect to the selected origin on the shell axis.

7. FAR-FIELD SCATTERED PRESSURE

The scattered pressure components associated with the bulkhead(s) forces as a function of the axial

Response Green’s function and acoustic scattering

wavenumber

363

are given by eqn (23). For the single bulkhead case,

X

-ASIF,

+ AUF,, + A,,Fttn - jEQMAS,(B,/a) AS,

+

A,,FL.

1

&znMxdK

(47)

where Fu,l, F,,,,,,F,.”and (B./a) are the junction ring loads per unit incident pressure. Instead of using the procedure of Section 3.3 to evaluate the inverse Fourier transform, since here the far field scattered pressure is of interest, a stationary phase approach [S] is used. That is

X

-AnFun + A,*E.”+ ADF,,.~- jmMA,,(B,/a) AS, + A,,FL,

(1 1L=Pm (48)

where Z = ,/G? is the normalized spherical distance from the location of the bulkhead, measured from the center of the shell, 0 is the angle of incidence of the excitation acoustic wave and $ is the scatter angle measured from the normal to the shell. For the double bulkhead, using the same approach,

-ASIF:_ + A,&!,, + A,&, - jmMA,,(Bi/a) AS, + A,,FL.

&Oh4

+Z*SM4

- A,&m + A,&

+ A,,E$ - jiXlMA,,(B,2/a) AS, + A,,FL,

1.G =31” I,

$* 1r-sin (49)

364

J. M. Cuschieri and D. Feit

where in this case, Z, the radial distance from the centre of the shell is measured from a point midway between the two bulkheads (Fig. 17) and

Z2=JZ2+X&2XoZsin*, $I = cos-‘[(Z cos $)/Z,], $2

=

cos-‘[(Z

cos

II/)/Z,],

and X0 is the distance of the bulkheads from the center of the shell, X0 = 1.0. Z and $ do not explicitly exist in all of the terms in eqn (49). Therefore, in the two bulkhead case, it is not possible to present the results normalized with respect to the radial distance from the shell center (Z). The far-field scattered pressure results are thus presented for a radial distance from the center of the shell of five diameters. A number of different distances were explored and the basic features of the far-field scattered pressure did not significantly change. The above expressions for the farfield scattered pressure do not include the component of the scattering associated with the elastic response of the shell without the presence of the bulkhead(s). This component is generally considered of lesser interest. Using the above expressions, bistatic and monostatic farfield scattered pressure results are generated. For the bistatic scattering, two angles of incidence are considered, 15” and 45” from the normal to the surface of the shell. The 15” angle of incidence is inside the coincidence angle of both the axial longitudinal waves and the tangential shear waves (for n greater than 0), while the 45” angle of incidence is outside the coincidence angle of both of these waves. The components of the scattered pressure associated with the different forces and moments at the bulkheads are separately presented, together with the total scattered pressure. By presenting the components of the scattered pressure separately, the character of the interaction between the bulkheads

I

I

I

location ofbdkhcod center ofsheN location

of bulkhead

Fig. 17. Schematic representation of the angular and radial dependencies of the scattered pressure contributions from the two bulkheads.

and the shell and between the two bulkheads on a component by component basis can be visualized. Furthermore, with this mode of presentation it is much easier to observe which of the junction load components are the dominating contributors to the far field scattered pressure. The results are presented in the form of contour plots with frequency along one axis and scattering angle along the other axis. The contour plots are of the ratio of the magnitude of the scattered pressure, multiplied by the normalized distance Z in the single bulkhead case and for a normalized distance of five shell diameters for the double bulkhead case, to the incident pressure. 7.1. Farjeld scattering from single bulkhead For the single bulkhead case, the bistatic scattering results for the n equals 0 are shown in Figs 18 and 19 and for n equals 2 are shown in Figs 21-23. The results for the monostatic scattering, n equals 0 and 2, are shown in Figs 20 and 24, respectively. For n equals zero, the tangential shear waves are not excited, and in-plane longitudinal waves exist for all frequencies. From the results in Fig. 18, when the incident acoustic wave is within the critical angle of the longitudinal in-plane waves (incidence angle less than approximately 16”), the component associated with the axial longitudinal waves dominates, and the contribution to the far field scattered pressure comes mainly from this component. As the angle of incidence, as measured from the normal increases, and becomes greater than the critical angle of the axial longitudinal waves, the contribution from the axial waves decreases (Figs 18 and 19) and the far field scattered pressure has approximately equal contributions from the axial waves and the dipole created by the radial component of the junction load due to the presence of the discontinuity. The junction moment is the least significant contributor for both angles of incidence. Superimposed on the general features of the results are the influences of the plate bulkhead out-of-plane and in-plane resonances. The distinct null in the scattered pressure, apparent in both the radial component and the total, around a normalized frequency of 2.0, is a result of the first in-plane resonance (Fig. 11~). At this frequency, the plate is not imposing any radial restriction to the shell motion, and thus, the radial junction force is very small, almost negligible. Since the scattering from the quasi-flexural waves comes from the presence of the if the radial constraint is radial discontinuity, negligible, then the scattering from the radial force at this frequency is also negligible. The components associated with the axial force and the radial bending moment show the influence of the out-of-plane resonances of the internal plate. However, in this case the dependency of the scattering on the resonances of the internal bulkhead is not as simple as in the case of the radial resonance. This is because there is

Response

Green’s

function

and acoustic

scattering

365

Bistatic, Inc. Angle = 15 om7d

(b)

0.15

a 0

-88.5

88.5

88.5

0

-88.5

Scattering Angle

Scat teriqq Angle From Normal

Fmm

Normal

Bistatic, Inc. Angle = 15

Bistatic, Inc. Angle = 15

Cd)

a 0

-88.5 Scatlerlng

Angle From Normal

88.5

-88.5

0 Scattering

Angle From Normal

Fig. 18. Bistatic scattered pressure for an incidence angle of 15” for circumferential mode n = 0. (a) Axial; (b) radial; (c) moment components; and (d) total scattering.

88.5

J. M. Cuschieri and D. Feit

366

0 0

-38.5

-86.5

as.5

Frm Nomd

Scattering A+

0

88.5

FromNormal

ScatterimgAngie

Bistatic~c.~gle

0

= 45

0 0

Scattering Angle

From Normal

0

-88.5 Seatterhg An&

88.5 From Normal

Fig. 19. Bistatic scattered pressure for an incidence angle of 45” for circumferential mode n = 0. (a) Axial; (b) radial; (c) moment components; and (d) total scattering.

Response

Monostatic Scattering

Green’s

function

and acoustic

scattering

367

(aj

Axial cmmmmt

2

2

0.1:

0.15 48.5

88.5

0

-88.5

0

88.5

Angle From Normal

Scattering Angle From Normal

Monostatic Scattering

Monostatic Scattering

Scattering

3 7.5 LZ 16.5 21 25.5 30 34 5 39 43.5 48 52.5 57 61.5 Hi 70.5

a

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Scattering Angle From Normal

Fig. 20. Monostatic

scattered

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Scattering Angle From Normal

pressure for circumferential mode N = 0. (a) Axial; (b) radial; components; and (d) total scattering.

(c) moment

J. M. Cuschieri

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Monostatic Scattering

and D. Feit

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d Scattering

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Fig. 27. Monostatic

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scattered pressure (c) moment

0

Scattering Angle From Normal

Angle From Normal components components;

for circumferential mode n = 0. (a) Axial; (b) radial; and (d) total scattering.

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Response

Green’s

function

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0

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Fig. 21. Bista.tic scattered pressure for an incidence angle of 15” for circumferential mode n = 2. (a) Axial; (b) radial; (c) moment components; and (d) total scattering.

88.5

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J. M. Cuschieri

and D. Feit

Bistatic, Inc. Angle = 45

Bistatic, Inc. Angle = 45

0

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Scattering

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Fig. 22. Bistatic scattered

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Angle From Normal

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pressure for an incidence angle of 45” for circumferential (b) radial; (c) moment; and (d) tangential components.

Angle From

Normal

mode n = 2. (a) Axial;

Response

Green’s

function

and acoustic

Bistatic, Inc. Angle = 15

scattering

(a)

0 -88.5 Scattering

i Angle

Scattering

Angle From Normal

88.5

From Normal

0

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88.5

Fig. 23. Total bistatic scattered pressure for incidence angles of 15” (a) and 45” (b) for circumferential mode n = 2.

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J. M. Cuschieri and D. Feit

Monostatic Scattering

a -88.5

0

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,. .

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FromNormal

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Fig. 24. Monostatic scattered pressure for circumferential mode n = 2. (a) Axial; (b) radial; (c) moment; (d) tangential components; and (e) total scattering. (Continued opposite).

88.5

Response Green’s function and acoustic scattering

Monostatic Scattering AllCO-am

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48.5 Scattering

Angle

88.5 Fmm

Fig. 2kContinued.

Normal

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J. M.

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interaction between the axial and the quasi-flexural wave components. The influence of this wave interaction can be better observed in the monostatic results. The results for monostatic scattering, for n equal 0 are shown in Fig. 20, and these exhibit the same type of behavior as observed for the bistatic scattering. That is, in the region of the critical angle of the axial longitudinal waves, the contribution from these waves to the far field scattered pressure is dominant. Outside of this angle, the contribution from the axial longitudinal waves becomes of the same order as that from the radial load. A feature that can be observed in the monostatic result, is the shift in frequency of the nulls and peaks in the scattered pressure associated with the out-of-plane resonances of the internal plate bulkhead, as the angle of incidence approaches the critical angle of the axial longitudinal waves, The axial motion of the shell, in general, does not couple well with the internal plate, because of the mobility mismatch between the shell axial in-plane motion and plate out-of-plane motion. Thus, the behavior of the internal plate is in general similar to that of an edge axially pinned plate. However, as the incidence angle approaches coincidence with the axial waves, these are more readily excited. The shell now has a significant motion in the axial direction, which couples with the out-of-plane motion of the internal plate. The edges of the plate now approach free boundary conditions, and thus the resonant frequencies of the plate are shifted to higher frequency values. This is the cause for the curving of the nulls and peaks in the scattered pressure components, associated with axial load and the bending moment as observed in Fig. 20. The peaks in the scattered pressure when the angle of incidence is close to the coincidence angle of the axial longitudinal waves, approximately matches the resonant frequencies of the internal plate with free boundary conditions. As the circumferential mode order increases, II equals 2, two distinct phenomena occur. First, the shell can now support tangential shear waves which are also supersonic, and secondly, both the axial longitudinal and the tangential shear waves have a cut on frequency. Figures 21-23 represent the bistatic scattering results for angles of incidence of 15” and 45”. The first observation that can be made is the “wine glass” shape in the scattering patterns, very similar to that observed in Ref. [7]. Because the axial longitudinal waves and the tangential shear waves have a cut-on frequency, the influence of these waves on the scattered pressure cuts on at this frequency. Below the cut-on frequency these waves do not propagate along the shell and therefore do not contribute to the far field scattered pressure. Below the cut-on frequency the scattering is controlled by the radial component of the junction force. For an angle of incidence below the critical angle of both the longitudinal axial waves and the tangential shear

waves, the contributions from these two waves dominate the far field scattered pressure. Which of the two is most significant depends on the angle of incidence. For angles of incidence larger than the coincidence angles of the axial longitudinal waves and the tangential shear waves, the significance of the contribution from the radial force increases and becomes of the same order as the contribution from the longitudinal and shear waves. The contribution from the junction moment is in all cases insignificant. A good comparison for the dependencies on the angle of incidence can be made from Fig. 23 which shows the total scattered pressure for the two angles for n equals 2. The monostatic results for n equals 2 (Fig. 24) show a similar dependency on the angle of incidence as the bistatic results. Peak scattering is obtained when the incident angle matches the coincidence angle of the axial longitudinal or tangential shear waves. Similar to the n equals zero case, the contribution to the scattered pressure associated with the radial force component exhibits a null at the in-plane resonant frequency of the internal plate structure, although there is a shift in this null close to the critical angle of the tangential shear waves. This shift is associated with the interaction between the axial longitudinal waves and the tangential shear waves. The null associated with the tangential resonant frequency is not as distinct, and it could hardly be observed in the result for the total scattered pressure. Nulls and peaks are also obtained in the axial and bending moment components associated with the out-of-plane resonances of the internal plate structure. The nulls in the axial and bending moment components of the scattered pressure do not show the shift in frequency with angle of incidence as was observed for the n equals 0 case. The null in the radial component still shows a shift with frequency as the angle of incidence approaches the coincidence angle of the tangential shear waves. A possible reason for this difference is the change in the coupling between the various in-plane and out-of-plane waves. For n equals 0, the in-plane longitudinal waves of the internal plate couple with the radial out-of-plane waves of the shell, while the out-of-plane waves of the internal plate couple with the axial waves of the shell. The in-plane and out-of-plane waves are uncoupled in the plate model (Mindlin), and only weakly coupled in the shell model by the curvature. Thus, there is little interaction between the waves as can be observed in Fig. 20. For n greater than 0, apart from the coupling as described for the n equals 0 case, there is coupling between the in-plane tangential shear waves in the internal plate and the tangential shear waves in the shell. Since these waves strongly couple with the other in-plane waves of the internal plate and the shell, strong interaction exists between the various waves. This strong interaction would tend to compensate for the nulls in the scattered pressure when a particular wave is highly excited.

Response Green’s function and acoustic scattering A comment which applies to all the results is the type of source that can be associated with each of the scattered pressure components. The radial bending moment component has quadrupole like characteristics, while the radial force component has dipole like characteristics. This is one reason why the scattered pressure from the radial bending moment is lower compared to the scattered pressure from the radial force. Both components have a relatively broad directivity pattern. The scattered pressure components associated with the axial force component which induces axial1 longitudinal waves in the shell, and the tangential force component, responsible for the tangential shea:r waves, have a very narrow beam pattern, typical of radiation from supersonic waves. The scattering from these two force components is mainly directed along the coincidence angle of the two respective waves. Along the coincidence angle the level of the sca.ttered pressure can be very high. The results for the far field scattering from the single bulkhead thlzrefore show that for n equals 0, the far field scattered pressure is dominated by the axial longitudinal waves, especially for incidence angles less than approximately 16”. The axial longitudinal waves are supersonic and couple well with the external medium. For angles of incidence greater than 16”, the far field scattered pressure has similar contributions from the radial discontinuity and the axial longitudinal waves. For higher order circumferential mo’des, n greater than 0, the far field scattered pressure is dominated by the axial longitudinal and tangential shear waves, above the cut-on frequency 01’these waves, especially for angles of incidence less than 30”. For greater angles of incidence, the cant ribution from the radial discontinuity increases and is comparable to that from the axial longitudinal a.nd tangential shear waves. These results match those obtained by Rummerman [13] and Guo [14], who also showed that the axial longitudinal and tangential shear waves play a significant role in the far field scattered pressure for high order circumferential modes. 7.2. Far field scattering from two bulkheads The results for the bistatic far field scattering for an angle of incidence of 15” for n equals 0 are shown in Fig. 25. Comparing these results with those for the single bulkhead (Fig. 18), the same general characteristics can be observed, except that in this case these characteristics are somewhat obscured by the superimposed interference pattern. For an angle of incidence less than the coincidence angle of the axial longitudinal waves, the contribution from the supersonic axial longitudinal waves is significant, when compared to the contribution from the radial and moment discontinuities. The contribution from the moment discontinuity is insignificant when compared to the other components. As the angle of incidence increases, the significance of the contribution from the axial components decreases, and the

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contribution from the radial components is enhanced, Fig. 26. Independent of the angle of incidence, the contribution from the radial forces is significant at certain spatial locations. This is due to the constructive interference of the scattered pressure from each of the two bulkheads. The character of the interference pattern is not the same for all of the components of the scattered pressure. The contribution from the axial force components comes mainly from the supersonic axial longitudinal waves and thus, the scattering is predominantly along the coincidence angle of these ‘waves. The axial contribution does not show the interference in the scattered pressure from the two bulkheads due to the specific angle of scatter. An interesting feature is that as the angle of observation varies from one side to the other of the normal, there are four predominant directions along which the contributions to the scattered pressure associated with the axial load components propagates. Each bulkhead generates two scattered waves symmetric about the normal at that bulkhead. As the angle of observation varies, the two waves generated by each of the two bulkheads are encountered, resulting in the four predominant direction of the scattered pressure as observed in the axial component of Figs 25 and 26. The interference is most distinct for the radial and moment components, although the interference is more readily observed in the radial component as compared to the moment component. These components have the characteristics of a dipole and a quadrupole, respectively, with broad radiation patterns, both of these components have a wide directivity pattern. The scattered pressure generated by one bulkhead will interact with the scattered pressure generated by the other bulkhead, creating a constructive/destructive interference pattern. The level of the moment component is very low compared to the other components. Apart from the acoustic interference pattern which mainly shows as areas of high and low intensity, at very specific frequencies one can observe strong scattering. An example would be the high intensity streak around a normalized frequency of 1.O (Figs 25 and 26). This strong scattering at a particular frequency can only be attributed to structural interaction. A form of resonant behavior may be induced in the portion of the shell between the two bulkheads, resulting in the high acoustic radiation at this frequency. The monostatic scattering results for n equals 0 are shown in Fig. 27. These results show a more regular interference pattern, symmetric about the normal direction. This is expected, since the incident and scatter angle are the same and therefore the results are independent of whether the incident wave is coming from the positive or negative axial direction. The basic features of the monostatic results are similar to those of the bistatic scattering. A feature which is more pronounced in the monostatic

J. M. Cuschieri

376

Bistatic, Inc. angle = 15

0.15 -88.5

and D. Feit

Bistatic,

, ,

Inc. angle = 15

0

0 Scattering

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88.5

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0 Scattering

Angle From Normal

Fig. 25. Bistatic scattered pressure for an incidence angle of 15” for circumferential mode n = 0. (a) Axial; (b) radial; (c) moment components; and (d) total scattering.

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Response Green’s function and acoustic scattering

Bistatic, Inc. angle = 45

a -88.5

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= 45

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Fig. 26. Bistatic scattered pressure for an incidence angle of 4.5”for circumferential mode n = 0. (a) Axial; (b) radial; (c) moment components; and (d) total scattering

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and D. Feit

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(b)

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Bistatic, Inc. angle = 15

”

Scattering

pressure for an incidence angle of 15” for circumferential (b) radial; (c) moment; and (d) tangential components

Angle From Normal

mode n = 2. (a) Axial;

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Response Green’s function and acoustic scattering

Bistatic, Inc. angle = 45

0

b

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Fig. 29. Bistatic scattered pressure for an incidence angle of 45” for circumferential mode n = 2. (a) Axial; (b) radial; (c) moment; and (d) tangential components.

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Bistatic, Inc. angle = 15

(a)

0.

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Fig. 30. Bistatic scattered pressure for incidence angles 15” (a) and 45”(b) for circumferential mode n = 2.

of

Response

Mono&tic

Green’s

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Scattering

0 0

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scattered pressure for circumferential mode n = 2. (a) Axial; (b) radial; (d) tangential components; (e) total scattering. (Continued overleaf.)

(c) moment:

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J. M. Cuschieri and D. Feii

0

-88.5 Scattering

88.5

Angle From Normal

Fig. 3 l-Continued.

scattering results is the null in the scattered pressure occurring around a normalized frequency of 2.0. This null corresponds to the in-plane resonance of the internal plate bulkheads. Under a bulkhead resonance condition, the severity of the discontinuity decreases resulting in a decrease in the scattered pressure. Acoustic radiation from the radial motion of the shell is only significant when the radial response of the shell is modified by the junction radial force creating a discontinuity. The bistatic scattering results for n equals 2, are shown in Figs 28-30. Again, the predominant scattering comes from the axial and tangential components, especially for incidence angles within the coincidence angles of the axial longitudinal and tangential shear waves. The contributions from the radial discontinuities are significant at the locations where the scattering from each bulkhead constructively interferes. For angles of incidence greater than the coincidence angle of the tangential shear waves, the contribution from the radial discontinuities becomes more significant. The contribution from the moment discontinuities is in all cases below all other contributions, even at locations where the moment contributions from the two bulkheads constructively interfere. Below the cut-on frequency of the axial longitudinal and tangential shear waves, the far-field scattered pressure is solely dominated by the scattering from the junctions’ radial force discontinuities. In this case as well, because of the presence of the cut-on frequencies of the axial longitudinal and

tangential shear waves, the scattering patterns show the “wine glass” shape similar to that observed in Ref. [13]. The main difference in the results for the bistatic scattering, between an angle of incidence of 15” and 45”, is the relative contributions between the in-plane waves (axial longitudinal and tangential shear) and the radial discontinuities. The contributions from the radial discontinuities and the longitudinal and shear waves are equally important for the larger angle of the incidence (Fig. 30). Finally, the scattering from each bulkhead from the axial longitudinal waves and the tangential shear waves, although occurring in preferred directions, when all these directions combine, the scattering within the cone centered about the normal and bounded by the coincidence angles of the tangential shear waves, is almost of uniform intensity, making it hard to distinguish between the contributions from the supersonic waves. Most of the observations made regarding the bistatic scattering results, apply equally well for the monostatic scattering results for n equals 2, which are shown in Fig. 31. The most significant feature of the results for the two bulkheads case is the interference pattern which is associated with the interaction between the contributions to the scattered pressure by each of the two bulkheads and the interaction through the fluidloaded shell structure. The characteristics of the interference patterns vary for the different components of the scattered pressure. 8. SUMMARY OF RESULTS

This paper presents results for the response Green’s function and the near field scattered pressure for a fluid-loaded cylindrical shell, with thickness to radius ratio of l%, internally loaded by a ring discontinuity. The shell is excited by an obliquely incident acoustic wave. The ring discontinuity creates axial, tangential and radial forces on the cylindrical shell which distort its response. The forces or moment generated by the ring discontinuity are specific to the characteristics of the discontinuity. To take these characteristics into account, far field scattered pressure results are presented when the internal discontinuity is a thick plate bulkhead. Finally, the far field scattered pressure results are extended for the case of two internal thick plate bulkheads. The solutions are based on a hybrid numerical/analytical approach which was presented in Ref. [l] for a fluid-loaded plate with a line discontinuity. In the hybrid solution, the response of the fluid-loaded cylindrical shell is evaluated using a numerical inverse Fourier transform based on a fast Fourier transform (FFT) algorithm. However before the transform is implemented, singularities in the integrand of the transform are removed. In the case

Response Green’s function and acoustic scattering of the fluid-loaded plate [l], below the critical frequency, only the real poles of the characteristic equation created singularities in the integrand, and for frequencies above the critical frequency, the near singularity created by poles close to the real axis, had to be treated in the same way as the singularities created by the real poles. In the case of the fluid-loaded cylindrical shell, real and near real poles exist for all frequencies and therefore, in the hybrid solution, the singularities associated with these poles are removed at all frequencies. That is, in this case both real and complex poles have to be located at all frequencies. This was not a problem since all poles of the characteristic (equation of the shell are located numerically. Having removed the singularities, the inverse Fourier transform is numerically evaluated. The contributions of the singularities are evaluated analytically. Combining the numerical and analytical contributions, the response Green’s function of the shell and the near field scattered pressure for excitation from axial, tangential and radial forces created by a ring discontinuity are evaluated for circumferential modes 0 and 2. The n equals 0 case is different from other circumferential modes because of the symmetry in the motion of the shell, however the n equals 2 case is representative of the shell behavior for order:5 of n greater than 0. No other results can be found in the open literature to compare the results obtained in this paper. However, for large enough frequencies, the radial response of the fluid-loaded shell under excitation from a radial discontinuity, approach that of a flat plate. At sufficiently high frequencies, the behavior of the shell starts to approach that of a flat plate. Furthermore, comparison was made with data obtained by other investigators using a modal summation approach (private communication to D. Feit from J. Cole of Cambridge Acoustical Associates, Inc.), but whi’ch has not been published in the literature. The results from the two approaches are very similar. The results show that for both n equals 0 and 2, at low frequencies, the response and scattered pressure are dominated by the subsonic membrane waves. The scattered pressure is predominantly from the ring discontinuity. At higher frequencies, the response and scattered pressure are more significantly influenced by the supersonic axial longitudinal and tangential shear waves, especially when the excitation is by a ring, axial or tangential discontinuities. This is shown by a scattering pattern which has preferred directions corresponding to the coincidence angles of these waves. For a radial discontinuity, the direct scattering from the discontinuity is significant. In the case where the discontinuity is a plate bulkhead, the results for the bistatic and monostatic scattering show that for n equals 0 and for most angles of incidence, the far field scattered pressure is dominated by the contributions from the supersonic axial longitudinal waves and the radial component of

383

the junction force. The exception occurs when the angle of incidence is equal to the coincidence angle of the axial longitudinal waves, in which case the contribution from the axial longitudinal waves exceeds that from the radial forces by as much as 20 dB. The difference in the level is frequency dependent, because of the presence of resonant frequencies of the internal plate bulkhead. The contribution from the junction bending moment is more than 20 dB lower when compared to the contribution from the radial component. This is expected, since the moment component creates a quadrupole like source, while the radial force component creates a dipole like source. For circumferential mode orders greater than zero, generalizing from the results for n equals 2, an additional component associated with the tangential shear waves is introduced. This component and the axial longitudinal component has a cut-on frequency, which gives rise to the “wine glass” shape in the scattered pressure results. The tangential shear waves play a significant role in the scattered pressure. If the angle of incidence is less than approximately 30”, then the scattered pressure is dominated by the contributions from the supersonic longitudinal axial and tangential shear waves, above the cut-on frequency. Below the cut-on frequency of these waves, the far field scattered pressure is dominated by the component associated with the radial junction force. For angles of incidence greater than approximately 30”, the radial junction force and the shear and longitudinal waves equally contribute. Finally, for very low frequencies, the in-plane response of the plate bulkhead is very small and the junction between the plate and the shell behaves similar to a radially pinned junction. As the frequency increases, the internal structure has in-plane resonances and the radial influence of the bulkhead is at a minimum. At these frequencies, the contribution from the radial forces to the far field pressure is minimal. When the cylindrical shell is internally loaded with two plate bulkheads, separated by a shell diameter, a more complex scattering pattern is observed. This is due to interference between the scattering from each of the two bulkheads. The interference pattern varies according to the component of the scattered pressure. For the components associated with the radial forces and bending moments, the interference pattern has the characteristics of a constructive/destructive interference pattern that one would typically observe when waves from two sources interact, with high and low intensity regions. The high intensity regions correspond to locations with constructive interference between the components of the scattered pressure by each bulkhead. The low intensity regions correspond to locations with destructive interference. The bending moment component is however small compared to the radial component. The scattered pressure components associated with the supersonic, axial longitudinal and tangential shear waves do not

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show much of an interference pattern and this is mainly attributed to the fact that the scattering from these supersonic waves has preferred directions of propagation. Additional to the interference pattern which can be associated with the interaction of the acoustic waves generated by each bulkhead discontinuity, enhanced scattering takes place at selected frequencies. This enhanced scattering does not show in the single bulkhead results, and is attributed to structural interaction between the two bulkheads and the shell. The scattering patterns for the two bulkhead case are certainly much more complex than those for the single bulkhead. Most of the added complexity comes from the new length scale introduced by the separation distance between the two bulkheads. This length scale influences both the acoustic and the structural interaction between the two bulkheads. Acknowledgmenrs-The authors would like to acknowledge ONR Code 4521 who funded this work through the Carderock Division of NSWC (DTRC). REFERENCES

1. J. M. Cuschieri and D. Feit, A hybrid numerical analytical solution for the Green’s function of a fluid-loaded elastic plate. J. Acousr. Sot. Am. 95, 1998-2005 (1994). 2. J. M. Cuschieri, Structural power flow analysis using a mobility approach of an L-shaped plate. J. Acoust. Sot. Am. 87, 1159-1165 (1990).

3. R. F. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic plates. J. appl. Mech. 18, 31-38 (1951). 4. D. M. Photiadis, The propagation of axisymmetric waves on a fluid-loaded cylindrical shell. J. Acoust. Sot. Am. 88, 239-250 (1990).

M. Junger and D. Feit, Sound Structures and Their Interaction. MIT Press, Cambridge MA (1986). P. M. Morse and K. U. Ingard, Theorefical Acoustics. McGraw-Hill, New York (1968). D. E. Muller, A method for solving algebraic equations using an automatic computer. Math. Tables Aids Compuf. 10, 29-37 (1956). 8. W. L. Frank, Finding zeros of arbitrary functions. J. Ass. comput. Mach. 5, 154160 (1958). 9. J. F. M. Scott, The free modes of propagation of an infinite fluid-loaded thin cylindrical shell. J. Sound Vibr. 125, 241-280 (1988).

10. C. R. Fuller and F. J. Fahy, Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. J. Sound Vibr. 81,501-518 (1982).

11. D. R. Mook, G. V. Frisk and A. V. Oppenheim, A hybrid numerical-analytical technique for the computation of wave field in stratified media based on the Hankel transform. J. Acoust. Sot. Am. 76. 222-243 (1984).

12. C. Hwang and W. S. Pi, Investigation of vibrational energy transfer in connected structures. NASA Contract Report, CR-124450 (1972). 13. M. L. Rumerman, Contribution of membrane wave reradiation to scattering from finite cylindrical steel shells in water. J. Acoust. Sot. Am. 93, 55-65 (1993).

14. Y. P. Guo, Sound scattering by bulkheads in cylindrical shells. J. Acoust. Sot. Am. 95, 2550-2559 (1994).