The analogy between acoustic waves and water surface waves with refraction

J. Sound Vib. (1971) 15 (l), 93-106

THE ANALOGY WATER

BETWEEN

SURFACE

ACOUSTIC

WAVES AND

WAVES WITH REFRACTION

L. WENT, R. J. SCHOENHALS ANDE. R. F. WINTER School of Mechanical Engineering, Pardue University, Lafayette, Indiana, U.S.A. (Received 23 April 1970)

The similarity between acoustic waves in an inhomogeneous medium and water surf= variable water depth is established. The two systems are analogous if they both are charm by the same index of refraction field. Based on this analogy, acoustic refraction problems are treated by utilizing a surface wave simulation achieved with a water wave tank. Experimental results obtained from the wave tank for various situations are used to deduce the behavior of the corresponding acoustic systems. It is concluded that there are two major advantages in employing a water wave tank. The first is that the entire wavefront configuration can visually be observed; the second is that water surface waves can be investigated in the laboratory much more readily and more economically than acoustic waves. waves with

1. INTRODUCTION The similarity of acoustic waves propagating in a homogeneous medium and shallow water waves propagating on the surface of a body of water with constant depth has been recognized for many years [l]. In an inhomogeneous medium acoustic wave refraction occurs, and this phenomenon is also exhibited by water surface waves in the presence of variable water depth. The refraction processes in these two systems are analogous if the wave source, boundary conditions, and index of refraction fields are the same. It is therefore possible to simulate an acoustic refraction problem by means of a water wave tank whose bottom surface possesses the appropriate topography. The wavefront configurations for both the acoustic and the water wave systems are described by the eikonal equation, and they exhibit identical wave patterns when the two systems are analogous. The actual amplitude distributions are not the same, however, when energy absorption effects are appreciable, since the viscous effects in the two systems are different. The analogy is further complicated by the fact that the absorption coe5cient for either system is variable in refractive situations. In spite of these apparent di5culties, the two different amplitude distributions occurring in the presence of spatially variable absorption phenomena can still be made analogous by the proper application of a single reference value of the absorption coefficient for each system [2]. In this paper the theoretical relationships describing acoustic and water surface wave propagation are presented, and the analogy between them is developed. An experimental water wave system is then described. Some typical measurements and a photographed wave pattern are given for an example situation exhibiting refraction. Finally, the analogy is used to deduce the behavior of the corresponding acoustic system. It is concluded that there are two major advantages in employing a water wave tank. The fist is that the entire wavefront configuration can be visually observed; the second is that water surface waves can be investigated in the laboratory much more readily and more economically than acoustic waves. t Present address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A. 93

L. WEN, R.

94

J. SCHOENHALS AND E. R. P. WINTER

2. ACOUSTIC WAVES IN AN INHOMOGENEOUS

MEDIUM

The conservation equations of mass and momentum and the entropy equation for an inhomogeneous stationary medium subjected to acoustic disturbances are

m

ap

aT,r

(2)

PDt --&+ax,’ a

DS pTx=~@+dx,

where p = po(xr) + 8(x,, t) is the density field, 6 is the acoustic condensation, uI is the particle velocity field, S is the entropy field and @ is the dissipation function. The equation of state for a medium undergoing acoustic processes can in general be expressed as P = P(p, S).

(4)

A particular example is that of a perfect gas, for which equation (4) can be written as (PIP,) = Wp0) y exp KS - ~0)ld

W

where PO, p. and So are the properties of the undisturbed medium, c,, is the constant volume specific heat, and y is the ratio of the specific heats. 2.1.

UNATTENUATED ACOUSTIC WAVES

an inviscid medium the viscous stress and dissipation terms vanish, and there is no transformation of mechanical energy into thermal energy. For small perturbations S/p0 --zz1 and lu,/C*l Q 1, where C* is the reference wave velocity. In this situation the continuity equation is well approximated by In

aP at=

a4 aP0 -Poax,-uiax,’

(5)

Similarly, the momentum equation is approximated by the first-order Euler equation [3],

au, ap

poat=-ax,.

(6)

The compression and rarefaction processes accompanying acoustic wave propagation take place so rapidly that each complete cycle can be considered to be essentially adiabatic for most media [4]. If the fluid is inviscid as well, then for each fluid element the time rate of change of entropy is zero, DS/Dt = 0. In general, the variation of pressure indicated in equation (4) can also be expressed as (7) which reduces to

(%)=(g)s2

(8)

for the acoustic situation described above. (aP/ap)if2 is the acoustic wave velocity, C. Accordingly, the local rates of change of P and p may be taken to be related by

ACOUSlKWAVESANDWATERSURFACE WAVES

95

Substitution of the continuity equation (5) into equation (9) and subsequent differentiation with respect to time gives a* U, a4 ab 1 a9

Sat2

----. = -po at ax,

at ax,

00)

Taking the gradient of equation (6) yields

apeau,

ah,

a*~

-asc,3i-pozqt==a,:

01)

The acoustic wave equation for an inviscid, inhomogeneous medium is obtained by combining equations (10) and (11). The result is vzp=l!z

(12)

c* at*

or, (13) where P,, is the acoustic pressure, P -PO. Here the undisturbed pressure, PO, is taken to be uniform throughout the system, since the medium is considered to be at rest. From this derivation the acoustic wave equation (13) has been shown to be the same as for the case of a homogeneous medium. The acoustic velocity, C, is a variable in the inhomogeneous situation, however. Similar derivations, based on slightly different approximations, have been given by Morse and Ingard [3] and by Bergmann [5]. 2.2. ACOUS’lKWAVESWITHENERGYABSORP’lJON In the foregoing analysis the effects of viscosity, heat conduction and other dissipative processes have been neglected. The solution of the wave equation (13) yields the unattenuated acoustic pressure, which is in phase with the acoustic condensation. In a real medium the acoustic pressure is attenuated somewhat, and there is a phase difference between acoustic pressure and acoustic condensation. This behavior has been treated by means of empirical relationships [6]. In order to include the absorption effect, the unattenuated acoustic pressure can be corrected by an absorption factor, (I, which takes into consideration the dissipative losses and heat conduction [3]. The energy loss per unit volume and unit time from the acoustic wave is equal to pT(DS/Dt) due to the dissipative processes. The absorption coe5cient, a, is defined by 2a ~ pTDS/Dt fraction of acoustic energy dissipated = per unit distance of wave propagation, I# where I# is the attenuated acoustic intensity. For a time harmonic wave in the presence of dissipation, the acoustic pressure can be expressed as P.# = A#exp [j(K*I;-

cd)],

414)

where A* is the attenuated acoustic pressure amplitude function, 2/2pc7-, F is the phase function and K* is the reference wave number. In a homogeneous medium, the attenuated intensity, I#, and the attenuated pressure amplitude, A*, are characterized by dZ# = -2aZ# ds

-

(1%

96

L. WEN, R. 1. SCHOENHALS AND E. R. F. WINTER

and

dA*

-

ds

= +A#,

(16)

where ds is the differential element of path length in the direction of propagation. In an inhomogeneous medium the pressure amplitude variation is determined by refraction effects in addition to absorption. For this situation equation (16) is now generalized to include refraction and is rewritten as

dA*

-

ds

= -+A#,

(17)

where K is the extinction coefficient, a + b, a is the absorption coefficient (always a positive quantity) and b is the refraction coefficient (which can be either positive or negative depending on whether the wave is divergent or convergent). Integration of equation (17) yields A # = A, e-lb& ae-lads, where A, is the pressure amplitude at the source along the initial wavefront. Note that the quantity A, e-Jab is just the unattenuated pressure amplitude, A, in the same refractive field, but in the absence of dissipation. Its variation is given by the solution to the wave equation (13). Thus, A # = A e-lads. (18) An expression for the absorption coefficient has been derived for gases [7]. The result is w* My - 1) a=2BC $L+------, [ CP I

m-9

where w is the angular frequency. For aninhomogeneous medium the coefficient of absorption is a function of position, which can be accounted for in the exponent of equation (18). In keeping with the inverse relationship of Q and C given in equation (19), a is taken to be proportional to the index of refraction n (defined as C*/C) where C* is the reference wave velocity. Since IZ= 1 at the reference condition, this procedure gives a=a*n, (20) where a* is the reference absorption coefficient. Inserting equation (20) into the exponent expression of equation (18) gives J ads = a* J rids. The path length, s, is related to IZby dF= ndr [2], where Pis the phase function which will be discussed in a subsequent section of the paper. Introducing this result into equation (18) gives A # = A e_a’F.

(21)

In this formulation a single reference value, a*, is used throughout the inhomogeneous medium to account for the variable distribution of the energy dissipation.

3. WATER SURFACE WAVES WITH VARIABLE WATER DEPTH 3.1.

UNATTENUATED WATER SURFACE WAVES

The continuity equation for an incompressible liquid is

ACOUSTIC

WAVES AND WATER

SURFACE

WAVES

97

Now a potential function, 4, is defined such that ui = grad$, which allows equation (22) to be written as

(23) The momentum equation for a quiescent, inviscid liquid subjected to small disturbances takes the form of the first-order Euler equation, &,/at = -(l/p) (@/8x,), which in component form can be written as

&J$=&(-j’ &(z)=&(-$)~ &($)=&(-y-9.

(24)

(25) (26)

where the x3-direction is taken to be vertically upward and where g is the gravitational constant. Integration of equations (24), (25), and (26) leads to

a+ p + e(r), -z=p+gxj

(27)

where 0(t) is an arbitrary function of time. At this point it is convenient to introduce the phase function F which characterizes the motion of the surface waves and is therefore a function of x1 and x2. It is then possible to replace the coordinates x, and x2 with the single coordinate, F. Equation (23) then becomes

(28) where

(29) Equation (29) is the well-known eikonal equation [7]. For harmonic waves it is assumed that $ = Wi,

x2, xJ)exp L/W* F-

wt)l,

(30)

where U is the amplitude of the potential function and K* is the reference wave number. Entering equation (30) into equation (28) gives

a2 u -= ax:

n2 K*2 U.

(31)

The general solution of equation (30) is U = c, cash (K* nxJ + c2 sinh (K* nx&.

(32)

If the bottom surface of the container is horizontal, the vertical component of velocity at the bottom is zero. In the case of a non-horizontal bottom, the vertical velocity component is small provided that the slope is considerably smaller than unity. Application of this approximation at the lower boundary corresponds to &$/ax3 = 0 at x3 = -h, where h is the depth and where x3 is measured from the equilibrium liquid surface level at the top. This condition leads to C#= Mcosh [K(h + x3)] exp [j(K* F - wt)], (33) where M is a potential amplitude constant and K is the local wavenumber, K*n.

98

L. WEN, R. J. SCHOENHALS AND E. R. F. WINTER

For small disturbances, the vertical velocity component at the free surface is well approximated by &)/at [8], where q is the vertical displacement of the free surface from its equilibrium position. Hence, at the free surface, u3 = (&#/lax,) x (a+%). The derivative of + is obtained from equation (33) which is related to 7, within the indicated approximation, by z = MKsinh Kh exp [j(K* F - wt)] for small amplitude waves (7 < h). The pressure change across the free surface at x3 = r) can be calculated from the law of Laplace for surface tension [9]. For a sinusoidal wave whose amplitude is small relative to the wavelength X, the radius of curvature is approximately X2/47r271 and the pressure difference across the free surface due to the surface tension, u, is 4~’ rr7/h2. Also, Lowell [lo, 1l] has shown that the pressure distribution in a thin layer of liquid is well approximated by the hydrostatic pressure relation, even under the dynamic conditions associated with surface waves. Thus, in the vicinity of the free surface, the pressure is given by p = P, + (47f2u/P) v + pg(r) -

x3),

(35)

where Pa is the ambient pressure. Substituting this relation into equation (27) gives (36) which is valid in the vicinity of the free surface. Equation (36) is now differentiated twice with respect to x1, and the procedure is repeated with respect to x2. The addition of the two resulting equations yields (37) near the free surface. In deriving this result, fractional changes in wavelength over a distance of one wavelength have been assumed to be small relative to unity. Now, from the continuity equation (23), (38)

-[$@)+&(:)]=&($).

($/ax:)@+/&) can be evaluated at the free surface for waves of small amplitude (q Q h) by employing equation (33). This makes it possible to rewrite equation (37) as

azrla27 G+p=-.i 1

Differentiation

2

MK2K*C* cash Kh exp [j(K* F - cot)]. 4& U/PA2+ g1

(3%

of equation (34) with respect to time gives

a27=--jMKK* __ at2

C*sinhKhexp[j(K*

F-

wt)].

(40)

Combining equations (39) and (40) yields (41) where (42)

ACOUSTIC WAVES AND WATER SURFACE WAVES

99

Equation (42) shows how the wave velocity C (celerity) varies for a liquid layer of variable depth. It can be seen that the celerity expression for variable liquid depth has the same form as for the case of constant liquid depth [12]. For variable depth, however, C is a function of position. 3.2. WATER SURFACE WAVES WITH ENERGY ABSORPTION In the foregoing analysis the viscous effects were neglected. In reality the dissipation effect occurs mainly within the viscous boundary layer. For time-harmonic motion, the oscillation boundary layer thickness, or “depth of penetration” [13] as it is sometimes referred to, is given approximately by 4*62/v/w, where v is the fluid kinematic viscosity and w is the angular frequency. When the water depth, h, is much greater than this value, equation (41) can be considered as a good approximation for the unattenuated wave displacement. If the oscillation boundary layer thickness is of the same order of magnitude as the water depth, then the inviscid model does not represent a good approximation. In this case it is appropriate to correct the unattenuated wave amplitude by using an absorption coefficient to account for the viscous effect. It has been suggested [14] that coefficient a for a progressive wave train can be predicted by v KW + sinh 2Kh (43) ‘=?% ( 2Kh + sinh 2Kh 1 ’ where W is the width of the channel. The attenuated wave amplitude, A*, is related to the unattenuated value by A# = A e-lads. When the water depth is variable, both the celerity and the absorption coefficient are variable also. However, from equation (43) it can be shown that when 2Kh is small the absorption coefficient, a, is approximately inversely proportional to the celerity, C, which is analogous to the case of acoustic absorption as indicated by equation (19). The reference absorption coefficient, a*, can then be used for surface waves according to equation (20), in the same manner as in the acoustic situation, such that the attenuated water wave amplitude is given by equation (21). 4. THE ANALOGY It can be seen that equations (13) and (41) are of the same form. Since the two systems are described by the same differential equation, they yield identical solutions if they possess the same index of refraction field and the same initial conditions and boundary conditions. Thus, the acoustic and water surface wave refraction phenomena are analogous for unattenuated waves for this situation. Note, however, that the water surface waves are twodimensional, while acoustic waves can be three-dimensional. Therefore, the water wave system basically is limited to two-dimensional acoustic situations. In spite of this limitation, the water wave analogy can be applied to three-dimensional acoustic systems which are characterized by stratified media sustaining wave propagation in which there is axial symmetry of the wavefronts [2]. It is convenient to use a characteristic length, L, to define a set of dimensionless coordinates, x and y, so that the wavefront configuration accompanying either the acoustic or the water wave propagation can be described by the same dimensionless eikonal function, F, which is the solution to the eikonal equation

Here, the wavefront family is represented by F(x,y) = constant. The amplitude distribution of the unattenuated wave propagation is described by the transport equation [15], V(lnA)-VF++V2F=0,

(45)

100

L. WEN,

R. J. SCHOENHALS

AND E. R. F. WINTER

where A is the amplitude function. For time-harmonic waves, the solution of the wave equation can be expressed by A exp [j(K* F - cd)]. Equation (45) should be corrected to account for energy absorption when this effect is appreciable. The dimensionless unattenuated amplitude distribution is defined by 5 = A/A,, where A, is the amplitude at the source along the initial wavefront. The absorption is treated by applying equation (21) to obtain <# = A #/As = 5 e-a*F,

(46)

where A# = A, along the initial wavefront at the source. If the reference absorption coefficient values are known for both the acoustic system and its corresponding analog consisting of the equivalent water wave system, then a direct measurement of the actual amplitude distribution for the water wave system can be used to calculate the attenuated amplitude distribution for the acoustic system. It is not required that the magnitudes of the reference absorption coefficients be equal. This is illustrated in the following way. First, if the two systems are analogous, the eikonal function, F&y), is the same for both, and the dimensionless unattenuated amplitude distributions are also the same. That is, ~,(x,y) = C,(X,JJ). Then, application of equation (46) to both the acoustic and water wave situations leads to (47) where x, y, a$ a.* and Fare all dimensionless quantities which have been scaled with respect to a suitable characteristic length, L. Thus, if F&y), a$ and a: are known and if the actual water wave amplitude distribution (a function of x and y) is obtained from measurements, then the attenuated acoustic amplitude distribution can be calculated using equation (47). 5. SIMULATION OF ACOUSTIC REFRACTION WITH A WATER WAVE TANK The celerity of the water surface wave propagation is a function of frequency, surface tension and water depth. If the frequency is high (short wavelength) the surface tension effect predominates [16]. Such short ripples are usually referred to as capillary waves. For capillary waves, the speed of propagation is inversely proportional to the square root of the wavelength, and as the wavelength decreases the speed of propagation increases. This can be illustrated by considering equation (42) in the limit as h becomes small. It is clear from this description that capillary waves cannot be used to simulate acoustic waves. On the other hand, for the limiting condition of very low frequency the effect of surface tension on the celerity is negligible. Thus, if the frequency is low enough (long wavelengths), the waves are essentially gravitational waves. Neglect of the surface tension term in equation (42) leads to the celerity expression for this case which is (48) For a specified frequency the relation between the local index of refraction and the water depth can be evaluated by solving the above equation for h. This procedure yields /I=&[ln(l

+E]-ln(1

-g)].

This equation can be applied to calculate the water depth variation, h(x,y), required to yield a prescribed index of refraction field, n(x,v), at a specified frequency w. A water wave tank designed to simulate acoustic waves is shown in Plate 1. The wave generator is composed of an oscillating plunger with adjustable amplitude and frequency

(facingp. 100)