acoustic scattering from a submerged plate. i. one reinforcing rib

Acoustic scattering from a submerged plate. I. One reinforcing rib

Barry Lee WoolIcy

Naval Ocean Systems Center, San Diego, California 92152 (Received 15 May 1979; accepted for publication 5 January 1980)

Theoretical calculations of the backscattering of a plane sound wave by a rib stiffened Timoshenko-Mindlin plate are presented. An exact solution is compared with two saddle point integral approximations. One saddle point integral approximation takes into account the near coalescence of the leaky wave pole and the saddle point. Mass loading of the rib and different rib constraints are discussed. The range of validity of the model is also given.

PACS numbers: 43.40.Dx, 43.30.Gv, 43.20.Bi

INTRODUCTION

This paper presents the results of a theoretical investigation of the backscattering of an acoustic plane wave from a rib stiflened infinite elastic plate. Kono- valyuk x has shown that the scattering of a plane incident wave by a rib attached to and behind an elastic plate can be characterized by a pair of impedances. We use these longitudinal and flexural impedances in our calculations which have been extended to the Timo-

shenko-Mindlin case which accounts for the shear and

rotary inertia of the plate. Our solution has been adapted to handle fluid-backed and mass-loaded thick ribs. 2 We can handle the structural damping of the rib and plate. In addition, we have implemented different and greatly improved methods of calculating the integral for the backscattered field. We have obtained an

exact evaluation of the integral which is compared with two saddle point integral approximations. The case of the incident plane wave being near the classical coincidence angle can only be accurately handled by an exact integral evaluation. One saddle point integral approximation, following Stuart's method, s takes into account the near coalescence of the leaky wave pole and the saddle point. Its range of validity is discussed.

The effect of the poles on the integrand are traced back to various components of the sound radiation. The proximity of the poles to the path of integration is given physical meaning. The theoretical possibilities for the behavior of these poles, and, therefore, the theoretical possibilities for the sound radiation, are presented as a function of the relevant parameters. The range of validity of the Timoshenko-Mindlin plate theory is discussed. 4,5

arrive from the liquid and impinge upon the plate perpendicularly to the line of attachment of the rib:

½o = exp(ikXsinO + ikY cosO), (1)

where 0 is the angle of incidence of the wave and k is the wavenumber in the liquid. The total field potential is represented as follows:

•=•o + Vexp(ikXsinO-ikY cos0) +•I,. (2)

Here 4, represents the field scattered by the rib and V is the plane-wave reflection coefficient of the elastic plate. V is3

v(o) = Zo)/(z + Zo) ,

for an air-backed plate and

v(o) = + 2Zo),

for a fluid-backed plate (when the fluid is the same as the fluid loading the plate), where

Zo=PoC/CosO and

Z•,=-iwHp( 1- Cff(sin20- k:)(sin•O - k•)) 1 + k•C• (sin•O- k•) ' with

•.• 0) = C2 -• , 00• (pH/D) •/2 ,

2

k•_[c•p(l_o•)] c •' = c • - =-*' c I

K•=tr•/12, C=S/2(l +cO.

(3)

I. MATHEMATICAL FORMUALTION

Consider an infinite plate of thickness H which has a perpendicularly attached rib of thickness h and length l. The rib may have a mass loading as shown in Fig. 1. Now let the plate be contiguous with a liquid-filled half- space (Fig. 1). Also let the rib extend downward into either a fluid-filled or an air-filled medium. The re- fleetion of a plane sound wave from this rib stiflened plate is the subject of this paper. The Timoshenko- Mindlin plate theory is used; this theory accounts for the shear and rotary inertia of the plate. The origin of our coordinate system is placed at the plate-liquid surface at the center of the rib. Let a plane sound wave

. •n•eY' NN• • .•Xaxi5

•,•,• IoaJ& 5

FIG. 1. Acoustic wave scattering from a plate with a rib.

1642 J. Acoust. Soc. Am. 67(5), May 1980 1642

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The constant K2G has been introduced by Timoshenko and Mindlin 6 to account for the fact that the transverse shear strains in the plate are not truly independent of the thickness of the plate. c z is the velocity of the compressional wave in the plate and c s is the velocity of the shear wave in the plate. D=EI•/[12(1-cf')] is the flexural rigidity of the plate, c is the speed of sound in the liquid, E, ct, and p, are the Young's modulus, Pois- son's ratio, and the density of the plate material, respectively, and Po is the density of the liquid.

The field potential ½ must satisfy the homogeneous reduced wave equation in the liquid above the plate:

(v • + •)• = 0.

The field potential must also satisfy the equation of motion on the entire surface of the plate except on the line of contact (the z axis in Fig. 1)6:

(4)

d d d

+ - + = 0. (5) In writing the above equation, the Timoshenko-Mindlin plate theory has been used with

F•.=pw•. 12D+HStc•'G .. 21pn w-.t t• 12•c 2 GD , F2 =p•w [' •-•c•'•-• , = 2 _poptt2co4 q• = u"2P':' Fs u., po and F4 ß D ' •2CH ' - 1'2K 2CD

The field potential • must also satisfy the following two boundary conditions:

+o -o ' XOYJ +o XOY/-o'

The following notation has been employed:

and

kaY"/ kaY"/ ' 6--•0 X "'0+6 +0

The preceding two conditions expressed as equalities of the derivatives of ½ are simply this physical condition: at the line X= 0, Y= 0, the plate displacements must be continuous and the angles of inclination of the cross section of the plate must be continuous, respectively. We also have

O2XOYJ +,o

83X0 +o

- k o2XOY/-o D O•'Y'' (6b)

[ 84• • =+i (6c) -- •83XOyJ -o D OY ' The preceding conditions are contact conditions for the

moments and forces at the contact line where Z• and Z a are the impedances of the rib in longitudinal and flexural vibrations, respectively. v

2• •'-•: $r•b•] tan 2p rib(1 - Urib)

. _/. Erib 3/2 Erib ZPribh/•.rib (•: Crr•b )){1 _ COS [02/(Pri b C osh[•/(p rib(1 -- (1

] W •: sin Finally, the radiation extinction principle {t infinity must be met.

To solve the above conditions and equations, 4, is sought in the form of the following integral which solves the homogeneous wave equation and the radiation extinction principle at infinity:

• = fr f(A) exp[i•X+ Y(X 2 - k2) 1/2 ] d'A. (8) The function f(A) must be defined from boundary and contact conditions. A contour F is used from _oo to + oo

in the X plane along which the integral will be evaluated. Contour F was selected as shown in Fig. 2. The integrand is not easily evaluated along this contour. As shown in Fig. 3, we have used a branch cut in the A plane (corresponding to a branch cut from + k 2 to _oo along the real X 2 axis in the X 2 plane) such that

Re(X 2 - k2) 1/2 >• 0

on the entire top Riemann sheet. This condition corresponds to the requirement that the field dies off at infinity (i.e., Y--•o); f(X) must die off at infinity faster than ];•]-s if both continutiy of displacement and the radiation extinction principle are to be satisfied.

We will use Cauchy's theorem as explained in Sec. IV in order to relate the integral along the contour r to the integral along the contour r + (or r-) and the residue of the integrand at the poles on the top Riemann sheet enclosed by r and F + (or F-). This will allow us to evaluate the integral with greater ease. The choice of whether to close the path above (i.e., F +) or below (i.e., r-) depends upon the requirement that the field die off at infinity. When computing the field at the points with X> 0 the integral in Eq. (8) must be computed by closing the integral above using the path F + so that it embraces all the singularities lying on the top Riemann sheet which are below the contour r while ex-

imaginary X axis

tie

FIG. 2. Integration contour.

real X axis

e->O

1643 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley: Scattering from submerged plate. I 1643


! !

smaginary ), axis

'X x F +

+k r_-._-_-_-_-9---- J real • axis

imaginary k axis

_• • l' real k axis I I ,

I / I I /

FIG. 3. Branch cuts and the F* and

F' paths.

cluding the lower branch cut.

Bringing the total potential in compliance with Eq. (5), an expression for f(X) is obtained (where the residue contribution is implied by the notation)-

f• f(X) [(•t2 -- k2)l/2(•t4_ ]t2F1 + F2 ) -- .F 5 _ •t2F3] x exp[iXX+ Y(X •'- k•')z/•']dX =0, (9)

for X• O, where

Fs = qS - P/ffW4Po/12K •' GD.

Since X is an arbitrary quantity, this equality is valid only if the function

F(•) :f(X) [(X 2 - ]•2)1/2(X4 -- X2F1 q-F 2) -- F 5 -•t2F3]

is regular over the entire complex plane, excluding the infinitely removed point where it can have a pole of finite order. Therefore, F(X) is some polynomial •. The requirement of continuity of the potential and its derivatives of the second order inclusively means that F(•) does not contain terms with powers above the first. Thus, the desired functionf(•) must be of the form

f(X) = (C O + CzX)/[(X •' - k•)l/2(X4 - •t2F1 +F2) -- F 5 -- •t2F3]o

(1ø)1

fr (CO + C!x) exp[ixx+ Y(X 2 - k•') •/•' dX = _ 4_ + - F,-

For •'s final definition it is necessary to find the co- effieients C O and C•.

C O and C• are found by substituting the total field, defined by Eq. (2), into the e.onditions of Eqs. (6b) and (6e). The following expressions are then obtained for C O and C•:

Co [-iwkZr(1 V) cosO]/(2 + + , = - DJa +wZFJo) (11)

C• = [-iwZ•k•'(1 - V) sinOcosO]/(2DJ] + wZ •J• ),

where

• 2tn(2t 2 -- k2)l/2 • J}= + (k 2 -- ]•2)1/2(k4-- X2F1 + F2) - F5 - X2F3 (12)

are computed in Sec. IV. The poles of the integrand in Eq. (12) and those of Eq. (10) are identical. They are discussed in See. III.

When the coefficients C o and C• are obtained and the explicit form of f(X) is defined, the methods of See. IV can be used to evaluate Eq. (8) for •. For X<0, given by

fo • - 2i(C o - iC•Z) exp(+ ZX)(Z 2 + ]•2)1/2(Z4 +Z2F! +F 2) Cos[Y(Z 2 + k2) 1/2] - (-F• q- Z2F3) sin[Y(Z 2 + ]•2)1/2] dZ = 2 2 (Z 2 + k2)(Z 4 + Z2Fx +F2) 2 + (- F 5 + Z

• - 2i(C o - CxZ) e•(-iZ•(- Z 2 + k2)l/2(Z 4- Z2FI +F•) Cos [Y(-Z 2 + k2) 1/2] + (F• + Z2F]) sin[Y(-Z 2 + k2) 1/2] dZ + (__ Z 2 + k 2) (Z 4 -- Z2F1 + F2) 2 + (F 5 + Z2F3) 2

(C 0 + C•X) e•[+ iXX+ Y(X 2 - •)•/2 X =poles on top •emann sheet enclosed + 2xi • a _ , x (• - 2XFx)(X 2- k2)x/2+X(X4-X2Fx +F2)(X 2- k2) -x/2 2XFa wit•n r and r-. (13)

II. IMPEDANCES FOR THE MASS LOADING OF RIBS

Consider a rib with a mass at the end which is taken

to act as a simple point mass under longitudinal vibra- tion. The end mass is assumed to have sufficient later-

al dimensions to require inclusion of rotational inertia effects. For purposes of illustration consider the case of the rib having a continuous weld: the case where the shear at the rib-plate junction may be nonzero whereas the transverse displacement is taken to be zero. The longitudinal impedance Z g for this case is given bye':

I

Eh•, EhT sin•,l + (1 - 0'2• 2 cos•// (14) Z•=iw(1 _(y2) Ehy cosy/- (1-a2•2sinyl ' where m is the mass per u•t volume for a unit length in the Z •reetion and

The flexural impedance Z• for the above ease with the plate •r backed is given by •-

- - o)(v + ,)½ + (0%- e)]},

1644 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley' Scattering from submerged plate. I 1644


where

a =- 0'[0' sinO'/+ (co'I/D) cosO'/],

•= 0'[- o'eosO'/ + (oo2I/O) sinO'l][D(O '• + qf) ]-• ,

6 =•[• sinh•/- (co'I/D) eosh•/],

½ =•[• eosh•/- (co'I/D) sinhcpl]{-[D(O '•' + ½•')]-•}, 5 =- 0 '• eosO'/+ (mco2/D) sinO'l,

r/= [0 'a sin0'/+ (mco2/D) eosO'l][D(O '2 + q•2)i-x,

t• = qb a eoshqb 1 + (rnco 2/D) sinhqb l,

and

v = [q•a sinhq• 1 + (rmo 2/D) eoshq• 1] {-[D (0 '•' + ½ •') ]-x},

with

0'= {[A + (A •' - 4F) •/•'1/2} •/•'

and

q• = {[- A + (A 2 - 4F)•/2]/2} •/2 ,

where

2(,_• h3) P h3w4 P hw2 A=pw +1'-• and F =12K2GD - D '

A =0 for the thin plate caseß I=mr 2, where r is the radius of gyrationß

A discussion of spot welding boundary conditions and the water-backed case will not be discussed here but

may be found elsewhere. 2

III. POLES IN THE X/k PLANE

The ten poles of the integrand in Eq. (12) and those of Eq. (10) are the zeros of their identical denomina- tors. The zeros or roots of that denominator are de-

termined by the following equation:

(X2 - k2)X/2 (X 4- •2Fx + F2) - Fs - X2Fa = O. (16)

Letting X 2 = X, a bit of algebraic manipulation gives a fifth-order equation in Xwhieh corresponds to the above tenth-order equation in X. This last equation can be put in the more convenient dimensionless form by dividing through by the normalization k •ø and defin- ing Z= X/k 2 =h2/k 2 to get

Z4 z • - (2F• + k•')• + (2F•. +r,' + 2 k•'F•) Z 2

- (2rxr• + 2F• • + F• + r•) •

Z (k F• + + (2r•r• • + r• - 2r•r• - •o = o. (• 7) A function of the form

f(Z) = g n + a•Z "• + ..' + an_xZ + a. = O,

where the a• are real nmbers, may be written as the determinate det(ZI-A)=O'where I is the nth order iden- tity matrix and

_.

.-- a 1 .- a 2 .- a 3 .... an_ • - a n

1 0 0 ... 0 0

0 1 0 ..' 0 0

0 0 1 ..' 0 0

0 0 0 ... 0 0

ß ß , ß ß ß

0 0 0 ... 1 0

'A is an upper Hessenberg matrix and its eigenvalues may be found by computer methods. We find the roots of Eq. (17) in this manner.

The paths of ten typical poles in the X/k plane as a function of frequency are schematically illustrated in Fig. 4. The four looped paths near the origin are greatly exaggerated in the direction parallel to the imaginary axis. The physical interpretation and quali- tative behavior of the effect on the integral and residue contribution of the specific ten poles used for purposes of illustration will now be discussed.

The two real poles, X• and X 6 where X6=- X•, which come in from infinity, at zero frequency, are on the top Riemann sheet. Their contribution to the acoustic ra-

diation corresponds to a slowly propagating wave at the plate surface that does not radiate sound into the farfield.

The four poles, •4, X10 and Xs, X 9 where •9-'--X4 and X•o=- As, which are near the imaginary axis to start with, i.e., when co = 0, move toward the origin as co increases. They then pass near the origin and later ex- hibit increasing imaginary parts while they move near the real axis as complex pairs. The imaginary parts increase in absolute value and then decrease again in absolute value as the real parts steadily increase in

imaginary X/k axis

FIG. 4. Schematic of pole paths in X/k plane.

real X/k axis



absolute value. The poles are on the top Riemann sheet before they start moving along near the real axis. After that, they remain on the top Riemann sheet. These poles describe the exponentially decreasing field near the rib when moving along near the imaginary axis. When moving along near the real axis after they have nearly merged at the origin, their contribution to the farfield is greatly enhanced through an increased contribution to the integral along the real axis and they represent the waves of the first thickness shear mode.

The four poles, X•.,X a and X7,•8 where •?=-•. and =-Xa, which are near the real axis to start with, i.e., when •0 = 0, move toward the origin as •0 increases. The four complex poles merge into four real poles. One set of poles moves toward the origin faster than the second set. The second set then catches up with the first set to create two sets of complex conjugate poles. The imaginary parts of these poles increase in absolute value and then decrease again in absolute value as the real parts steadily decrease.

These pole pairs, which begin near the real axis, correspond to a resonant vibratory response and represent the flexural or first antisymmetric mode propagating in the Mindlin plate. At very high frequencies the flexural wave propagates like a Rayleigh surface wave in the plate. These poles appear in the bottom Riemann sheet. This implies that these poles stop contributing directly. However, it is one of these poles which is the leaky wave pole that produces the coincidence peak in the radiation pattern. Leaky wave poles are defined as those zeros of Eq. (16) that are located on the bottom Riemann sheet of the complex wavenumber plane which appear near the saddle po.int of the complex path of steepest descent. The peak in the integral contribution along the real axis due to the proximity of the pole to the real axis path of integration is easily identified when we use our exact integral evaluation method discussed in Sec. IV. Thus, for the first time we have the existing mathematical technique to justify our claim for a physical reality for the modes we herein describe. Such a physical reality had been questioned • because the usual asymptotic integral evaluation techniques can- not be used to identify any contribution of a pole other than its usually insignificant residue contribution.

Figure 4 is a schematic plot of the paths of a typical set of poles of Eqs. (12) and (10) when Young's modulus is real. Taking into account absorption of acoustic energy by the plate and rib by using a complex Young's modulus instead of a real one may be don• by introduc- ing the structural damping coefficient 7 > 0 such that E = (1- iT)E, where • is the complex Young's modulus. When this is done, the sum of the poles in the Z(=h•'/•) plane gets an interesting imaginary part as a function of increasing 7. This can cause all the poles of Eqs. (12) and (10) to rotate counterclockwise as a function of increasing 7. It can also, for different material parameters, cause the •, •., ha, X6, h•, and ks poles to rotate counterclockwise while the X4, ks, X9, and hxo poles rotate clockwise in the h/k plane as a function of increasing Furthermore, when •. and h a are both real, all the poles except X•. and •? can rotate counterclockwise while

and hv rotate clockwise as a function of increasing These are just some of the more typical possibilities for combinations of pole rotations. One may obtain just one simply stated general rule for pole rotations. Using Descartes' rule of signs from the theory of equations 9 and requiring (for physical reasons) that the leaky wave pole •a rotate counterclockwise, it can be shown that and •6 rotate counterclockwise for frequencies below six or seven times the classical coincidence frequency while they rotate clockwise for higher frequencies.

So far we have only discussed typical cases for the ten poles. Now we will point out other possible physical cases. Examining Eq. (17) and using Descartes' rule of signs from the theory of equations, we see that the pole pair • and • will always be purely real. There will always be a slowly propagating wave at the plate surface that does not radiate into the farfield. The pole

pairs •o and X4, and •9 and k s (Fig. 4) may combine to form four purely imaginary poles representing the propagating waves of the first thickness shear mode. This may occur with •., ha, X?, and ks all either complex or purely real. Also when pH•'c•'=•r•'G, =0 and •a=•. and •?=• with X•., •a, •, and •8 all purely real poles. (Radiation loading must be taken into account in this last case.) These cases exhaust the possible physical cases.

The Timoshenko-Mindlin plate theory has been used throughout this investigation. This plate theory only takes into account the first two antisymmetric or flexural plate modes. No higher flexural modes are con- tained in the theory and all compressional or symmetric modes are absent from the theory. Furthermore, the Timoshenko-Mindlin plate theory only accurately pre- dicts the first two flexural modes up to a frequency- thickness product of approximately 2.5 kHz-m. 4's This can be seen from a comparison with the exact equations of elasticity. For an accurate description of higher flexural modes one must extend the Timoshenko-Mind-

lin plate theory: this would be reflected in the intro- 'duction of four more poles per flexural mode incorpo- rated into the theory. Similarly, one would have to use an extended Lyamshev plate equation to accurately pre- dict the symmetric or compressional plate modes. TM

IV. EVALUATION OF INTEGRALS (EXACT INTEGRAL EVALUATION METHOD)

We now evaluate the se• of integrals of the form (note that •,• =

+• X"(X 2 - k2) •/2 exp(+/X Ix, - x•, i) dX (18) (x ß The method of contour integration in the complex X/k plane is used to evaluate these integrals. The integration is done along a line • above and parallel to the real X/k axis for X<0 and ½ below and parallel to the real X/k axis for X> 0 where • is greater than zero but smaller than any given positive real number no matter how small. This modification of the contour was taken

because any pulse of finite duration (represented by a complex •o) • will cause the branch points and those por- tions of the cuts along the real axis to move in a counterclockwise direction off of the real axis. The integral

1646 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley' Scat(ering from submerged plate. I 1646


over that part of the real axis shown in the contours in Figs. 5 and 6 is related to the integral over the entire real axis by the following considerations.

Application of Cauchy's theorem to the contours shown in Figs. 5 and 6 yields

I•,,tm + I2.t., + Ia,,t., = 2rri( • Ri) , (19) where I•.,m is the integral along the real axis, I2.,m is the integral around the circular arc, Is.tin is the integral down and up the branch cut due to the factor (X •'- k•') •/•' in the integrand, and •i R• is the sum of the residues within the contour on the upper Riemann sheet. As the

radius R- oo, I•,•- I•, and, according to Jordan's theorem/•' I•..tm-O when the integrand is as in this case: one which falls off faster as R-oo than X -• [or (X/k) -• for Fig. 4]. Hence, to evaluate I,, m we must be

able to calculate Is. tin and • R•. Having found the poles in the previous section, the

residues of the integrands at the poles of Eq. (18) are obtained using the fact that the residue of p(z)/q(z), wherep(z) and q(z) are analytic andp(z) •0 at z=z• and where there is a first-order pole at z =zi, is

where the prime denotes differentiation with respect to z. For

P (X) =X• exp(+zX ix, and

q(x) =x • - x•'F• +F•. - F•/(x •' - }•')•/•' - x•'F•/(x •' - •)•/•', the residue at X• is

• x" exp (+ix I xt - x m I ) I -- .- -- 3 2 - 3/2 4A a 2XF• +XFs(X 2 - k•') -•/2 2XFa(X 2 k2) -•/2 +X Fa(X - k •') (20)

where, for the typical case of pole locations used in Sec. III, Xi =X•, 2,4, and X•o or Xs, X6, and Xo, depending on whether the path of contour integration is closed above or below, respectively, and the negative sign goes with X• =2`s, 2`6, and 2`0. We now evaluate the integrals of 3rl • m' Careful evaluation shows that the radical (2, 2- k2) •/•

in the integrand must be evaluated in the various re- gions of the contour integration as follows (Fig. 7)•:

I

A. Case of contour closed above

•+ = •x+ []•x+ []•x+ []•x; 3nl m

• (x •' _ •-)•/•. = _ i(r •. + •.)•/•., (• (x •' - •-)•/•. =_ (x•-_ •.)•/•., © (x •' - •-)•?. = + (x•-_ •.)•/•., • (x •' _ •.)•/•. = + i(• •- + •-)•/•-,

where X = X+iY and [ ] is the integrand.

! /

imaginary • axis

! ?',. '2

! \

•œ .......... x%-: ! • real X axis

I1

FIG. 5. Contour closed above in the

X plane.



imaginary X axis

FIG. 6. Contour closed below in the

X plane.

B. Case of contour closed below

= []dx + []dX + []dX + dX; J•3rll m -i •o k (X) (x"- F.)u•.= i(z •- + k•-),/•. '

• (x•'- •),/•-= + (x •- - F-),/•- '

© (x•'- F-)•/•.=_ (x •. _ F-p/•- ' •) (x•' - F')•/•'=- i(r •' + •)•/•',

where X = X+ iY and [ ] is the integrand. For the case of the contour integration closed above, the integral over the upper branch cut is equal to

(21)

For the case of the contour integration closed below, the integral over the lower branch cut is equal to

fo ø (-1)n+ti"2yn(y2+ k2) 1/2 exp(-Y Ix• - %n I)](y2F• - Fo)dY I3'-•' m = + (y2 + k 2) (y4 + y2Fx + F2)2 + (y2F3 - F5)2

•0 n n 2 2 1/2 2 •(-1) 2X (X -k ) exp(+iXIx!-xml)(X Fa+Fs)dX 2 2 4 2 2 2 2 ' + -(X - k )(X - X Fx +F2) + (F5 + X F3) (22)

We now can write an expUcit expression for I,•,,,. We use Eq. (19) in the limit of R--oo to get

I,.,,, =- •a,,, + 2rri • Ri + , (23) m i

where the I•nlm are given by Eqs. (21) and (22), for the upper and lower sign, respectively, and the •i Ri are given by Eq. (20). We use a mathematical method, which is a generalization of Romberg integration, to evaluate the integrals which come from the integrals along the real and imaginary axis. It should be noted that the contribution of the integrand to the integral

along the imaginary axis dies off very rapidly for n < 4, especially when l and m are not equal to each other.

The method that we have just employed to evaluate the I, tm's may be used to evaluate ß of Eq. (13). It should be noted that the residue contribution to ß is generally only important at near grazing incidence. Neither the integral contribution to • from the integration along the real X/k axis nor the integral contribution to ß from the integration along the imaginary X/k axis can be dis- regarded.

Examining Eqs. (21)-(23), and noting that the poles to

1648 J. Acoust. $oc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley' Scattering from submerged plate. I 1648 ß


argument of (X2-k2)«=-•r/2

imaginary X axis

II

• argument of (X2-k2)«=+;ff2 II I I II II

tl real :X axis

I argument of (X2-k2)«=+•r/2 • argument of (X2-k2)«=-•r/2

II I I I I II

I I

FIG. 7. Phase of CA 2--k2)1/2 on different sides of the branch cuts.

be used for the evaluation of the residues when the path is closed above are just the negative of those to be used when the path is closed below, we may write the general relationship

I;,+• = (-1)"I;,=

Furthermore, from the contact conditions for •b, Eq. (6), one obtains the following conditions-

CoJo + C1J• = CoJ •' + C•. J•'

and

CoZf + c•.z• = cozœ + c•.z[.

Since it follows from J• (-1) J,that J•- + = -Jo and J• =J• it also follows from the above two conditions that • :5œ =0.

V. SOLUTION BY SADDLE POINT INTEGRAL APPROXIMATION

We evaluate the integral for • in Eq. (13) by the method of stationary phase. First we use a conformal transformation fo put the integral into a suitable form for use of the stationary phase approximation. Then the path of integration for the stationary phase approximation will be found. Following this, the stationary phase approximation calculation will be made.

4• is evaluated along the path F in the X plane. Under the conformal transformation X: k sina the path and the • plane are transformed as shown (Fig. 8). The • plane is transformed into the a plane which goes from -•/2 -< ax-< •r/2 with a r ranging from

(I)= (X -- kz)I/2(X4-X2F1 +F2)-Fs-X2F3 (24) is transformed into

(D = •c(e ) x (Cø+C'ksina)exp{-ikR[eøs(a+O)]} kcøsada

+ ikcosa(k 4 sin4a - k•-F• sinZa + F•.)- Fs- k2Fa sin•'a ' (25)

where X= + R sin0 and Y = - R cos0.

The equation for the path of integration for the method of stationary phase C(0) satisfies two conditions placed on the exponential function cos(c• + •). The first condition is that Im{cos(c• + •)} is a constant equal to its value at the saddle point and the second is that Re{cos(c• + •)} < 0 along the entire path C(0). At the saddle point {the point where d/dc•[cos(c• + and o•7 = 0.

Applying the above two conditions, the path C(O) is given by the equation

cos((•, + •) coshc• 7 = 1. (26)

The path is illustrated in Fig. 9. Since the path C(0) results from an analytically continuous deformation of the original path, it crosses the L-shaped branch cut (the conformal transformation X = k sin(• is such that the branch cut remains unchanged in shape) and passes into the firs[ quadrant of the bottom Riemann sheet. The space between the original path, the imaginary axis,

Xyaxis

• =•i•y• )•X axis

path F in the • plane

axis C•y

•x=ksinc•xcoshc•y •y=kcos•xsinhc• Y

2 2 •X axis

path [• in the • plane

FIG. 8. Conformal transformation

from • to (• plane. /



ey axis

branch

cuts

FIG. 9. Path of stationary phase.

I [ C(O) :cos(a X +O)coshay=l I I I

l

a x axis

I I

eX = + •+0

and the path C(0) can contain any of the transformed poles which are located on the bottom Riemann sheet in the strip Re(X/k)<1, 0<Im(X/k)<oo. If such poles appear close to the saddle point, they violate the condition that the major contribution of the line integral along C(0) occurs at the saddle point and one must modify the stationary phase approach accordingly. Before investi- gating this, the stationary phase solution is presented.

Consider the original path F and the path of steepest descent F' (Fig. 10). Now close the path F and the path -F ' at infinity. This means, according to the theory of residues, that an integral of an integrand over the path I', plus the integral of the same integrand over the path LF•, is equal to 2•ri times the sum of the residues at the poles of the integrand enclosed in areas I and 3

y axis

I"

_5

FIG. 10. Paths F and F'.

a,=C•x+ic• Y

a,=sin -1 •k

, o• x axis

minus 2•ri times the sum of the residues at the poles of the integrand enclosed in area 2 (Fig. 11). The poles in area 2 are encircled in the opposite manner of those in areas I and 3. The poles must be on the correct Riemann sheet: the poles must be on the upper Rie- mann sheet, except in the region formed by -F' and the positive c• x and c• axes where the poles must be on the bottom Riemann sheet. This is true when • is large enough for -F' to pass into the first quadrant of the bottom Riemann sheet. The path -F' is dependent upon • through Eq. (26). Hence, the poles that may be enclosed depend upon •. The residues we have to consider at any time also depend upon •. A pole, such as, for example, pole C•xo corresponding to the pole Xxo of Sec. Ill, will lie inside area 2 for • •< •o where

O•o:- a•ox+ cos-•[cosh(a•o•)] -• ,

with a•o=a•o • + iago •, Analogous equations hold for other poles. Therefore, using the saddle point approximation, one obtains:

• =•ø+ • •i ' where

and

-k(Co+ C•k sinO) cosO\•-•] (1 + i) exp(-ikR) +ikcosO( k4 si n40- k2F• sin20 +F2)- (• + k2Fasin20)

+2•i(Co +Cxk sina i) ctnaf exp[-ikR cos((•i + O)] i(-k 4 sin40t! + 3k2F• sin2at - F2 + k 4 sin22a• - 2k2Fx) - 2kFs cosa i

•I, o is the contribution due to the saddle point integral calculation. •I,a• is 2•ri times the residue contribution at the pole (•t. If c• lies in region 2, then the minus sign should be used.

Now the saddle point is at the juncture of the path of steepest descent and the real a, axis. When the frequency is high enough, the poles a•. and a3 (corresponding to poles X•. and X3 from the example of See. III) may

I

get close enough to the real axis to significantly affect the method of steepest descent calculation. In fact, when the frequency is high enough, they may both fall on the real ax axis. When this occurs the saddle point will coincide with the poles for specific values of 0. The case of coincidence, or the ease where the pole is in close enough proximity to affect the saddle point calculation, will be considered in the next paragraph. When absorption by the plate and rib is taken into account by

1650 J. Acoust. Soc. Am., ¾ol. 67, No. 5, May 1980 Barry Lee Woolley: Scattering from submerged plate. I 1650


ay axis

•---•x+ie•y

•=sin -l•./k

FIG. 11. Riemann sheet location of poles.

aX axis

the introduction of a complex Young' s modulus, the poles are affected. Such absorption causes the leaky wave pole a 3 to rotate in a counterclockwise direction around the origin; it also may cause the pole a,. to rotate in either a clockwise or counterclockwise di-

rection around the origin.

Now consider the specific case of Eq. (25) when a saddle point a = a s is near a simple pole a = ao, with a o = ao• +iao•. It can be shown TM that for real U> 0, where

v=

= ]-k cos(2e)]'/"(as - aox -

(• -• h (a s) exp [-iRg(as) ]

1- } x (1 + i)•r •/" exp(-YR •/" U+ iY"/2) dY

+ 2•ih (C•o)exp[+

where

h(as ) = (C• + C•k sinO)k cos(O - ao• - ia9• ) + ik cosg(k 4 sin49 - k" sin•'OF• + F,.) - F• - k•'F3 sin"9

and g•(a•)=kcos(2•). For real U<0,

• _• h(as)exp[-iRg(as)]

x (1 + i)•r •/•' e•p(+ YRZ/•'U+ iY •'/2) dY ,

where U, h(as) , and g(a•) are as previously defined. It should be noted that there is no residue contribution

from a o because it lies on the wrong Riemann sheet; the stationary phase path does not have to cross ao to be deformed into the original path.

When U=O, (I, :•rih(ao) , as can be shown from either the limit of the expression when U<O or the limit of the expression when U> O. This can also be calculated di-

rectly using

VI. EFFECTIVE SCATTERING DIAMETER AND PEAK RETURNS

When the acoustic field (I, is evaluated using the stationary phase approximation without taking into account the effect of leaky wave poles, it has two parts. The first is

_ -k(C o+ C•k sin•) cos•(•r/kR)•/"(1 +i) exp(-ikR) •o-+ ikcosO(k 4 sin40 _ k•.F• sin•.0 +F•.) - (F• + k2F3sin20) ' This is the contribution from the integration along the path of stationary phase. It has the form of a diverging wave. The second is

2•i(C0 + C, sina m) ctnan• exp[-.ikR cos(a m + 0)] E •,• = • .+ i(_k 4 sin4a,, + 3k2F• sin2a, _ F2 + k 2 sin22a• - 2k2FO - 2kFa cosa• '

T•s is the sum of the contributions of the residues from the relevant poles. It has the form of a surface wave w•eh does not e•end into the farfield. Now focus attention on the contribution to the acoustic field labeled •o.

The effective seaffering •ameter for the •vergi• wave •o is defined as the ratio of the mean energy seaffered per unit time W to the mean density of the energy of the incident wave J:

=

For a phne wave,

The mean •ssipated energy is written in integral form:

ea0, where •o is the pre•ously •ven amplitude of the wave scattered by the rib. So

f 2 2 2 +•/2 O[ICol2+2ksinO(ReCoReC•+ImColmCO+ IC•l k sin O]dO • = 2•kR cøs2 -v/2 k2 cøs20( k4 sin40 - k2F[ sin20 + F2) 2 + (Fs + k2F3 sin20) 2

The effective scattering diameter is made up of es- sentially two parts. One part is due to the l Col 2 term in the numerator of the integrand, while the outer part is due to the I Czl •' term in the numerator of the integrand. The presence of the odd function sin0 in the in-

i

tegrand causes the remaining terms in the numerator to contribute insigrdfieantly to the total integral when compared with the integral of the I C o I •' and I C• 1•' terms. The ICo] •' term is due to the longitudinal vibrations of the rib while the I C• 1•' term is due to the flexural vibra-

1651 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee WoolIcy: Scattering from submerged plate. I 1651


10 0

--

Q 10 -•

Z

• 10 -2

tu 10 -2

10-4

4x10-s

RIB THICKNESS

15

,o 5

0

-5

-10

m, -15

• -20 ,'• -25

-30

-35

-40

-45

-50

-55

-60 0.0 1.0 2.0 3.0 4.0 5.0

OMEGA

70-

65

6O

55

5O

45

,,• 40

m 35

I.- 3O 25

2O

15

10

5

0

0.0

ß , ! .... i .... , , • , , i , i , , i

1.0 2.0 3.0 4.0 5.0 OMEGA

FIG. 12. (a) Schematic of the effective scattering diameter versus rib thickness. (b) Return for a rib stiffened plate as a function of normalized frequency f•. (c) Angle of incidence for peak return as shown in Fig. 12 (b).

tions of the rib. Note the dependence of C O on ZF, the impedance of the rib in longitudinal vibrations, and that of Cz On Z,•, the impedance of the rib in flexural vibrations. As a function of increasing rib thickness, there is a gradual rise in the contribution to the effective scattering diameter by the ]C•]•' term while the ]Co] •'

term passes through maxima. This is schematically illustrated in Fig 12(a) for rib thicknesses only up to a. ß 30

of the plate thickness and for a frequency below the classical coincidence frequency of the plate. The maxima due to the ]C•I •' terms cluster near zero rib thickness.

Consider the maximum diffracted return off of a rib

over all possible incident angles. The magnitude of this maximum diffracted return is a function of many parameters. For purposes of illustration, we present in Fig. 12(b), 'a plot of the maximum diffracted return for an air-backed steel plate stiflened by a spot-welded, simple steel rib of width equal to the plate thickness and length equal to 21 times the plate thickness of 0.009525 m. The saddle point approximation was used in the calculations. The peak return at 0.9144 m is plotted against •, the frequency of the incoming wave divided by the classical coincidence frequency of the plate. The broad peaks in the curve are due to flexural resonances in the rib while the narrower peaks are due to compressional resonances in the rib. As frequency (or, equivalently, plate thickness) increases, there is typically a gradual rise in the level of the broad reso- nance return due to rib flexural scattering with super- imposed peaks due to rib compressional scattering. This pattern of peaks and dips in the maximum return versus frequency or plate thickness will be shifted as a function of rib length. The values presented in the graph are a function of rib attachment and rib loading. An experiment has been run to check this type of curve and the theory and experiment are in excellent agreement. •5 (Direct transmission through the plate followed by reflection off of the rib and retransmission through the plate is a competing scattering mechanism which is prominent at small grazing angles' this corner reflec- tor mechanism may be the dominant scattering mechanism for certain rib, plate, and signal parameters.) It might be feasible to determine the structural damping coefficient of the plate due to internal damping by comparison with exact integral evaluation runs of curves which include various amounts of internal damping. The incidence angle • at which the maximum occurs for the case illustrated in Fig. 12. is given in Fig. 12(c).

VIII. DISCUSSION AND RESULTS

A typical example of the theoretical calculation of the monostatic backscattering 4, of a continuous wave plane wave due solely to the presence of a rib on a Timo- shenko-Mindlin infinite plate is shown in Fig. 13. The rib is 0.0088 m thick and 0.1131 m long. It is attached to a 0.0278 m-thick plate insonified bya 37-kHz signal. The return is observed at a distance of 50 m. The re-

turn is plotted for every degree of aspect from grazing the plate from the left (0 ø aspect) to normal incidence (90 ø aspect) to grazing the plate from the left (180 ø aspect).

The solid line shows the results of the exact integral calculation. The dotted line displays the results of the saddle point calculation taking into account pole-saddle point near coalescence. Finally, the dashed line represents the return as obtained by the saddle point calcula-



-14

-24

-.• -54

"' -64

-74

-104 , ' 0

• I I • I • I I I I I

20 40 60 80 100 120 ASPECT ANGLE

140 160 180

FIG. 13. Rib scattered acoustic return using exact and approximate solutions.

tion without taking into account the effect of the' proximity of a pole to the path of stationary phase.

As expected, the exact integral calculation gives greater returns than the saddle point integral approximation calculations do when one is away from the coincidence peaks. The integral approximation, taking into account the effect of the nearness of the pole to the path of stationary phase, is in closer agreement with the exact calculation than the integral approximation which ignores this effect.

The exact integral evaluation method is computation- ally cumbersome compared to the integral approximation methods. But a straightforward application of the exact integral evaluation method can be used for the determination of the acoustic backscattering of a plane sound wave from a rib stiflened Timoshenko-Mindlin

plate stiflened by N arbitrarily situated, arbitrarily loaded ribs. This is the advantage of the method. This problem will be discussed in a future paper.

ACKNOWLEDGMENTS

The author is grateful to P. A. Barakos and S, L. Speidel of the Naval Ocean Systems Center for helpful discussions pertaining to this investigation. This effort was supported by the Naval Sea Systems Command, Code 63R1.

lI. P. Konovalyuk, "Diffraction of a Plane Sound Wave by a Plate Reinforced with Stiffness Members," Sov. Phys. Acoust. 14, 465-469 (April-June 1969).

2B. L. Woolley, "The Effect of Mass Loading on a Stiffening Rib," TR 286 Naval Ocean Systems Center (September 1978).

3A. D. Stuart, "Acoustic Radiation from Submerged Plates, Part I: Influence of Leaky Wave Poles," J. Acoust. Soc. Am. 1160-1169 (May 1976).

4A. Freedman, "Reflectivity and Transmittivity of Elastic Plates, Part I: Comparison of Exact and Approximate The- ories," J. Sound Vib. 59, 369-393 (1978).

5S. L. Speidel, "Wave Propagation Through Thick Flat Layers of Isotropic Material," Masters thesis, San Diego State Uni- versity (1979).

6R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates," J. Appl. Mech. 18 (March 1951).

7V. N. Krasil'nikov, "Some Properties of Wave Processes in a Fluid Half-Space Bounded by an Elastic Layer," in Prob- lems of Wave Diffraction and Propagation (Leningrad U. P., Leningrad, 1965), No. 4.

8D. G. Crighton, "The Free and Forced Waves on a Fluid- Loaded Elastic Plate," J. Sound Vib. 63, 225- 235 (1979).

9j. V. Uspensky, Theory of Equations (McGraw-Hill, New York, 1948).

løL. M. Lyamshev, "Reflection of Sound from a Moving Thin Plate," Sov. Phys. Acoust. 6, 505-507 (1960).

llW. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957), pp. 44- 61.

12G. F. Carrier, M. Krook, and C. E. Pearson, Functiens of a Complex Variable (McGraw-Hill, New York, 1966).

13p. R. Nayak, "Line Admittance of Infinite Isotropic Fluid- Loaded Plates," J. Acoust. Soc Am. 47, 191-201 (1970).

14D. S. Jones, The Theory of Electromagnetism (Macmillan, New York, 1964).

15Private communication from Professor S. I. Hayek of the Applied Research Laboratory of the Pennsylvania State Uni- versity.



acoustic scattering from a submerged plate. i. one reinforcing rib

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