acoustic scattering from a submerged plate. ii. finite number of reinforcing ribs

Acoustic scattering from a submerged plate. II. Finite number of reinforcing ribs

Barry Lee Woolley

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 stiflened Timoshenko-Mindlin plate are presented. Equations are derived for the calculation of the influence of N arbitrarily situated, arbitrarily loaded ribs on the amplitude of the reflected wave. Calculated results for two, nine, ten, and an infinite number of equally spaced ribs are given.

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

INTRODUCTION

This paper presents the results of a theoretical investigation of the backscattering of' an acoustic plane wave from an infinite, elastic Timoshenko-Mindlin plate. This plate is stiflened by any number of ribs. The ribs need not be uniform in geometry, spacing, or material parameters. The ribs may be loaded by dif- ferent masses. The ribs may be fluid or air backed. This investigation can handle the structural damping of the rib and plate.

In a previous paper the effect of one rib on the backscattering of an acoustic plane wave from a Timoshen- ko-Mindlin elastic plate was given. • In the present paper we extend the solution to the more general problem of an arbitrary number of not necessarily identical ribs which may be arbitrarily spaced. The solution is found by the same method used for the case of one rib.

I. MATHEMATICAL FORMULATION

Consider an infinite plate of thickness H which has perpendicularly attached ribs of thicknesses h i and lengths li, respectively. The ribs may have mass load- ings as shown in Fig. 1. Let the plate be contiguous with a liquid-filled hag-space. Also let the ribs extend downward into either a fluid-filled or an air-filled med-

ium. The origin of the orthogonal coordinate system is placed at the plate-liquid surface at the center of the extreme left rib. This extreme left rib is designated as the zeroth rib. The positive Y axis is taken to be below the plate, the positive X axis is taken to be to the right of the zeroth rib along the plate, and the positive Z axis is taken to be into the page along the center of the zeroth rib. Let X,•be the distance of the ruth rib from the zeroth rib.

Now let a plane sound wave arrive from the liquid and impinge upon the plate perpendicularly to the line of at- tachment of the rib:

•o=exp(ikX sinO+ikY cos0),

where 0 is the angle of incidence of the wave measured from the negative Y axis and k is the wavenumber in the liquidø The total field is represented as follows:

•= •o + V exp(ikX sin0 - ikY cos0) + cI,. (2)

Here • is the field scattered by the ribs, and V is the plane-wave *reflection coefficient of the elastic plateø The coefficient V is

v(o) = Zo)/(z + Zo) for an air-backed plate and

v(o)=z/(z+2Zo) for a fluid-backed plate (when the fluid is the same as the fluid loading the plate), where

Zo=poc/cosO and

(1- 12•'(sin•'O- k•)(sin•'O-k•)) Z•=iwHp 1 + ks2•'(sin•' 0 - k•) ' with

C 2 2 2 ,

K C s

(3)

E •2=•r2/12, G =

2(1+•) '

The constant •2G has been introduced by Timoshenko and Mindlin 2 to account for the fact that the transverse

--X axis FIG. 1. Acoustic wave scattering from a ribbed plate.

1654 J. Acoust. Soc. Am. 67(5), May 1980 1654

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imaginary X axis

I i-

tie real X axis

tie

FIG. 2. Integration contour.

shear strains in the plate are not truly independent of the thickness of the plate. The term c• is the velocity of the compressional wave in the plate; c s is the velocity of the shear wave in the plate; D-EHS/12(1- 0 '2) is the cylindrical stiffness of the plate; c is the speed of sound in the liquid; E, c•, and p are the Young's modu- lus, Poisson ratio, and density of the plate material, respectively; and Po is the density of the liquid.

The field potential • must satisfy the reduced homo- geneous wave equation in the liquid above the plate:

(V 2 + k2)½ =0. (4)

The field potential must also satisfy the equation of mo- tion on the entire surface of the plate except on the lines of contact (the lines X=Xm, Y =0 in Fig. 1). In writing the equation

d__ V4½+F1 d• '2½'1'F2 •Y ½-qS½+FaV2½+F4½:0 dY '

the Timoshenko-Mindlin plate theory has been used with

(12D+HaK2G) tPH2co •' 12K2G ) F• = pw 12•2GD , F 2 = pHw 2 - 12•GD '

qS = c02po D Fa= w2Po F4= POP H2c04 ' •GH ' 12K2GD

.1

The field potential ½ must also satisfy the following two boundary conditions

kaY } •,,.o •,.-o ' \aXay •m.o aXay •.,-o' (6a)

The following notation has been employed

lim/ø• =(0• o¾" \at" ' • --} 0 •:0- • -0

The preceding two conditions expressed as equalities of the derivatives of • are simply this physical con- dition: at the lines X =X•, Y = Y• =0 the plate displace- ments must be continuous and the angles of inclination of the cross section of the plate must be continuous, respectively. We also have

- = -i • (6b) \ a2XaY •m.o \a2XaY •m_ o D oXaY'

, aSXOY xm.o •aSXOY xr•_ o D aY ' The preceeding conditions are contact conditions for the moments and forces at the contact line where ZF• and Z M• are the impedances of the ruth rib in longitudinal and flexural vibrations, respectively. Finally, the ra- diation extinction principle at infinity must be met.

The ribs are excited by the incident wave in the phase described by the factor exp(ikX,, sin0). The cylindrical waves propagating from each rib into the adjacent liquid can be described analytically in the same form as the wave excited by one rib on the plate. • Thus, by the linearity of the problem, the field generated in the liquid by the vibrations of all the ribs can be written as follows:

*= Z fr [(Cø)•+ (C')mX]exp(ikXø' sin0)exp[iX{X-X, + Y(X 2 - k2) •/2] m:o (>,2_ k2),12(X4_ X2F, + F2 ) _ F5_ X2Fa dX

= • [(C0) • + (C•),,X]exp[ixX+ Y(x 2 - k2)X/2]exp[-iX•n(X- k sin0)] ax, (7)

where Fs=q • - popH2w4/12K2GD=q •- F 4. The path r is shown in Fig. 2.

Using expression (7) in Eq. (2) for the total potential and substituting this into conditions (6b) and (6c) for each rib, we obtain the following equations which may be used to determine the unknown coefficients (Co) • and (Cx)•:

• 2J](Cx),exp(+/Xtksin0) - wZ.,t E [(Co),,l•,•+(Cx),,I],,,]exp(+i•.,ksinO) -wZ•' (Cx),d•exp(+i•tksinO) • =o D • =o D

and

wZ•, 2 [-(Co)• 11, + (Cz)mI•,•] exp(+/X•k sin0) - i D "• rn=/+l wZ•t k2 sin0 cosO(1 - V) exp(+ ikXt sin0)) = 0 (8a) D

•.. 2iJ](Co)texp(+iXtksinO)-i wZr, •. [(Co)ml;t•+(Cl)•Iit•]exp(+iX,,ksin0) •=o D ,._-o

+(Co),J•exp(+iX, ksine)+ • (Co).•J•exp(+iXmksine)+ikcos•(1- V)exp(+iX,ksine =0, m=/+l

(8b)

1655 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley: Scattering from submerged plate. II 1655


i ! i

!

imaginary 3• axis

I i \

t +k ...... -J real X axis

imaginary X axis

k I' ; .... _% ...... real X axis

t I

/ !

FIG. 3. Branch cuts and the r* and

r paths.

where

frX"(X* - k*) u* • = , (X 3-- •2)I/2(X4 -- X*'F, +F•.)- F s - X*F 3

fr X"X* - k2) u* exp(+iX IX; - X•l) d•

are discussed in Sec. IV of Part I of this paper. It should be noted that J*. and I•.tm have residue terms which have not been explicitly indicated. Also note that I,,.,., = J,,. In Sec. IV of Part I of this paper the following relationships were proven' J;,=(-l)"•,J;=J•.=O, and I•,t•= (-l)"•t=. These relationships have been used in obtaining Eqso (8a) and (8b). The paths r* are shown in Fig. 3. The impedances Z•t and Z•t of the/th rib in longitudinal and flexural vibrations respectively, are given by the following expressions:

Z•'t=iptht 2pt(l- cr] tan It 12pt(l- cry) ,

ip ,ht[E t/P,(1 - (r•)]3/•(1 - cos{•v/t [E t/pt(.1 - (r •)]"/*'•f cosh{•vlt [E t/Pt (1 - (r •)]4/,.})

(9)

The mathematical procedure outlined in Part I, Sec. IV, indicates how to convert the integration for J*,or I e along the path F* or F- to line integrals of a simpler nature. This same procedure may be applied to the expression for •, Eqo (7)ø First of all, Cauchy's theorem is used to convert the path of integration for ß from the path F to the

r ß path F* or F-. The choice of which of the two paths, or F', to use depends upon whether X is positive or negative, respectivelyo In the case of X negative, • is

ß = • exp(+ix,,k sin0) •=0

(fo •- 2[(C0), - i(C•),,y]exp[+ y(X -X•)]{(y 2 + k2)•/2(y 4 +y2F t +F•)cos[Y(y 2 + k2) •'2] - (-F• +y2F•) sin[Y(y 2 + x (y, + ½.) (y• _ y,• + •,), + (• _ y,•), •y

fo • - 2i [(C0) • - (C•)• x] exp[-ix(X- X•)]{(k •' - xZ)UZ(x • - x•'Fz +}'a)cøs [Y(k2 - x•')•/•'] + (Fs +xZF• ) sin [Y(k2 - x•')u2] } '{' (k 3- X2)(X 4'-- x2.F1 '{' F2 )2 '{' (Fs + x2Fs) 2 dx

+ • 2rri (X•'-k2)U•'[(Cø)"+(Cz)r"X]exp[iX(X-Xr")]exp[+¾(X2-k2)•/2]) 5X s - (3.F• +4k2)Xa+(F2+2k2F•)X- 2X•3(X•' - k2) •/•' ' (10) X=X5, •[6, and X 9

where the poles Xs, >,•, and •t 9 of Eq. (7) canbe seen schematically for a typical case as a function of frequency in Fig. 4 along with the other seven poles.

The solution of the problem is complicatedby the intricate form of Eqs. (Sa) and (Sb). It is greatly simplified for a small number of ribs or for an infinite number of equally spaced identical ribs.

II. EXAMPLES

Now let us first consider; the case of an infinite number of equally spaced identical ribs. In this case, the conditions for excitation of the ribs and, therefore, the processes of generation of cylindrical waves in the liquid differ only in phase. Because of this, there are only two unknown coefficients, Co and C•, as in the case of only one rib. Takingan arbitrary rib as the designated zeroth rib, the field produced in the liquid by the vibration of all the ribs can be written as

1656 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley: Scattering from submerged plate. II 1656


•= • fr (Cø+C•'X)exp[iXX+ Y(X"- k")•/"]exp[+imXo(X- ksin0)] m=• ( }t•'-- k•')•/•'( X4- }t•'F• + F•.)- F s - •t•'F• dX

fr (Cø+C•X) exp[+iXX+ Y(X"- k") •/"] "fr (Cø+C•X) exp[+iXX+ Y(X2- k")•/"]exp[-imXø(X- ksin0)] + (X •'-- k•')v•'(X 4- X•'F• + F•.)- •'s- X•'F• dX + • (X •'- k•')v•'(X 4 X•'Fz + F•.) F s •=1 .... dX,

(11)

where Xo is the distance at which the ribs are spaced.

Equation (11) may be rewritten in the more convenient form

4• = f(X) exp[+iXX + Y (X2- k2) •/2] l _ exp[+iXo(X_ k sino)] dX + (X) exp[+iXX + Y(X2 _ k2) •/2] 1

1 - exp[-iXo(>,- k sin0)]

- frf(X)exp[+iXX + Y(X•'- k•')x/•'], where

f(x) = (Co +c•x)/[(x '• - k'•)•/"(x • - x'-•'• + •.)- •',- x'-•'o].

It can be easily seen from the above equation that • = 0. This is not surprising for a problem of such great symmetry, although another author has obtained adifferent result. 3 The form of Eq. (7) reveals that the scattering from an array of ribs can be physically thought of as a product of a measure of the scattering from a single rib and a measure of the scattering from an equivalent array of line scatters. However, extreme care must be taken in physically reason- ing from such a point of view since the rib interaction terms significantly affect the calculations.

Now let us consider the two-rib case. For the two-rib case, Eq. (11) reduces to 1

•= •' Jr [(Cø)n+ (C•)mX]exp[iXX+ Y(X2-k2)•/2]exp[-iXr•(X-ksinO)] dX •=o (x '• - k'•)•/'•(x • - X"F• + F,.) -F• - where Coo , Co• , C•o, C n are found from Eqs. (8a) and (8b). LetX o = 0 andX• = •o. Then the C i i' s are given by the following formulas'

-i wZ r• (1 - V) cosO Cøx - 2DJ• + w(Z •o + Z Fx )J;

(12)

Cll = -i{coZMxk"(1 - V) sin0 cosO[2DJ• + co(Z ro + Z F•)J•>] - co'•Z•to Z •xk(1 - V) cos0I[o•}

Coo= -iwkZro(1 - V) cosO 2DJ• + ooZ Fo,J5

oozed; exp(+i•ok sin0) Co • 2Dd• + coZ sodg

(13)

Czo= -iwZgok•'(1 - V) sin0 cos0 2DJ• + WZ MoJ•

wZstoI•o: exp(+iXok sin0) Co x _ 2DJ• + wZ •toJ• coZ,•oI•o • exp(-iX_-ok sin0) Ctt

2DJ• + wZ •toJ• '

imaginary }k/k axis

x

X7 X 3 X? X 3

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

real }k/k axis

-4

-14

-24

-74

g ß ..•

%!

0 10 20 30 40 50 60 70 80 90

ASPECT ANGLE

FIG. 5. Two-rib case.

1657 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley' Scattering from submerged plate. II 1657


-20

-30

-40

co,, -50

• -$0

-70

-80

-90

-lOO o

I

1•0 20. I I I 610 I I I 30 40 50 70 80 90

ASPECT ANGLE

FIG. 6. Nine-rib case.

-17

-27

-37

-, -47

m -57

-67

-77

-87

-97 0 10 30 40 50 60 90 I ASPECT ANGLE

FIG. 7. Ten-rib case.

In the absence of the second rib, Coo and C•o reduce, respectively, to Co and C• of the one-rib case. The first term on the right-hand side of the equations for Coo and C•o represents the longitudinal and flexural impedance contribution, respectively, for the first of the two ribs. The next terms represent the interaction con- tributions of the impedances of the second rib on the first rib. Because of the boundary conditions (6a), which cause the vanishing of J• and J•, Co and C• of the one-rib case are only dependent upon the longitudinal and flexural impedance, respectively; the Coi'S are not dependent upon the flexural impedances of those ribs which are to the right of the ith rib. This last state- ment is true no matter how many ribs there are.

A typical example of the theoretical calculation of the backscattering ß of a plane wave due to the presence of two ribs on a Timoshenko-Mindlin infinite plate is shown in Fig. 5. Both ribs are 0.0088m thick and 0.1131 m long. Both ribs are attached to a 0.0278-m-thick plate insonified by a 37-kHz signal. The return is ob- served at a distance of 50 m. The return is plotted for every degree of aspect from grazing the plate from the left (0 ø aspect) to normal incidence (90 ø aspect). The angie is measured as shown in Fig. i from the center of the left rib. Target strength at 0.9144 m, (1 yard) is plotted.

The solid line shows the results when the ribs are

5.7323 m apart. The line of long dashes displays the results when the ribs are 2.8661 m apart. Finally, the line of short dashes represents the return when the ribs are separated by a distance of 0.5468 m.

In conclusion, two more examples are displayed in Figs. 6 and 7. Figure 6 shows the backscattering • for nine equally spaced ribs placed 0.5732 m apart. The ribs, plate, and signal have the same parameters, except for spacing, as in the above two-rib case. This is

also true of the ten-rib case displayed in Fig. 7, where the equal spacing between ribs is now 0.5468 m.

The incident plane waves for the examples of Figs. 5-7 are continuous waves.

II I. CONCLUSIONS

It has been shown that it is possible to calculate the backscattering of a plane sound wave by a rib stiflened Timoshenko-Mindlin plate due to N arbitrarily situated, arbitrarily loaded ribs. The procedure is complicated by having to find the inverse of a 2N by 2N matrix [Eqs. (Sa) and (Sb)] and by computationally difficult integrals, Eq. (7) and the J]'s and I,•m s. It is apparent from the calculations, from the form of Eqs. (Sa) and (Sb), and from knowledge of the magnitudes of the I•m s that the longitudinal and flexural impedances of any given rib can be significantly affected by the presence of other ribs. Thus, inclusion of rib interaction effects is nec- essary to correctly determine scattering levels from rib stiflened plates.

ACKNOWLEDGMENT

This effort was supported by the Naval Sea Systems Command, Code 63R1.

lB. L. Woolley, "Acoustic Scattering from a Submerged Plate. I. One Reinforcing Rib," J. Acoust. Soe. Am. 67, 1642--1653 (1980).

2R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates," J. Appl. Mech. 18, 31- 38 (1951).

3p. R. Stepanishen, "The Acoustic Transmission and Scat- tering Characteristics of a Plate with Line Impedance Di s- continuities," J. Sound Vib. 58, 257-272 (1978).

1658 J. Acoust. Soc. Am., Vol. 67, No. 5, May 1980 Barry Lee Woolley' Scattering from submerged plate. II 1658


acoustic scattering from a submerged plate. ii. finite number of reinforcing ribs

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