backscattering from a plate with a thickness discontinuity

Backscattering from a plate with a thickness discontinuity Barry Lee WoolIcy

U. $. Naval Ocean Systems Center, San Diego, California 92152 (Received 3 January 1980; accepted for publication 20 March 1980)

The general theory of backscattering from the impedance discontinuity presented by a plate made up of two semi-infinite sections of different thicknesses is given along with some illustrative calculated results.

PACS numbers: 43.20.Fn, 43.20.Bi

INTRODUCTION

This paper presents the results of a theoretical in- vestigation of the backscattering of an acoustic plane wave from the line impedance discontinuity presented by a plate made up of two semi-infinite sections of different thicknesses. To the author's knowledge, no previous work has been done on this subject. Timoshenko- Mindlin plate theory x is used in this analysis. The solution can handle either the liquid-backed or the vacuum-backed cases.

I. MATHEMATICAL FORMULATION

Consider an infinite plate with two semi-infinite sections of thickness h and ha, respectively (Fig. 1). The plate is liquid-loaded and may be vacuum or liquid- backed. The origin of the coordinate system is placed at the upper plate-liquid interface above the thickness discontinuity of the plate. Now let a plane sound wave arrive from above and impinge upon the plate perpen- dicularly to the line of the thickness discontinuity of the plate. Its contribution to the field potential is as follows:

, (1)

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

•2 = d2o + VzH(-X)e i•x $ tnS-i/• Yco$8

+ VaH(X)eikXstne'ikrcøae+ •.

Here • represents the field scattered by the thickness impedance discontinuity and Vx and V•. are the plane- wave reflection coefficients of the semi-infinite plate

LIQUID

ha •

LIQUID OR VACUUM

- • X-AXIS

Y-AXIS

FIG. 1. Incident plane wave on an impedance (thickness) discontinuity.

sections of thicknesses h and ha respectively. H(x) is the Heaviside step function'

H(X)={O, X<0; 1, X>0

Vx and Va are given by

,.(0)= z o . (3) Zt, + Zo '

for a vacuum-backed plate and

Zp V• o• 2(0) = Z•+2Zo ;

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

Zo=PoC/CosO and

Zt, = -iwp 1- 9•'(sin•'O- k])(sin•'8- k•) 1 + k•9•'(sin•'O- •) times h or ha,

for Vx or V•., respectively, with •=w/w½, w½=c•'(p/ Dx)X/2h•/2, or c2(p/D2)X/2ha •/•, for Vx or V2, respectively,

k• = c"p(1- •r •') c" c"p _ c" = 2 ' ks 2- K2G 2 a , E c t K C s

v 2 E K 2 - and G-

12 ' 2(1+•r)'

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

shear strains in the plate are not truly independent of the thickness of the plate. c t is the speed of the com- pressional wave in the plate (or the speed of p waves in an infinite medium made of the same material as the

plate) and c s is the speed of the shear wave in the plate. D = E/[12(1 - a•')] times h a or ha s (for the right or left semi-infinite sections of the plate, respectively denoted by Dx or D•.) is the flexural rigidity of the semi-infinite section of the plate, c is the speed of sound in the liquid, E, a, and p are the Young's modulus, Poisson's ratio, and density of the plate material, respectively, and P0 is the density of the loading liquid.

The field potential • must satisfy the reduced homogeneous wave equation in the liquid above the plate'

(V •' + k•')½ =0. (4)

The field potential must also satisfy the equation of mo- tion on the entire surface of the plate except on the line of the thickness discontinuity (the z axis in Fig. 1);

345 J. Acoust. Soc. Am. 68(1), July 1980 0001-4966/80/070345-05500.80 ¸ 1980 Acoustical Society of America 345

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 11:39:58

d V4•+ F• d d dY •-• v•'• + F•. • • - qS• +FsV•'½+ F4•b=0 , for X <0

+F•Va•+F•=O, forX>O.

In writ•g the above equations, Timoshe•o-Mindlin plate theory has been used with •

(12D•+hSgaG) (12Dah asgaG ) Fx = pw • • 12gaGDx F• = pw • , , 12gaGDa

pw•'h (ph•'w •' _ 12K•G ) ( pha •' w •' - 12•'G) , F•a = pw•.ha 12•.GD•.

qS waPo qO5 waPo Fs= waPo • = waPo , = Dx ' = D a ' •aGh ' •aGha '

pQph•' ('04 and F•4 = pøpha•' ('04 F4= 12•:•.GD, 12•:•'GD•. ß The field potential ½ must also satisfy the following

two boundary conditions,

, +o \a Y ] -o \OXa I/] .o aXa Y / -o' The following notation has been employed:

.•o OY"x:o_. = \a-•-!-o

(6a)

and

lim \a-•] \a Y"] ' •--* 0 X=0+• +0

The two boundary conditions expressed in Eq. (6a) as equalities of the derivatives of • represent the followi• physical conditions: at the line X =0, Y =0, the plate displacements must be eont•uous (first condition) •d the •gles of •elin•ion of the cross section of the plate must be eont•uous (second condition). We also have:

and

/ 04• • / 04½ ) (6C) \axar/.o: o, Dxa r -o ' These are contact conditions for the equal moments and forces at the contact line of the two semi-infinite plate sections of different thicknesses.

To solve all of the above equations ß is sought in the form of the following sum of integrals which solves the homogeneous wave equation and the radiation extinction principle at infinity:

• = fr H(-X)f<(A)exp [iXX + Y(X •' - ka)x/•']dX

+ fr H(X)f>(A)exp [iXX + Y(A •' - k•')'/•']dX. (7) The functions f>(x) andf<(X) must be defined from bound-

ary and contact conditions. A contour r is used from _•o to +oo in the X plane along which the integral will be evaluated. Contour F was selected as shown in Fig. 2. The integrantis are not easily evaluated along this contour. As shown in Fig. 3, we have used a branch cut in the X plane (corresponding to a branch cut from +k a to _•o along the real X •' axis in the X •' plane) such that, Re(X •' -k•') '/•' >•0 on the entire top Riemann sheet. This condition corresponds to the requirement that the field dies off at infinity (i.e., y__oo); f<(X) andf >(X) must die off at infinity faster than I x] -s if both continuity of dis- placement and the radiation extinction principle are to be satisfied.

We will use Cauchy's theorem as explained in previous work •' in order to relate the integrals along the contour F to the integrals along the contour F* (or F-) and the residues of the integrands at the poles on the top Riemann sheet enclosed by F and F + (or F-). This allows easier evaluation of the integrals. The choice of whether to close the paths above (i.e., U +) or below (i.e., F-) depends upon the requirement that the fields f<(X) and f>(X) die off at infinity. When computing the fields at points withX >0the integrals [only the one withH(X) is nonzero] in Eq. (7) must be computed by closing the integrals above using the path F + so that it embraces all the singularities lying on the top Riemann sheet which are below the contour F while excluding the lower branch cut.

When the total potential is brought into compliance with Eq. (5), expressions forf<(X) andf>(X) are obtained (where the residue contributions are implied by the notation):

ft. f< (>,) [(>,•' - ka)z/a(X 4 _ XaFz + F a) - F s - XaFa] x exp[iXX + Y(X •' - k•')x/•']dX =0

for X, 0 (8)

- + )- - ] x exp[iXX + y(x2 _ k2)Z/•.]d x =0

for X•0,

where

F5 = qS oh2 C04p0- and F• ø = qOS P ha2 C04p0 - 12•:•.GD - 12•:•.GD ß

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

F<(X) =f<(X)[(X •' - k•')z/•'(X 4- X•'Fz + F•.) •- F s - X•'Fa],

F>(X) =f>(X) [(X 2 - k2)l/•'(X4- X•'F• + Fga)- Fs ø - X2Fg]

IMAGINARY XAXIS

_[

FIG. 2. Integration contour.

REAL kAXIS

346 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Barry Lee Woolley: Backscattering from a plate 346


IMAGINARY X. AXIS

/ /

i I L

IMAGINARY X. AXIS

' +k ', -k '--- - - "-- • '- •r -- -.•

REAL ', XAXIS '-,

k F

// REAL /

_ •. •_ x AXIS

with powers above the first. Thus, the desired functions f<(X) and f>(M must be of the form

y<(z) = (x•. F.),/•.(x , Cø + c•x - - Z•'F• + F•.)- F 5 - X•'F3 (9)

FIG. 3. The r + and F' paths.

are regular over the entire complex plane, excluding the infinitely-removed point where they can have a pole of finite order. This means that F<(Z) or F>(X) is some polynominal in X. The requirement of continuity of the potential and its derivatives of the second order inclu-

sively means that F<(X) or F>(X) does not contain terms

For (I,'s final definition it is necessary to find the coefficients Co, Cx, C•., and C s.

Two equations for the four coefficients Co, C•, C•. and C s are found by substituting the total field into the conditions of Eqs. (6b) and (6c). An additional two equations are found from Eqs. (6a). Solving the resultant four equations, the four unknowns are found to be

o+ s sinS0) i(V•. - V•)k cos0(d• + - o r 0 k Co= or Te] oO+ _ or ;or 3o+ ,

C 1 -- 1 D2jO k 3 cosO sin20(D2V2- D• Vz)J2 ø+- ik •' cos0 sin0(V•.- Vx)D•.J ø+ 3

i(V•.- V•)k cos0(J ;-J •k s sinS0) C•. = j,•oro+ j 7:)jo+ , 0 -- 3

(lO)

C3 = 1 ( o+ - _ ik3 ' - D•.j 3 j•. D•j•j20+ cosOsin20(D•.V2-D•Vz)J2 - ik 2 cosOsinO(V•.- Vz)D•J 3

where

fr X"(X2- k2)Z/2dX fr X"(X2 k2)Z/2dX J*. = (X 2 ka)•/a(X 4 XaF• + F2 ) F5 XaF3 and jo, = - _ _ _ ,o ß _ nß - _X•z + o_ 2 o

C• be calculated as they have been in previous work. 2 The poles of the integr•d in Eqs. (11) and those of Eqs. (9) are identical. They have been discussed in previous work. 2

When the coefficients Co, C•, C2, and Ca are obtained •d the explicit forms off<(X) andf >(X) are defined, previous methods 2 may be used to evalu•e Eq. (7) for •. For X <0, ß is given by

fr X)H(-X) exp [i;L + r(x - (Cø(5[k2)l/2(X 4 X2Fi+F2)_Fs X2/;s

= fo" -2i(Cø- iCzZ)e*ZX{(Z2 + k2)Z/•'(Z4 + Z•'I•'I +/•'•) cøs[r(Z•' + k•')z/•']- (-F$ + Z•'Fa) sin[Y(Z •' + k•')z/•']}dZ (j2 q. k 2) (Z4 q. Z•.Ft + F•.)•. + (-F• + Z•'Fs) •'

fo • -2i(Co- C•Z)e-'ZX{(-Z2 + k•')•/2(Z 4- Z•'Fi + F•) cos[Y(-Z •' + k•')•/•'] + (F• + Z•'F•) sin[Y(-Z •' + k•')x/•']}dZ q' (-Z 2 q' k9') (Z 4- Z•'F• + F•.) • + (Fs + Z•'Fs) •'

(11)

• (co+c •) exp [i M( + Y(X 2 - k2) z/2 ] + 2•i (4x a - 2XFz)(X 2 - k2) •/2 + )[()[4 ._ X2F1 q. F2) (•t2 ._ k2)-1/2 ._ 2XFs where

poles on top Riemann sheet enclosed within r and r-.



For X >0, ß may be obtained from the physical equiva- lence shown in Fig. 4 and stated as follows-

q,(•,h,ha) = q,(-•,ha, h) .

This is an exact integral evaluation of •. ß may also be evaluated by a stationary phase method. •'

II. DISCUSSION

Since the boundary conditions Eqs. (6b) and (6c) refer to the median planes of the two sections of semi-infinite plates, our solution for scattering from thickness impedance discontinuities may be inaccurate at high ratios of the thicknesses. Also our theory is limited by the accuracy of the Timoshenko-Mindlin plate equation •"4 which only describes the first two antisymmetric modes of plate vibration. Furthermore, Timoshenko-Mindlin plate theory only accurately predicts the behavior of these modes at lower frequency-thickness products. The Timoshenko-Mindlin plate equation gives very inaccurate modal angular positions for the second antisymmetric mode for steel plates above a frequency that is approximately ten times the classical coincidence frequency for the plate. These inaccuracies of the Timoshenko-Mindlin plate equation at lower frequencies and the introduction of the higher antisymmetric modes described by the Rayleigh-Lamb equation 5 at higher frequencies can be accomplished by suitable modification of the Timoshenko-Mindlin plate equation (as will be shown in a future paper) if one wants to do scattering problems at frequencies higher than ten times the classical coincidence frequency of a steel plate. The introduction of the description of higher antisymmetric modes complicates the algebra [e.g., in Eq. (5)] but does not change the arguments of this paper. Sym- metric modes are not taken into account at all in the

Timoshenko-Mindlin plate theory. This is a serious defect because the first symmetric mode of plate vibration is present down to zero frequency and has a prom- inant peak centered at about three times the classical coincidence frequency of a steel plate. At about ten times the classical coincidence frequency of a steel plate, the first symmetric mode is the most prominent mode of plate vibration in a steel plate and is highly dispersive. The author hopes to show in the future that all these defects may be corrected by suitable modification of the Lyamshev equation. 6 Such a modification would allow the backscattering from a plate with a thickness discontinuity to be calculated for the symmetric modes. The calculation would start with a mod-

he

x<o x>o

FIG. 4. Physically equivalent situations.

-27

-37

Z•-47

-57

-•7

-77

-87 0 20 40 60 80 100 120 140 160 180

ASPECT ANGLE

FIG. 5. Scattering from a thickness discontinuity, 6.7-kHz signal, air-backed.

flied Lyamshev equation in place of Eq. (5) and proceed in a manner exactly analogous to the development in this paper. The value of the present analysis lies in the fact that the .first antisymmetric mode is the dominant scattering mode in the range of interest for most current investigators.

III. CONCLUSIONS

The backscattering from the line impedance discontinuity presented by a plate made up of two semi-infinite sections of different thicknesses has a peak which is primarily a function of the frequency of the incoming acoustic wave and the thickness of the thickest section

of the plate. As would be expected, when the ratio of thicknesses of the two plates is held constant, the peak backscattering level increases with increasing frequency-thickness product and the angular window over which the backscattering level is significant decreases with increasing frequency-thickness product. The peak backscattering level drops rapidly when the incoming wave is below the coincidence frequencies of both plates. When the ratio of the thicknesses of the two semi-infin-

ite plate sections becomes 1, the scattered return from the thickness impedance discontinuity becomes nonex- istent. When the ratio of the thicknesses of the two

semi-finite plate sections is allowed to go to infinity,

28

O -12 O

-22

-32 ß ' 100 120 '1•0

ASPECT ANGLE

FIG. 6. Scattering from a thickness discontinuity, 80-kHz signal, air-backed.

348 J. Acoust. Soc. Am., Vol. 68, No. 1. July 1980 Barry Lee Woolley: Backscattering from a plate 348


x

x

x x x

x

•3 x + + + + x +

Z + + x •'" x x x •) + x x • x

o

-17 •

+--X>0

,:7• , •--X <0 - 7 '- ß • - , ß 0 10 20 • • • • 70 • • 1•

ASPECT ANGLE

FIG. 7. Peak scattering returns as a function of frequency.

the scattered return from the thickness impedance discontinuity has a finite upper limit. Slight asymmetries in the backscattering are evident when grazing angle comparisons are made between the case of insonification from the thinner plate section side and the case of insonification from the thicker plate section side. The difference in backscattering target strength levels for the vacuum-backed over the liquid-backed case is not a simple constant.

IV. EXAMPLES

In Figs. 5-8 we plot the scattering from a thickness impedance discontinuity for a vacuum-backed plate. The plots show target strength at I yard (0.9144 m) ver- sus aspect angle. The thicknesses of the two adjoining semi-infinite sections of plates for all the plots except Fig. 8 are 2.858 cm (on the side of 0 ø aspect angie, where X< 0) and 1.588 cm (on the side of 180 ø aspect angle, where X > 0). Since the calculations were made by using stationary phase integral approximations, the peaks are lower than what one would obtain from an exact integral evaluation.

Figure 5 shows the return for a 6.7-kHz insonifying wave. The frequency of this wave is below the coincidence frequencies of both plates. Note the low returns and the absence of coincidence peaks. Figure 6 shows the return for an 80-kHz insonifying wave. The first antisymmetric or flexural peaks for both plate sections appear in the figure, and the second antisymmetric or shear peaks for the thicker of the two plate sections appear near 80 ø and 100 ø aspect. The shear peaks for the thinner plate section first appear at a higher frequency. Figure 7 plots the peak scattering returns from the thickness impedance discontinuity as a function of the

25

• 15

n- 5 o

z) -5 o

-15

-25

20 40 60 80 100 120 140 160 180

ASPECT ANGLE

FIG. 8. Scattering from a thickness discontinuity in which one of the two plate sections is infinite in thickness, 37 kHz, air- backed.

frequency of the insonifying wave. The "+" denotes the highest of the peaks for aspect angles between 0 ø and 90 ø and the "X" denotes the highest of the peaks for aspect angles between 90 ø and 180 ø. The pronounced dip in the peaks for aspect angles between 0 ø and 90 ø at 50 kHz is not typical of such plots and is due to the specif- ic ratio of the plate thicknesses that was chosen. Fin- ally Fig. 8 illustrates the scattering target strength for the thickness impedance discontinuity when the thicknesses of the section of the plate with X < 0 is infinite and the thickness of the section of the plate with X >0 is 1.073 cm with the insonifying plane wave having a frequency of 37 kHz.

ACKNOWLEDGMENT

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

1R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotopic, Elastic Plates," J. Appl. Mech. 18, No. 1, 31-38 (1951).

2B. L. Woolley, "Acoustic Scattering from a Submerged Plate, Part I: One Reinforcing Rib," J. Acoust. 67, 1642-1654 (1980).

3A. Freedman, "Reflectivity and Transmittivity of Elastic Plates, Part I: Comparison of Exact and Approximate The- ories," J. Sound Vib. 59, 369-393 (1978).

aS. L. Speidel, '•Vave Propagation Through Thick Flat Layers of Isotopic Material," Masters thesis, San Diego State Univ- ersity (1979).

5R. D. Mindlin, '%Vaves and Vibrations in Isotopic Elastic Plates," in Structural Mechanics, edited by J. N. Goodier and N. Hoff (Pergamon, New York, 1960), pp. 199-232.

6L. M. Lyamshev, '•Reflection of Sound from a Moving Thin Plate," Soy. Phys. Acoust. 6, No. 4, 505-507 (1960).



backscattering from a plate with a thickness discontinuity

Documents