acoustic radiation from fluid-loaded elastic plates. ii. symmetric modes

Acoustic radiation from fluid-loaded elastic plates. II. Symmetric modes

Barry Lee Woolley a) Naval Ocean Systems Center, San Diego, California 92152

(Received 10 September 1981; accepted for publication 10 April 1982)

A mathematical method for extending equations of motion to include higher order symmetric modes is presented and discussed. This method is illustrated by the development of equations of motion for the first two and the first three symmetric modes of plate vibration. The form of the Lyamshev plate equation of motion is used as a starting point for the development of the new equations. Transmission through an infinite elastic plate is calculated for developed plate equations of motion incorporating four antisymmetric and three symmetric modes of plate vibration.

PACS numbers: 43.40.Dx,43.20.Bi

INTRODUCTION

As noted in a previous paper, • it is difficult to obtain solutions of problems of wave propagation in bounded elastic media using the three-dimensional theory of elasticity. Solutions have only been obtained for infinite trains of har- monic waves traveling along infinitely long plates, semi-infinite media, and bodies bounded by circular or elliptical cylindrical surfaces. 2 If an additional boundary is introduced, such as an impedance discontinuity on an infinite plate (for example, a rib or change in plate density or thickness), a superposition of an infinite number of modes is generally required in order to satisfy the additional boundary conditions. The complexity of such problems as well as those asso- ciated with the treatment of transients has led to the estab-

lishment of various approximate plate theories. These strength-of-material theories of plates involve a finite number of modes and result in simpler frequency equations. These theories have to be used to describe the wave motion of

plates if one wants to calculate the acoustic scattering from liquid-loaded plates with impedance discontinuities. But these approximate plate motion equations have their limita- tions:

We have previously dealt with the case of plate equations of motion for antisymmetric modes of plate vibration. • Now we will concentrate on plate equations of motion de- scribing symmetric modes of plate vibration. We will improve and extend existing plate equations of motion for symmetric modes of plate vibration. In particular, we will improve and extend the Lyamshev plate equation of motion 3 for the first symmetric mode of plate vibration. (The system of approximate, two-dimensional equations of extensional motion of isotropic elastic plates derived by Mindlin and Medick 4 cannot be used for our work. This is because no

single equation of motion relating vertical plate displacement to incident pressure is capable of being derived from Mindlin and Medick's work.)

The Lyamshev plate equation of motion for the first symmetric mode of plate vibration gives inaccurate modal angular positions above a frequency that is approximately

a) Present address: McDonnell Douglas Astronautics Company, 5301 Bolsa Avenue, Huntington Beach, CA 92647.

six to seven times the classical coincidence frequency of the plate (Fig. 1 ). That is, it does not describe the dispersion of the first symmetric mode of plate vibration. The Lyamshev plate equation of motion is quite inaccurate in predicting the magnitude of the first symmetric mode at frequencies of only approximately two to three times the classical coincidence frequency of the plate. Furthermore, the Lyamshev plate equation of motion predicts too narrow of a modal peak and too small of a peak at lower frequency-thickness products (i.e., frequency of the insonifying wave X thickness of the plate) and too narrow of a peak and too large of a peak at higher frequency-thickness products.

Now the Lyamshev equation's lack of precise predictive power is undoubtedly due to defects in Lyamshev's derivation. Precisely, his ignoring of transverse inertia by ignoring one equilibrium condition and his assumption of a nonvary- ing tangential stress are defects in his derivation. Neverthe- less, at low frequency-thickness products the correct modal angular positions are predicted by the Lyamshev equation. Also at one point at least the correct magnitude of the modal peak is given. This is good enough to be able to employ our method to use the form of the Lyamshev equation to build more accurate theories containing more symmetric modes. It is the purpose of this paper to improve upon Lyamshev's work by obtaining new and more accurate equations of mo-

35 ø

30 ø

25 ø

Z_ 150

(9 10 ø Z

5 o -

PLATE PARAMETERS

E = 2.168663 x 10" N/m 3

_ p -- 7.8 x 10 -• kg/m • O -- 0.283629

_ 1st SYMMETRIC MODE (EXACT.• '••-- _

2nd SYMMETRIC MODE (EXACT)• _

120 140 160 180 200 220 240

fh (kHz-in,)

FIG. 1. Symmetric modal transmission peaks in an infinite steel plate.

859 J. Acoust. Soc. Am. 72(3), September 1982 859

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tion containing two and three times the number of symmetric modes now contained in the Lyamshev equation of motion.

The method we employ to derive equations of motion for symmetric modes is forced upon us by mathematical ne- cessity. Mindlin and Medick's approach cannot be used to obtain a single equation of motion for three or more symmetric modes of plate vibration. The reason for this is that the second and third and higher modes would have velocities which would all be asymptotic to the shear wave velocity in the plate. This would lead to a set of inconsistent equations when one attempted to determine the adjustment factors derived from asymptotic velocities. This would be true for any strength of material approach. One would always obtain an asymptotic theory which could not be used to obtain a single equation of motion for three or more modes. Hence one must abandon any idea of an asymptotic or strength of material theory in favor of a nonasymptotic theory which would be directly derived from mathematical and physical considerations of the differential equation of motion. Pre- cisely, the nonasymptotic theory would be derived by making the possible form of the differential equation of motion give the same answers for transmission through or reflection from a plate as those given by the exact equations of elasticity for a specific set of material parameters. There are several objections that can be raised about this method of deriving equations of motion.

One objection about the method would be that the physics or mechanics of the situation may not be uniquely repre- sented by a differential equation of motion even though that differential equation of motion was mathematically tailored to give the same answers for transmission loss in a plate as would be given by the exact equations of elasticity. This problem could arise if one were allowed enough mathematical freedom (in the form of undetermined coefficients) to compensate for unphysical values for one term in the differential equation by unphysical values for another. In practice, such a situation can be completely avoided by doing two things. First, one would require the differential equation one wrote to reduce to a known strength of material equation at low frequency-thickness (of the plate) products. Next, one would have to be mindful that the physical values or mean- ings of the terms of the differential equation of motion were not allowed to become unreasonable. This second considera-

tion is far easier to avoid than one might a priori realize. A final objection about our method would be that it has

no built-in assurance of the derived equation of motion working for any other set of material parameters than those chosen for the fit of the equation of motion. This objection could not extend to our determination of the cutoff frequencies since they are precisely determined for any given set of material parameters just as they can be in Mindlin's theory. Furthermore, we have built our theory [see Eqs. (6) and (71] on an expansion in terms of a dimensionless expression involving material properties. Explicitly, we have expanded in terms of the only relevant set of material properties: those having to do with the Rayleigh or shear wave velocities. This insures that the coefficients we derive are material independent excluding some multiplicative factor or function of the

Poisson ratio. This function of the Poisson ratio, if it is not identically equal to one, can probably be written as

f(cr) = C + f•(rr),

where C is large in value compared to f•(cr),a function of the Poisson ratio or. So this objection is either overruled or miti- gated and easily modified.

I. MATHEMATICAL FORMULATION

The methodology for extending the Lyamshev plate equation to include more modes is the exact analog of the methodology for extending the Timoshenko-Mindlin plate equation to include more modes. 1 One just starts with a dif- ferent equation. The methodology is motivated by a desire to solve problems involving scattering from impedance discontinuities on infinite plates. 5'6 Nevertheless, the resulting equations apply to plate problems with or without impedance discontinuities. Indeed, the methodology could just as easily be motivated by a desire to solve transmission through plates without any reference to impedance discontinuities.

Abbreviating the discussion of our paper on the extended Timoshenko-Mindlin plate equations, we just note that a total field potential for scattering problems may be written to contain a term ß which represents the field scat- tered by an impedance discontinuity. This term ß is sought in a form that satisfies the reduced homogeneous wave equation:

rp = f f(A )e ixx + 'v(X:- k :)l/:dA ' where f0[ ) is found to be a function of A whose denominator • 0 [ ) is determined by the equation of motion used to char- acterize the plate. There is a one-to-one correspondence between plate equations of. motion and the .• 0[ )'s. We are simply working with an algebraic characteristic equation rather than directly with a differential equation of motion. k is the wavenumber of the insonifying plane wave in the liquid (Fig. 2). For the Lyamshev plate equation we have 3

(x ,

h(l_o-2)po(.O2 ( 2 2 (.o 2 ) - 2E 1 - a 2 c• ' (1)

• PLATE FLUID TP-

FIG. 2. An insonified infinite plate.

860 J. Acoust. Soc. Am., Vol. 72, No. 3, September 1982 Barry Lee Woolley: Acoustic radiation 860


A slightly modified but nearly equivalent form is 7

_q(x ) = (x - k - o.,VCc

- - 2 , 12} 2E (1 -- or) 2 C c •- E/p(l a:) is the where Po is the density of the liquid, Cc -- --

compressional wave speed, c• = E/p, • is 2• times the frequency of the insonifying wave, and the other constants are as previously defined. We will hencefo•h only work with

Now the modes of plate vibration are known from the Rayleigh-Lamb equation. s Hence if we wish to create im- proved Lyamshev plate equations with additional modes, we are present• with a mathematical problem of tailoring the behavio• of the poles in f(Z ) or the zeros in • (Z) as a function of the frequency and the physical and geomet•cal parameters to the known behavior of the modes from the Ray- leigh-Lamb equation. The introduction of additional modes into the approximate plate equations is accomplished by the introduction of additional poles in f(Z ) or more conveniently additional zeros in • (Z). Now let us extend the Lyamshev plate equation to include first the second symmet•c mode and then the third symmet•c mode of plate vibration. In the process of introducing these two additional modes, the behavior of the first symmet•c mode desc•bed by the Lyam- shev plate equation will be made more realistic.

Each new mode introduces four new zeros into • (Z) because of the physical symmet• of scatte•ng denoted by ß (•) = •( -- • ) (Fig. 3). This symmetry is reflected in the introduction ofa pole's negative and the introduction of both of their complex conjugates as poles. Hence the most general equation we may w•te for • (Z) which is an extension of the Lyamshev plate equation and which includes the second symmet•c mode is

h(1 - a 2) poco 2 (QA 5 + RA 4 _•_ S,• 3 2E

+ •"A 2 + TA_ (1- 2or)6 w co•2 ) (1 -- rr) 2 ' (3) However, the previously stated physical symmetry, •P (0); •P ( -- 0 ), requires all the odd terms to be identically zero. That is D; M = Q; S; T•0. Hence we are left with

o • -o IMPEDANCE IMPED DISCONTINUITY DISCONTINUITY

FIG. 3. Physically equivalent situations.

h(1 -- a 2) poro 2 (RA 4 •_ •to,• 2 2E

(1 -- 2or) 6• co 2 ) (1 -- rr) 2 • . (4)

We must now determine the six unknown coefficients C, ct •, /g •, R, •, and 6w.

In the next section we will use the following facts and procedures to determine the six unknown coefficients. At low frequency-thickness (i.e., coh ) products, C and R must be small compared to the other coefficients and a •, fl. •, •, and 6• must all be very nearly equal to 1. Actually the values of C, a •, jg w, R, •, and 6 • are determined by the exact solution and not by the Lyamshev approximation to the exact solution. Nevertheless, roughly it can still be stated that at low frequency-thickness products they are very nearly equal to the Lyamshev coefficients. Furthermore at the frequency- thickness product at which the second symmetric mode is introduced, jg • and 6 • must be identically equal to zero in order to obtain the correct cutoff frequencies for these modes.

Using the same reasoning employed in the preceding two paragraphs, we can write the equation for .• ()t) which is the extension of the Lyamshev plate equation which includes the first three symmetric modes.

..•(,• )= (,• 2 __ k 2)1/2( _,•,• 6 •_ CW,•, 4

(5)

Physical symmetry was used in writing Eq. (5). We must now determine the eight unknown coefficients

,4, C •, ct wø, • Wo, p, R •o, •o, and 6•o. In the next section we will use the following facts about the behavior of the coefficients. At low frequency-thickness products ,4, C •, P, and R w" must be small compared to the other coefficients and a•o, •o, •o, and •5 •ø must all be very nearly equal to 1. Furthermore at the frequency-thickness products at which the second and third symmetric modes are introduced, • •o and •5 •ø must be identically equal to zero in order to obtain the correct cutoff frequencies for these modes. The Rayleigh- Lamb equation can be exploited to give the exact frequency- thickness products at which these two modes are introduced. The relationship between the roots of Eq. (4) and its coefficients and the roots of Eq. (5) and its coefficients will be used in the next section to complete the determination of the unknown coefficients in the respective .• (,•)'s. More particu-

861 J. Acoust. Soc. Am., Vol. 72, No. 3, September 1982 Barry Lee WoolIcy: Acoustic radiation 861


larly, by this method the values of C and ,4 can be unambi- guously determined for any frequency-thickness product above the point at which the second and third, respectively, modes are introduced.

II. DETERMINATION OF CONSTANTS

First of all we want to decide upon a suitable analytic form for the two sets of unknown coefficients introduced in

the last section. We need a suitable form for the six unknown

coefficients in the ease of the Lyamshev equation extended to include two symmetric modes and a suitable form for the ten unknown coefficients in the ease of the Lyamshev equation extended to include three symmetric modes. We assume the coefficients can be written as a dimensionless expansion in the square of the frequency-thickness product. Let us examine this assumption.

Since the exact equations of elasticity can be used to find the values of the modal peaks, we can precisely determine what the values of all the coefficients in Eqs. (4} and (5} should be at any given frequency-thickness product above the cutoff frequency of the highest mode. How this is done will be shown later in this section. Hence it is possible to use our dimensionless expansion in the square of the frequency- thickness product to fix all modes in their precise location at as many frequency-thickness products as we wish. The only drawback of this procedure is the fact that every time we fix a new point we add a term in the expansion of most of the coefficients. This may get us even further from what might be the easily seen physics of the problem. Also this does not account for the low-frequency behavior of the theory or the behavior of the theory at frequencies just below the highest cutoff frequency. The low frequency behavior is taken into account by building our work upon an existing strength of material theory. The correct behavior of the theory can be automatically insured at low frequency-thickness products due to the higher order terms being negligible at these lower freque'neies. We can obtain as accurate a theory as the one we build upon at low frequency-thickness products. We also can theoretically obtain as accurate a theory as we wish above the point of introduction of the highest mode if we fix enough points. But it is reasonable to try to introduce as few as possible additional terms in our expansions for each coefficient. The fewer the terms, the greater the computational ease. Fewer terms may also lead to greater physical insight. And, most importantly, fewer terms due to high frequency-thickness points being fixed will allow one more freedom in cor- rectly matching the low frequency strength of material theory to our high frequency solution. Our assumption of a dimensionless expansion is certainly one that can be made to work: it is simply a problem of making it work with a mini- mum of effort. Now we need to examine just one more point before we present a suitable analytic form for the two sets of unknown coefficients introduced in the last section.

Since C, R, A, C •, and R •" must all be small compared to the other coefficients at low frequency-thickness pro- duets, it is necessary that their expressions be multiplied by some factor of either h or to or both. So we may tentatively write:

C = h 2C o ph 2oa2

a •=1-- ph2to2

R =h2Ro • ph 2oa • •G

or h 2• Ph 2to2 ( Ph 2co2 ) ' •G 1-- •r2•Cgo ,

• 1 ph 2092 ph 2092

for the coefficients in the extension of the Lyamshev equation which includes the first two symmetric modes. And we may tentatively write:

A = h 4A 0 ph 202 ph 2092 ( ph 2092 rr2G or h n•ll rr2G 1- •G

C•=h2•6ph202( ph202 ) rda 1- ,

,)

or

h2C,,Oh2c02[ ,oh 2c02 (,0h2c02)2C, a•o=l_ P h202 (ph2•2) 2 +

•w,,=l_ P h2•2 (ph2o2) 2 c;,

P = h 4P 0 ph 202 ph 202 ( ph 202 •G or h nP• •G 1- •G

R•,,=h2R,ph2m2( P h2m2 ) 0 , or

h 2R' ph 202 ph 202 • •G 1 - •"= 1 - ph 202

6•" = 1 -- ph 2•2

,)

•2 G

•C;2 q-( Ph 2cø2) 2 rdG C•,

•.2G 15,

(7)

for the coefficients in the extension of the Lyamshev plate equation which includes the first three symmetric modes.

.... s and R "s, Pi's and The C i's and C i S, (• i'S and cK i'S' Ri i •1i's are constants which have to be determined. The factor of h 2 Which appears in the expressions for C, C •, R, and R •o, and the factor of h 4 multiplying the .4i's and Pi's were obtained by assuming the correctness of the kinematical as- sumptions used by Mindlin in his systematic method of ex- panding the displacements as power series in the thickness coordinate. 9 Now that we have analytic forms for the two sets of unknown coefficients we will proceed to determine them explicitly in detail.

The cutoff frequencies for the first three symmetric modes of plate vibration are given by the Rayleigh-Lamb equation. They are

0• 1 ( E(1 -- o') •1/2 '•- p(l +tr)(l -- 2tr)/ '



and

1( E) m '•- 2p(1 + tr)' ' /• w,/• Wo, •, and •o must be zero at the nonzero cutoff frequencies of the modes contained in our approximate theory. From this it can be determined that

1 - 2rr G G , c,- = = • C6 = 14 --

2(1 -- and

1 - 2o- # C # C7 • 15 • '

1 -- 2rr

2(1 --

(8)

We have determined/• • and •./• Wo and •o will have to be modified as explained below.

Now because the second and third cutoff frequencies are so close together, the polynomial fit determined by the above process will give unphysically large values for/• •o and •o at frequency-thickness products above the cutoff frequency of the third mode. To avoid this/• •o and •Wo will be modified in order to take into account the fourth cutoff fre-

quency given by

2( E )•/2. h 2p{1 + rr)

We thus obtain for/• •o and •o:

,O •Oo = 6•,,, = 1 - ph 202 rdG

_ (ph2co2) 3 rdG C;",

with

5 (1 -- 2tr) 1 5(1 -- 2tr) c;'= c;'= 16 2(1 -- rr) 64 32(1 --

and

,, (1 -- 2rr) (9) C? • o 128(1

We have determined/• Wo and •o. One should keep in mind that the polynomial fits for the

./• 's and • 's will show greater deviation from the values they should give the further one is from a cutoff frequency.

R, R •o, and P will only be significant in comparison with y• or y•o if Ro or R l, R/• or R •, and Po or P1 are large. But if these latter constants are large they will violate the known condition that R, R •o, and P must be small compared to the other coefficients {i.e., a •, a wø,/• •,/• •o, y•, y•o, 6•, and 6 •ø) at low frequency-thickness products. Hence R, R Wo, and P are either insignificant or zero. We take them to be identically zero. We have determined R, R •o, and P.

The relationship between the ten roots of Eq. {4} and its coefficients and the relationship between the 14 roots of Eq. {5) and its coefficients can now be used to complete the determination of the unknown coefficients in D {A ). First we must state a theorem we will use. lo In an integral rational equation of degree n, with the coefficient ofx "unity, the coefficient of x n- • is the sum of the roots with the sign changed; the coefficient ofx "- •- is the sum of all possible products of the roots, two at a time; the coefficient of x • - 3 is the sum of all possible products of the roots, three at a time, with the sign changed; and so on, the constant term being the product of all the roots multiplied by { -- 1)n. Now if we set D {A ) = 0 in Eq. {4) and perform a little algebra, we get the following integral rational equation in X = A 2:

X5+(2 •w h 2(1 - ø'2)2po2(-ø4R • X4_• - • 2co2p/g •(1 - cr 2) 2k 2a• EC C

4p2• •ø)2(1 - or2) 2 2co2pk 2a•]• •'(1 - cr 2) + , + h 2(1 -- 0'2)2p2oO)4(1 -- 2cr)2(6•)2co4p2(1 -- oa) 2

h 2(1 -- d}2Do2(_l)4•wR ) ••'• ' X 3

2E 3C 2( 1 -- O') 2

=0. (10) w4p2• W)2k 2(1 _ o2) 2

4E 4C 2( 1 -- O') 4 E 2C 2

Similarly if we set • {A ) = 0 in Eq. {5) and perform a little algebra, we get the following integral rational equation in X = A 2.

( -2c• _k 2 h2(1--0'2}2p2ow4p2,)Xa(--20tWøl{CW}2 2k2C • h 2(1 -- 0'2)2p200)4pR wø.)X5 2C •'a •'o 2co2p/5 ' •o(1 -- cr 2) + 2k 2a•'ø k 2(C •,)2 -I- •' -I- •'A A '• A :2 h 2(1 -- d)2D2oO)4(]• 2 + 2Py•'ø) ) 4

[ (aw") 2 2co2pC •øl• •øø(1 -- a 2) 2k 2C •a •o + •' EA 2 -- A2 2co2pk 2l• Wo(1 _ cr 2)

EA

h 2(1 - d)2p200) 4 +

2EM 2



2(1 -- 2cr)6•øøca2pR (1 -- o'2).) (1 -- o')2E

+ aot5 _ o EA 2

h 2(1 -- 0'2):•p2oca4(1 -- 2Cy)2(6wø)2ca4,O2( 1 -- 0'2) :• 4E 4A 2( 1 -- 0') 4

q- h 2(1 -- 0-2)2p•ca4(1 -- 2a)6oo,ro(1 - x 2E 3A 2(1 -- O') 2

ca4p2 k 2• Wo)2(1 _ 0.2)2 =0. (11)

Now the constant term in Eq. (10) is the product of all five roots in ,• 2. Likewise, the constant term in Eq. (11) is the product of all seven roots in ,• 2. But these roots may be un- ambiguously determined from the exact equations of elasticity at frequencies which are greater than the cutoff frequency for the second symmetric mode in the case of Eq. (10) and greater than the cutoff frequency for the third symmetric mode in the case of Eq. (11). Hence, since we already have ?g • and ]g •", one may use the constant terms to determine the value of C or A at any given point above the cutoff frequency of the second or third symmetric mode, respectively. Given C or A at any given point above the cutoff frequency of the second or third symmetric mode, respectively, we may use the coefficient of the X 4 term in Eq. (t0) or the coefficient of the X 6 term in Eq. (11) to determine the value of a • or C •, respectively, at that point. Having determined the values of C or A and a • or C • at any given point above the cutoff frequency of the second or third symmetric mode, respectively, we may then use the coefficient of the X 3 term in Eq. (10) or the coefficient of the X • term in Eq. (11) to respectively determine ]g • or a •o. This procedure can be continued to determine the values of the remaining coefficients f' and f'". Now we may choose the analytic form of any of these coefficients so that it has the precise values it should have (as determined from the exact equations of elasticity) at as many points above the highest cutoff frequency as we wish. If this is done well, we will have determined the analytic form for the coefficients of Eqs. (6) and (7j above the highest cutoff frequencies contained in the theorY. It is then only a not- necessarily-trivial matter of insuring that the low frequency- thickness product behavior given by building our theory on a given strength of material theory is properly joined to our high frequency-thickness product theory. The proper joining is pragmatically judged to be the joining that gives the correct modal angular and magnitude behavior.

The procedure we have outlined above is straightfor- ward and is limited only by the inaccuracies inherent in the analytic fit to the cutoff frequency dependent coefficients- /• w,/• •o, 6•, and 6 •ø. The procedure matches the angular modal dependence of our theory to that given by the three- dimensional equations of elasticity. The magnitude dependence is matched by appropriate choice of 7 w or 7 wo. There is an alternate approach which has the potential of yielding simpler coefficients. This approach would be to modify and complicate as necessary the coefficients in Eqs. (6) and (7) in order to match the angular modal dependence as given by

the three dimensional equations of elasticity. This approach would be one which extends the existing strength•of material theory to higher frequency-thickness products from lower frequency-thickness products rather than fitting the existing strength of material theory to an analytic fit which was derived so as to be good at higher frequency-thickness products. We will now give examples of results using both procedures.

The determination of the remaining coefficients of Eq. (6) by the low frequency-thickness product approach gives'

C = + 3.232849845 X 10 -6 ph 2092

X(1--5.771698482X 10 -2 ph 2092) a • = 1 -- 4.182035764 X 10 -1 ,oh 2092 rr2G , (12) yw = 1 -- 4.112335168 X 10-1 ph 2092

rdG '

The final equation of motion •or th e first two symmetric modes of plate vibration is [with/g • and 6 • given by Eq. {6) with the coefficients C2 and Cs given by Eq. (8); with R = 0; and with C, a •, and y• given by Eq. (12)]:

C2•5. w c

-- 2E (1 -- or) 2 C• 2 • •' (13) Equation (13)is used for Fig. 4. In Fig. 4 we display the

peak transmission in an infinite steel plate as a function of the angle of incidence 0 of an insonifYing plane wave of speed c and circular frequency ca versus the frequency-thickness product {i.e., frequency of insonifying plane wave X thickness of the plate). The frequency-thickness product is related to the dimensionless normalized frequency g2 =ca/cac, where cac = c2(P h/D }•/2.D = Eh 3/12(1 -- oa). cac is the classical coincidence frequency of the plate. {To obtain an approximate value for/2, divide the frequency-thickness product expressed in kiloHertz-inches by 10.) As in Fig. 1, the liquid loading the plate is water. The first two symmetric modes of plate vibration are shown as predicted by the exact equations of elasticity and as predicted by Eq. (13). The quality of the angular agreement between the behavior as predicted by Eq. {13) and the behavior as given by the exact equations of elasticity can be easily seen in Fig. 4. Above /2 = 19 for the steel plate we are using as an example, the

864 d. Acoust. Soc. Am., Vol. 72, No. 3, September 1982 Barry Lee Woolley: Acoustic radiation 864


angular behavior as predicted by Eq. (13} smoothly diverges from the behavior as given by the exact equations of elasticity. This is due to Eq. (13} not being an asymptotic theory. The magnitude behavior of the mode will be discussed in the

l

next section.

The determination of the remaining coefficients of Eq. {7} by the high frequency-thickness product matching approach gives:

•4=_1.454006X 10-3h4ph:(0: ph2(02 rr:G 1--0.2280815 rr:G d- 1.815443 X 10 -2 ph2(02 2 - 4.950063 X 10-4( Ph 2(02) 3] rc2G '

C•o +6.718279X 10-2h2ph2(02 [ (ph2(02) 2 rr2G 1 -- 0.71485950 ph 2(02 rr2G + 0.13002360 rr2G

--9'436399X 10-3(ph2(02) 3 ( )] rr2G + 2.441888 X 10 --4 ph 2(02 4 rc2G '

ot•o,,=l_O.4898674Ph2(02 ( Ph 2(02) 2 ( Ph 2(02) 3 rr2G - 0.33952120 rr2G d- 0.21200950 rr2G -- 3.494811 X 10 -2 ph 2(02 4 rr2G

d-2.381743X 10-3(ph2(02) 5 ( rr2G -- 5.914968 X 10 -5 ph 2(02 6, ff '"'=1-0.603465663ph2(02 () () rr2G + 0.1373990760 ph 2(02 2 ph 2(02 3 rr2G -- 8.04867797 X 10-3 rr2G

+ 9.77854033 X 10-5( Ph 2(02) 4.

(14)

The fihal equation of motion for the first three symmetric modes ofplate vibration is [with/g w,, and 6 •" given by Eq. (9); with R •"= 0 and P = 0; and with •4, C •, a •", and ?,•o given by Eq. (14}]:

C2•- 2. w c

(15)

Equation {15} is used for Fig. 5. In Fig. 5, the first three symmetric modes of plate vibration are shown as predicted by the exact equations of elasticity and as predicted by Eq. { 15}. Otherwise Fig. 5 is the same as Fig. 4. The quality of the

l

angular agreement between the behavior as predicted by Eq. { 15} and the behavior as given by the exact equations of elasticity can be easily seen in Fig. 5. Above/2 ---- 24 for the steel plate we are using as an example, the angular behavior as predicted by Eq. (15) rapidly diverges from the behavior as given by the exact equations of elasticity. This is due to Eq. {15) not being an asymptotic theory and to the extra cutoff frequency incorporated into/g w" and 6•ø; the cutoff frequency of the fourth symmetric mode serves as an absolute upper limit to the validity of Eq. {15}. The magnitude behavior of the modes will be discussed in the next section.

Now the double-valued behavior displayed by the second symmetric mode at its cutoff frequency (Fig. 5} is impos- sible to duplicate by a differential equation. Therefore any theory with a finite number of modes with correct cutoff

35 ø

30 ø

25 ø

(.) 20 ø z

Z_ 15 ø

O

O 10 ø

0 o

PLATE PARAMETERS

E -- 2.168663 x 10" N/m 3

_ P = 7.8 x 103 kg/m 3 O: 0.283629

- 1st SYMMETRIC MODE • (EQ. 13 OF TEXT). •

- • _/ 2nd SYMMETRIC MODE • ,•' (EXACT)•

1St SYMMETRIC MODE (EXACT)

2nd SYMMETRIC MODE . I ." •' I

I ' I I I i •i ß i i i ß _ (EQ. I3OFTEXT, •// .•••TAi•T• ) _

20 40 60 80 100 120 140 160 180

fh (kHz-irt)


35 ø

30 ø

A 25ø

(.) z 20 ø

Z - 15o

O

Z 10ø

0 o

PLATE PARAMETERS

E: 2.168663 x 10" N/m 3

p: 7.8 x 103 kg/m 3

O: 0.283629 • .... --- ' '"'-- 1st SYMMETRIC MODE

•/" (EQS. 14 & 15 OF TEXT) 1st SYMMETRIC MODE

(EXACT)

• 2nd SYMMETRIC MODE • (EQS. 14 & I5 OF TEXT)

2nd SYMMETRIC MODE (EXACT)

/ •

3rd SYMMETRIC MODE (EXACT)• I // /••///

• I '""•f 3rd SYMMETRIC MODE

• / (EQS. ,4 & 15 OF TEXT) 20 40 60 80 100 120 140 160 180 200 220 240

fh (kHz-irO




frequencies will be faced with large angular discrepancies in the vicinity of a cutoff frequency which displays such double-valued behavior. Hence it might be advantageous to have a theory which duplicates the angular behavior of such a double-valued mode at the expense of obtaining a correct

cutoff frequency. Such a theory would be valid only over a limited range of frequency-thickness products. We have pro- duced such a theory. It is displayed in Fig. 6 and given by Eq. (15) with the following coefficients differing from those given in Eq. (15):

,d = -- 1 237756 X 10-3h 4 ph :co: [ ph :co: ' •G 1 - 0.2668152 •G 2.663795 X 10 -2 (ph 20.)2 \

--9.492795X 10-4( Ph 2co2) 3] C•'= -- 1.693895 X 10-2h 2 ph 2co2 [ •.2G 1 -F 8.336130 X 10 -2 ph 2co2 -1.233778X 10-:(Ph:co:) 2

•.2G -F 1.069050 X 10 -4 ph 2(.02 4 •G '

a•"' = 1 -- 2.0605960 ph 2(-02

•.2G - 5.835813 X 10 -3 ph 2(-02 5 • ?"ø = 1 - 0.644595042 ph :co: ( ) ,n.2 G + 0.1532301720 ph :0.): •

•r2G - 0.42688960 ( ph ß •G

+ x 0-4 /

rr2G + 1.80750435 X 10 -4 ph 20.)2 4.

(16)

Figure 6 uses the same plate, fluid, and geometrical parameters used in Fig. 5. The three modes in Fig. 6 were tailored at a number of points to have exactly the values they should have as given by the exact equations of elasticity. The lowest frequency-thickness product at which this was done was 140 kHz-in. The highest frequency-thickness product at which this was done was 200 kHz-in. The angular behavior can be seen to be excellent for the limited frequency-thickness product range that was chosen. The cutoff frequency of the second symmetric mode serves as an absolute lower limit to the range of validity of the theory. Above/2 = 20 for the steel plate we are using as an example, the angular behavior for the modal peaks as predicted by Eqs. (15) and (16) rapidly diverges from the behavior as given by the exact equations of elasticity. The differential equation used in Fig. 6 is exceed- ingly simple to generate because all points are above the cutoff frequency of the highest mode contained in the theory. We will not go on to discuss the magnitude behavior of our equations.

III. TRANSMISSION LOSS COMPARISONS

In a previous paper • we presented a nonasymptotic plate equation of motion for the first four antisymmetric modes of plate vibration. In that paper we' examined the angular accuracy of those modes. We also examined the magnitude accuracy of those modes by comparing the transmission loss as predicted by the exact equations of elasticity with the transmission loss as predicted by our nonasympto-

•z

tic antisymmetric theory. We found that such comparisons were complicated by the symmetric modes coalescing or canceling the antisymmetric modes as predicted by the exact equations of elasticity. We have presented several nonasymptotic plate equations of motion for symmetric modes of vibration in the preceding sections of this paper. Therefore, we now are in a position to compare transmission loss as predicted by our equations of motion for antisymmetric and symmetric modes with transmission loss as predicted by the exact equations of elasticity.

35ø I PLATE PARAMETERS 1st SYMMETRIC MODE (EXACT). / E = 2.168663 x

30ol _ p = 7.8 x 103 kg/m 3

250_ 1st SYMMETRIC MODE (EQS, 15 & 16 OF TEXT)

.....

0 20 o_ z

-- 2nd SYMMETRIC MODE (EXACT)• _z 15o ,,•

•;-• ..... O 2nd SYMMETRIC MODE (EQS. 15 & 16 OF TE

m 3rd SYMMETRIC MODE (EQS. 15 & 16 OF TEXT_) _..• O 10 ø-

5 ø

14( 1 155

fh (kHz-irt)


200



Consider a theory for rn antisymmetric modes given by

(Fm V 2m q- ... -+. G wVS -- E wVa -I- 15gwv 4 -•' F•'V 2 q- F•')w = _ F,v)q/fo

where the form of the terms is defined as the obvious exten-

sion of Eq. (22)in our previous paper. • Now consider a theory for n symmetric modes given by

where the form of the terms is the obvious extension of Eq. (15). Now the plane-wave transmission coefficient for an elastic plate with rn antisymmetric modes and n symmetric modes taken into account is given by the following simple extension of existing analysis• •:

----- i

(2pocP'•Z •" cos 0 -pocQ "Z •'• cos 0) (Z •" cos 0 + poCQ")(Z • cos 0 + 2pocP •)

where

D Dk 2 sin 2 0 p,• _ F •, •+ F•',

Q n _ (1 - 20') •w,,_ E sin 2 0 - (1 - o 2) c2P( 1 _ or2) •,o, -- _ 2iE ( Ek 2(n- 1)sin2. 0 Z •" -- o(1 - 0.2)h ( - 1)" - 1,t• n c2p(1 _ 0.2) + '"

+ ,4 Ek 4 sin 6 0 _ C •' Ek 2 sin4 0 c2p(1 -- •) c2p(1 -- •)

E sin 2 0 ) -a•'ø c2p(1 _0.2) Z '•'" =(iD/co)[(- 1)'"F,,k •'" sin 2'" 0 + ... + G•øk • sin s 0

+ E•'k 6sin6 0 +a•'k 4 sin40 -F•'k 2 sin20 +F•'], with k being the wavenumber of the insonifying plane wave and i being the unit imaginary number.

When Eq. (17} is used for the case of two symmetric and four antisymmetric modes or the case of three symmetric and four antisymmetric modes, we get excellent transmission loss behavior for/2 < 8. This includes excellent behavior

for grazing incidence of the insonifying plane wave. Every antisymmetric theory gives perfect transmission at grazing incidence. Yet at grazing incidence the exact equations of elasticity give perfect reflection as can be seen in Fig. 7 for the case of a water-immersed steel plate. A slight amount of attenuation has been assumed for the steel plate in Fig. 7. The combination of symmetric and antisymmetric modes included in our theory has corrected this defect of all purely antisymmetric theories. However, above/2 = 8 we get an increasingly narrow region of excellent reflection near grazing for our combined antisymmetric and symmetric theories. From an/2 of approximately 18 to an/2 of 24, our combined antisymmetric and symmetric theories have near-grazing transmission peaks of transmission losses less than 15 dB. These can be as wide as one-half degree.

90 ø

80 ø

70 ø

m 50

Z 40 ø

m, 30 ø z

'< 20 ø

1.0 dB

3.0 dB

6.0 dB

1.0 dB WATER-IMMERSED 15.0 dB INFINITE STEEL PLATE

10 ø

2 10 100 300 fh (kHz-ir•

FIG. 7. Transmission loss contours for an infinite steel plate as predicted by the exact equations of elasticity.

The transmission loss or magnitude behavior of the symmetric modes in our various theories and the antisymmetric-symmetric mode interaction will now be discussed.

Below/2 = 7 or 8, the transmission loss or magnitude behavior of both the 2 symmetric-4 antisymmetric mode theory and the 3 symmetric-4 antisymmetric mode theory is excellent. For the 2 symmetric-4 antisymmetric mode theory the first symmetric mode shows too great of transmission loss for/2 = 7-10 (i.e., the peak in transmission is too narrow compared to results from the exact equations of elasticity), too low of a transmission loss for/2 = 13-18, and excellent antisymmetric-symmetric canceling behavior above/2 = 19. The second symmetric mode for the 2 symmetric-4 antisymmetric mode theory shows too low of a transmission loss above /2 = 14. The second symmetric mode for the 3 symmetric-4 antisymmetric mode theory also shows too low of a transmission loss above/2 = 14. The

third symmetric mode for the 3 symmetric-4 antisymmetric mode theory has too high of a transmission loss from /2 = 14-20 and too low of a transmission loss for/2 = 22-24.

And finally, the first symmetric mode for the 3 symmetric-4 antisymmetric mode theory has too high of a transmission loss for/2 = 9-16, 18-19, and too low of a transmission loss for/2 - 23-24.

Some of the disparity between the combined antisymmetric and symmetric mode theories and the results given by the exact equations of elasticity is due to slight angular discrepancies in both the antisymmetric and symmetric modes. When examining our 3 symmetric-4 antisymmetric mode theory given by Eqs. (15) and (16) we find an improvement in its transmission loss predictions (as compared with the exact equations of elasticity) over the transmission loss predictions given by our 3 symmetric-4 antisymmetric mode theory given by Eqs. (14) and (15). This is for the first symmetric mode which has better angular behavior in the theory given by Eqs. (15)and (16) than in the theory given by Eqs. (14) and (15). The second and third symmetric modes actually have slightly worse behavior in the theory given by Eqs. (15) and (16) than in the theory given by Eqs. (14) and (15). The Lyam- shev theory, upon which we have build our theory, has a very narrow peak in transmission which also has unreasonably



high values of transmission. This may be affecting our results.

IV. BOUNDARY CONDITIONS

We now wish to consider the uniqueness of any boundary value problem we may encounter using the operator L2 from Eq. (13) or the operator L 3 from Eq. (15). Let u2(x} and v•(x} be functions with four continuous derivatives and u3(x} and v3(x} be functions with six continuous derivatives. Let the boundary points or surfaces be denoted by a and b. Then there will be a unique solution ifP•(u•,v•)[P3{u3,v3} ], given by

= (u2L2v 2 - v2L2u2)dx ,

(P3(u3,v3) = •ø(u3L3v3 - v3L3u3)dx), (18)

is zero. 12 That is, L 2 and L 3 must be self-adjoint. We inte- grate by parts to obtain P2(u2,v2) and P3(u3,/)3}. For self-adjoint boundary conditions u2 or u3 must satisfy the same boundary conditions as v2 or v3, respectively. If we let u2 or v2 = w, where w is the vertical displacement of the plate, we obtain three distinct sets of the two boundary conditions that must be satisfied at both a and b in order for P2(u,v) to be zero and the problem to have a unique solution. They are

SetI:w=O, w' = O,

SetII:w=O, w"=O,

Set III: w' = O, w'" = O,

where the prime denotes differentiation with respect to x (without loss of generality}. Alternately, each of the three sets may be made up of two linearly independent combinations of the two conditions presented in the set.

Now if we let u3 or v3 = w, where w is again the vertical displacement of the plate, we obtain four distinct sets of the three boundary conditions that must be satisfied at both a and b in order for P3(u,v) to be zero and the problem to have a unique solution. They are

SetI:w=O, w' = O, w"= O,

SetII:w=O, w' = O, w'" = O,

SetlI:w=O, w"= O, w •" = 0, Set IV:w'=O, w'"=O, w * =0,

where again the prime denotes differentiation with respect to x. Alternately, each of the four •ets may be made up of three linearly independent combinations of the three conditions presented in the set. Before going on, it is perhaps worth- while to note that the combination w = 0 and w' = 0 charac-

terizes a fixed end, the combination w = 0 and w" = 0 characterizes a hinged end, and the combination w'= 0 and w'" = 0 characterizes a so-called free-fixed end.

Finally, if we consider the problem of a line impedance discontinuity at c

P2(u,v) = (uL2v -- vL2u)dx -[- .(uL2v - vL2u)dx , or {19)

P3(u,v) = (uL3v -- vL3u)dx -[- +(uL3v -- vL3u)dx. where the boundary conditions are self-adjoint at a and b, we obtain the following three distinct sets of boundary conditions that may be satisfied by u = w (the vertical displacement of the plate) in order to make the solution of the problem unique for Le:

Set I: w(c+) - w(c_)

Set II: w(c+ ) -- w(c_ ) = c•, w" (c+) -- w"(c_) = c4,

Set III: w'(c+) - w'(c_) = c•, w'"(c+) - w'"(c_) = ca,

where the prime denotes differentiation with respect to x. Similarly we obtain the following four distinct sets of boundary conditions that may be satisfied by u = w (the vertical displacement of the plate) in order to make the solution of the problem unique for L3:

Set I: w(c+) - w(c_)

Set II: w(c+)- w{c_)=Clo, w'(c+)- w'(c_)=ell, w"'(c+) - w'"(c_) =

Set III: w(c+) - w{c_) =el3, w"(c+)- w"{c_) =el4, w (c + ) - w (c_) =

Set IV: w'(c +) -- w'(c_) = ClO w'"(c +) -- w"'(c_) = c17 ,

where again the prime denotes differentiation with respect to x. The ci's are constants or zero. The conditions physically represent statements on the continuity of the slope, and re- strictions on the higher order moments and forces which have to be derived for the particular problem under consi- deration.

V. SUMMARY

Lyamshev has proposed a theory 3 which has one symmetric mode of plate vibration in it and one antisymmetric mode of plate vibration in it. The antisymmetric mode was taken by Lyamshev to be governed by the classical or La- grange plate equation of motion. This poor choice does not concern us here. Nor do the inadequacies of Lyamshev's theory vitally concern us since we just abstract the form of his equation to build two and three symmetric mode theories. We vastly improve the theory in its ability to reproduce the dispersion of the first symmetric mode by introducing a second [Eq. (13)and Fig. 4] and then a third [Eqs. (14}-{ 16) and Figs. 5 and 6] symmetric mode of plate vibration. We thereby also vastly improve the predictive ability of the theory as far as transmission loss comparison [Eq. ( 17}] with the exact equations of elasticity is concerned. Our three symmetric mode theory is--and has to beta nonasymptotic theory. The inconsistency introduced by all but the lowest symmetric mode being asymptotic to the same limit makes it mathematically necessary to have a nonasymptotic theory for any theory with three or more modes in it. We make our theory



have the correct cutoff frequencies and the correct modal angular behavior at a number of points above the introduction of the highest symmetric mode in the theory. Thereby we insure that the theory will work for any set of material parameters for an elastic plate. The applicability of our approximate symmetric mode theories is limited to a frequency below the cutoff frequency of the lowest neglected mode or to a frequency below the cutoff frequency if the lowest neglected mode is strongly coupled to one of the lower modes. All asymptotic theories are also limited by these same upper bounds.

ACKNOWLEDGMENTS

The author wishes to thank Dr. P. A. Barakos of the

Naval Ocean Systems Center for his encouragement of this project and S. L. Speidel,/also of the Naval Ocean Systems Center, for the transmission loss calculations shown in Fig. 7 and used throughout this work in calculating modal peaks using the exact equations of elasticity. Dr. A. Kalnins of Lehigh University is especially thanked for his comments on this manuscript. This work was supported by the Indepen- dent Research program of the Technical Director of the Na- val Ocean Systems Center.

lB. L. WoolIcy, "Acoustic Radiation from Fluid-Loaded Elastic Plates. I. Antisymmetric Modes," J. Acoust. Soc. Am. 70, 771-781 ( 1981 ).

2A. E. Armenakas, D.C. Gazis, and G. Herrmann, Free Vibrations of Cir- cular Cylindrical Shells (Pergamon, New York, 1969} and references cited therein.

3L. M. Lyamshev, "Reflection of Sound from a Moving Thin Plate," Sov. Phys. Acoust. 6 (4}, 505 (1960).

hR. D. Mindlin and M. A. Medick, "Extensional Vibrations of Elastic Plates," J. Appl. Mech. 26, 561-569 (1959).

•B. L. WoolIcy, "Acoustic Scattering from a Submerged Plate. I. One Rein- forcing Rib," J. Acoust. Soc. Am. 67, 1642-1653 (1980).

6B. L. WoolIcy, "Acoustic Scattering from a Submerged Plate. II. Finite Number of Reinforcing Ribs," J. Acoust. Soc. Am. 67, 1654--1658 (1980).

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

8K. F. Graff, Waoe Motion in Elastic Solids (Ohio State Univ., Columbus, OH, 1975).

9R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Mo- tions of Isotropic, Elastic Plates," J. Appl. Mech. 18, 31-38 {March 1951 }.

IOj. V. Uspensky, Theory of Equations {McGraw-Hill, New York, 1948}. IlK. F. Graff, C. A. Klein, and R. G. Kouyoumjian, ,4 Study of Seoeral ,4pproximate Theories for Calculating the Reflection of ,4coustic Plane Y•a•esfrom Elastic Plates, The Ohio State University Electro Science La- boratory Technical Report 4720-2 {7847201, Contract No. N66001-77-C- 0195 WCJ {15 December, 19781.

•2P. M. Morse and H. Feshbach, Methods of Theoretical Physics {McGraw- Hill, New York, 19531, Vol. 1, p. 870.



acoustic radiation from fluid-loaded elastic plates. ii. symmetric modes

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