RAD-Ai40 377 NUMERICRL RESULTS OF ACOUSTIC RADIATION TRANSMISSION i/iAND REFLECTION OF ST..(U) ADMIRALTY MARINE TECHNOLOGYESTABLISHMENT TEDDINOTON (ENGLAND. E J CLEMENT FEB 94

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NUMERICAL RESULTS OF ACOUSTIC RADIATION, TRANSMISSIONAND REPLECTIOM OF STEEL PLATE IN TTER

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E.J. CLEMENT

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>Plots of acoustic power versus frequency, and acoustic pressure direc-tivity at selecteA frequencies, are given for the separate cases of thin,thick and exact plate theories in which the excitation is a vertical time-harmonic point force; they illustrate the range of applicability of theapproximate theories. Plots of acoustic power and pressure directivity aregiven for exact plate theory in which the excitation is a longitudinal force.Plots of pressure transmission and reflection coefficients are given for exactplate theory. Contour and perspective plots of pressure-frequency-angle data,obtained from exact theory, provide vivid illustrations of the acousticproperties of the plate.

ANTE( Teddinqton)30 pages Queen's Road17 figures TEDDINGTON Middlesex TWil OLN February 1984

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Pestell and James (1] have calculated far-field sound radiation from ..layered media, excited by time-harmonic forces, by the method of dynamicstiffness coupling in which each element of the system is represented by aspectral, dynamic stiffness matrix. Based on this work, Spicer (2] has

simlified the mathematical derivation of the stiffness matrices and has

included uniform mean subsonic flow in the acoustic fluid elements. FortranprogrI arising from the aforementioned studies have been used to predict V'

sound radiation from, and free-wave propagation in, a variety of layeredsystem. Additionally, a Fortran program has been written to calculatereflection and transmission coefficients due to plane-wave excitation.

The acoustics of fluid-loaded infinite plates has been studied exhaus-tively in the literature, but only a restricted number of graphical presen-tations have been published. The mathematics of the 'thin' and 'thick' platetheories can be found in the standard text on fluid-structure interaction [3].Evseev et al (4] and Sergeev [5] have studied sound radiation using platetheories which include the effects of longitudinal vibrations as formulated byLyamshev. Freedman (6] gives a limited comparison of the transmissioncoefficient obtained from exact and approximate plate theories and discusses°',_the effect of including the first symmtric longitudinal wave notion in thethin plate model. Fiorito et al (7] have used a resonance formalism theorywhich provides physical insight into the peaks and dips that occur in plots oftransmission and reflection coefficients versus angle of incidence. Woolley(9] has extended thick plate theory to include symmetric modes of vibration.

It is the purpOse of this report to extend the range of published numeri-cal results of sound radiation, reflection and transmission of an infinitesteel plate which is imrsed in water. Subsequent reports will includenumrical results of the acoustics of layered media and will also discusswavenuier versus frequency plots obtained from the dispersion relation of thelayered system. The nurical results contained herein will help to illustrateand aid understanding of the acoustics of a steel platet they will alsoprovide reference plots for comparisons with plots obtained from simplifiedtheories.

Section 2, below, gives no details of the mathematics needed for botheit and approximate plate theories, the exact theory being based on themultilayer theory of Spicer (2]. The numerical results are discusse inSection 3 and they are followed by some concluding remarks in Section 4.

2.."

(a) 19D Yamiao and Reflection.

In Figure 1A a time-harmonic plane-wave is incident on a layered systemwhose elements are governed by the equations of acoustics, elastodynamics andviso-dyNics. The incident, reflected and transmitted waves must be of theform

P, " Piexp(iQox+o W-iYs Z)

Pr = Pi IR.exp( ii+i90y+iY z ) (2.1)

Pt - Pi7 ' exp(L n+i1f-iY2[z+H])

where Pi is the amplitude of the incident wave; R is defined as the complexpressure reflection coefficient; T is defined as the complex pressure trans-itted coefficientl a, 0, Vi and Y2 are wave-numbers which must satisfy the

constraints

YIM -i ,(2 -- 2 2)

Y2- W(k 2 -a2 -13 .)

because the pressures must satisfy the acoustic wave equations in their re-spective domaina; k, and k2 are acoustic wavenumbers, w/c1 and w/c2g thetime-hazuonic factor exp(-iwt) is omitted throughout.

Spicer (2] has presented the mathematics needed to generate the matrixrelation connecting interface displacements and forces, viz.

(Z(a,13)J(U(a,j3)] - [Z(a,)] (2.3)M*N Nw1 N'!

where M-3N+3 with N being the number of layers; (Z(a,13)] is the systemdynamic stiffness matrix;

(U( ,A ) ] -- (u X , Uy l U z i, . . Uxn Uyn Uzn ]IT

is a column vector of displacements at the n layer interfaces, where n-N+l;

E(aA)j - i , ... ,,0] T

is a column vector of interface stresses, assumed to be positive when actingin the positive s-dlrectlion, that takes the above form for plane-waveexcitation because the excitation is simply the 'blocked pressure' (3],-(pi +pr]zmo; the factor ezp(ci+iy) has been omitted from both sides of theequation.

The normal displacements of the upper and lower surfaces of the system,viz.,

U (a,13) -u (a~jl(a,)eP(io4y)I ZI (2.4)

W U (a,,) u (a,3)eXp(iCoc+i3y)

are found by matrix inversion of equation (2.3).

-4-

The boundary conditionPi/,s apr/aZ]Z. o - p1 (a2U(a,3)

that requires continuity of fluid particle and interface displacements at theupper surface gives

R-lz+p U u,3)iySuZ(a,)/iylP (2.5)

and a similar boundary condition applied at z--H gives

2T - -P2 W uzn(,13)/"y2p i (2.6)

For the particular case shown in Figure 1, the wavenumbers and angle ofincidence, 0i , are related by the simple equations

a klsin(ei)

Y, k COO(O)

y +vfk2-k 2sin 2 (e)

The angle of transmission, e t , can be found from Snell's law, viz.

x2 sin(eO - k 1sin(0i ) (2.7)

which gives

0t - tan [klsin9i/(k2 -k2sin 20) 1/2 (2.8)

It may be deduced from equation (2.8) that when c1tC 2 , there is always a realangle of transmission; and when COlC2 there is a critical angle of inci-den*e, * 1-sin- 1 (01/02), beyond which there is no transmitted wave. Thisphenomenon is discussed in numerous texts, for example (9].

The pressure reflection and transmission coefficients in decibels aredefined as

d -20.0 x 1o 10 IRI (2.9)

Td - 20.0 x log10 ITIde

- - -,,d . ... 4

(b) soun 3i ion. m

In Figure 23, time-hazonic point forces and sources ezcite the layeredsystm, whose fax-field pressure in the upper half-space is

2p(R.,O) - -pl0 u (a.0)exp ik R)/2fR (2.10)

in which the ,stationery phase' (3 ] vavenumbers are

a- k sin(e)Cos(*)

(2.11)3 = k sin(e)sin(*)

The displacement u z l (a,13) is found by matrix inversion of equation(2.3) in which

(20,13)] - (FP lop IF if ... IF IF IF +5i (z~~aJ)] - [r l~rl z, .. lxn yn, zn+Szn]

A vertical point force, Po, located on the upper interface at (x-xw, yyo ), isrepresented by, for example

pz t Focxp(-iax0-iyo ) (2.12)

and a point source of sound, p~ep ik 2 R)/R, located at dstance XO below thebottom Interface gives

S - 4wiPOeX-i"oO-i3Yo4iIZI1Y2 /Y2 (2.13)

"W sound level in do ref. 1 gI/U2 at 1- is defined as

43 - 20.0 X log t0 P(R'1,, )1 + 120.0 (2.14)

For the case of point force excitation with peak amplitude Po, the decibels

as defined above must be regarded as for force F0 rm.

(C) Sound Radiation of Plate*. Anooimate

For the special cases of thin and thick (Timoshenko-Mindlin) platetheories the sound pressure is again given by equation (2.10) in which, forpoint force eoitation

u (Ga)- PoeXp(-iaxo-i.y o )/z(*a, ) (2.1S)

where

h Z(a,,) - D(a 2 +1 )2 -w 2 ph-ip 1W 2/yl-L.P2 2/2

or (2.16)

-6-

,i,

2 2 3 2 2 2 2 2Z(a,D) ([(D(a +,a )-P h Id /12)(a +a -p /KG)-- pSh]

/[ 1+D(a 2 2 ) / M k p 2Wz / l 2 =G ] )-. ,,2/Y1-P2 2/Y2

respctiv*ly. D Is the plate flexural rigidity; p, is it. density and h

its thickness; G is the shear modulus and K z- #r2/12 is a shear correction

factor; Y1 and /2 are given by equation (2.2). The plate theories are

discussed in some detail in the standard text E3] on fluid-structureinteraction.

A.Md Power Radiation

The radiated power, in the upper half-space, is defined as

2 2/2

P = (1/PC) Ip(R,O,*)I 2 R sin(e)dWd) (2.17)

0 0

which becomes, on using equation (2.10)

2ff f/2

P- (PJ /4f2C) lUzla,O)l sin(e)dedi (2.18)

0 0

For the special case of aXiByMMetric excitation, it is relatively easy to show

that

ff/2

P " (PWI/21wc) z lzta (a-k IsinO)l 2 sin(e)de (2.19)

0

The sound power level in dB ref lpW is defined as

dB - 10.0 x logtolPI + 120.0 (2.20)

For the case of point force excitation with peak amplitude FO, thisdefinition of power level must be regarded as for force P0 rms; hence, thecustomary multiplication factor of one-half has been omitted in equation(2.17).

3. *NJICAL - ,

(a) 9fux"

In Figures 2-17, which show transmission, reflection and radiation

levels, the following SI constants were used:

-7-

Steel plate: A , 10.437x10 ;L , 7.558x10 1 0

E " 19.5X1010 a " 0.29

PS W 7000.0 h 0.05

Water: p - 1000.0 C - 1500.0

Plate damping was included in the computations by setting the elasticconstants to omplex values, for example ZEE(1-i)), where 71 is called thehysteretic loss factor.

(b) Acoustic Power

*Figures 2 and 3 show acoustic power radiated into the upper half-space bya plate that is excited by a IkN point force. The lower half-space contains avacuum. The power levels were obtained via numerical integration of equation(2.19). Small irregularities in the plots are due to the low, but adequate,accuracy of the integrations.

Figures 2A, 2B and 3A form a comparison of power radiated by thin, thickand exact plate theories in which the excitation is applied at the lower sur-face in the vertical direction. The effects of increasing the damping factorfrom 0.01 to 0.1 are quantitatively similar on all three plots, viz., a verysmall effect at frequencies below 5k/z rising to an OdB reduction at 50k~z.The levels, however, depend on the particular plate theory used; the thinplate theory is inadequate above 5kz and the thick plate theory is wantingabove 40kHz.

A plot of acoustic power radiated by a longitudinal force applied to thelower surface is given in Figure 3B. A comparison of Figures 3A and 3B showsthat while the power radiated by a longitudinal force is significantly less atfrequencies below 10k~z * it is only about 6dB loe than the power radiated bya vertical force at higher frequencies. Because this plot was also obtainedvia equation (2.19), allowance had to be made for the fact that the pressurevaries in the azimuthal direction as cos(*), i.e. this equation was multipliedby one-half.

(C) Radiated Pressure- Comarison of TheoriesFi Pgures 4-9 show the variation with angle • of the far-field pressure,

at selected frequencies, of te three plte .theories. The lower half-spacecontains a vacuum. The excitation in a lX point force which is appliedvertil~Bly at the loffr surface.

At 2kHz, Figure 4 ound levels of the three theories are virtually

identical. They arise from the first antisymmetric branch motion of the platesurface, viz., simple plate flexure. The small notch at 170 in the plot usingexact plate theory marks the initial *developmnt of significant sound radi-ation from the first symmtric branch motion of the plate, which is not builtinto the thin and thick plate equations.

At 5kfz, Figure 5, the 'hump' at 0-680 marks the development of coinci-

---

V dence lobe radiation: thin plate theory is now no longer suitable. At lOkHz,Figure 6, there is still good agreement between the thick and exact platetheories, except in the region of 0-170 where the first symetric branchmo tion of the exact theory continues its development. At 20kHz, Figure 7, theN! . increasing inadequacy of thick plate theory is evident at 9=70 and at all

angles greater than 450. At 30kHz, Figure 8, thick plate theory is totallyinadequate, except over a small range of angles near to the coincidence lobe

At 40kfz, Figure 9, the emergence of sound radiation due to the secondantiszyitric branch motion of the plate is evident at 9=90 in both thickand exact plate theories. However, while the thick plate theory is stillsatisfactory at the coincidence lobe, 9-350, it does not predict correctly thelevel of the lobe at 0=90.

*(d) Radiated Pressure. Exact Theory. Lonoitudinal Force

Figures 10-11 show the variation with G of the far-field pressure atselected frequencies for the case of longitudinal point force excitation onthe lower surface, which is in contact with a vacuum. The pressure, whichvaries in the azimuthal direction as cos(40), is plotted at the angle 0-00.

The sharp peaks at 9-170 are dominant. When allowance is made for theinadequate resolution of the peaks, the maximum levels are seen to be close tothose levels that would have been obtained in the absence of the plate, viz.20.0 x logl 0(Ff/c) + 120.0, which range from 182.5dB at 2kHz to 208.5dB at40kHz. The levels of the broader coincidence lobes, which occur from 10kfz,are only 3-6b less than the corresponding lobes due to vertical excitation.

(e) Exact Theory. Excitation on Eoler Surface

Comparisons have also been made between power and pressure plots obtainedfrom point force excitation on the upper and lower surfaces, respectively, butthe plots obtained from the former condition are not included herein. Powerlevels show only small differences, whereas the pressure levels may differconsiderably in the continuum background but not too significantly in regions

-close to the resonant peaks.

(f) Reflection and Transmission. Exact Theory

Figures 12 and 13 show the reflection and transmission coefficients,respectively, as a function of the angle of incidence, 9. The reflectioncoefficient is close to 0dB, except at dips which occur at the coincidenceangles of the symmetric and antisymmetric branch motions. At 20 and 40kHz,

A. there are well defined peaks in the transmission coefficient at the coinci-dence angles. The reflection and transmission characteristics of fluid-loadedplates is discussed in some detail in the papers of Freedman [6] and Fiorito(7].

(g) Perseoive and Contour Plots. Eact Thery

Figures 14-17 show perspective and contour plots of sound radiation,. ."-

V. r~...... 7,17- W.W

reflection and transmission in which the loss factor was chosen as 0. 1l. Theywere obtained via the University of Bradford's SIHPLEPOT graphics package,fro a 128x129 array of equispaced decibel data. The perspective plots arefor a user-selected angle of view of 600 with the horizontal; their appearanceis enhanced by setting a dynamic range of 40d, that is, values less than themaxim minus 40dB are set to this base level. The perspective plots enable aquick qualitative assessment of the acoustic properties of the steel plate in Iterms of the two antisymmeric and one symetric branch motions of the plate.The contour plots present the same information =u=h less vividly, but are morehelpful if precise magnitudes are of concern.

Numerical results obtained from exact linear theory have been presentedfor sound radiation, reflection and transmission of a steel plate in water.Some of the plots are new to the literature and should aid physical under-standing when studied in conjunction with som of the papers referencedherein. The comarison of sound radiation predictions of thin, thick andexact plate theories have illustrated the range of applicability of the

te plate theories. The predictions of sound radiation due tolongitudinal excitation serve as reference plots for use in the future whenpredictions obtained from more refred plate theories become available.

E.J. CLEm" (HBO)

.- 1.-

-10- -

1. PESTELU J.L., JAMES J.B., Sound radiation from layered media, AdmiraltyMarine Technology Establishment, Teddington, AMTE(N )TM79423.

2. SPICER W.J., Free-wave propagation in and sound radiation by layered media

with flow, Admiralty Marine Technology Establishment, Teddington, AMTE( N)TM82102."''

3. JUNGER N.C., FEIT D., Sound, structures and their interaction, KIT Press, .- 4

1972.

4. EVSEEV V.N., et al, Sound radiation by an infinite thin plate driven by a

longitudinal force, Sov. Phys. Acoust., 23(5), Sept-Oct. 1977, pages 418-421.

5. SERGEEV Y.D., Influence of shear, rotory inertia, and longitudinalvibrations on the radiation of sound by a point-driven plate, Soy. Phys. .Acoust., 25(2), arch-Aril 1979, pages 174-176.

6. FREEDMAN A., Reflectivity and transmitivity of elastic plates. I. Compari-son of exact and approximate theories.

7. FIORITO R., et al, Resonance theory of acoustic waves interacting with an

elastic plate, J. Acoust. Soc. Am., 66(6), 1979, pages 1857-1966.

8. WOOLLEY B.A., Acoustic radiation from fluid-loaded elastic plates. II.Symetric modes, J. Acoust. Soc. Am., 72(3), 1982, pages 959-869.

9. DOWLING A.P., PFONCS-WILLIAMS J.E., Sound and sources of sound, Ellis ..

HOrWood, 1983.

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H -C. OS E'rAo. 01 ,FREOUENCY-,OOOO. 0

FI..TANMSIN pIIETO TELPAE.NW N

-220

w

u

-40 _ __

TRANSMISSION COEFFICIENT OF STEEL PLATE. EXACT THEORY.

0.05 ET A-C. 01 FREQUENCY-40000. 0

0

-I 0

V)~* -20- ---J

-30 -- I I

9 t 2' 30 45 54 83 2 al g

ANCLE %

FIG-13 TRANSMISSION COFFICINT OF STEEL PIA2TE IN WATR

-25-

VERTICAL POINT FORCE EXCITATION I

DYNAMIC RANGE =40dB. MAX -206dB.

p

LOBGITUDINAL POINT FORCE EXCITATION vcu

9-9

-~ -S.

i.e .-

FIG.14 PERSPECTIVE PLOTS OF SOUND RADIATION

EXCITATION IS ON LOWER SURFACE

-26-

VERTICAL EXCTATION. COTUSI MLI7IPLES OF10.

*190

LONGITUDINAL EXCITATION. COUMTUMS IN WMtXIP1TS OF 10.MAXIM LEVEL~ 205.7 0B. MfINIMUM LEVEL 112.2 dB.

5580

1710ENY ~

FIG 15 CNTOU PLO OF OUNDRADITIO... CIT..ION I1ON 1W3SRFC

16

13 907

0-9

water

TRANSMISSION COEFFICIENT.-AT

DYNAMIC RA14CE 40dB.

:4

®r:5.k4

,~. .%

F..6PRPCIE LT FTASISIN&RFETO

4F Vp7 W..- I-

* RWETIoN C00fICIM.f GOMUOWS -1 -5 -10 -20 -10 -40 -30. 0

-3 -1 ~

- TRAmISSION c0apTicIEN. OUTWS -1 -5-10 -20 -30 -40 -50 -60 -?0

90- -10 -20 -0 .40 -0

-12

-20 201

.L0' -.0

1 20FREQUJENCY kflz

FIG.J17 CONTOUR PLOTS OF TRASMSION AND REFLECTION

-29-

rip' 46

, 4 4KfQ:-* ,- .Ole!

4Y

4 V4~ ~12~'