sound-structure interaction

Introduction

This is the second of a two-part tutorial on structuralacoustics written for Acoustics Today. The firstappeared in the October 2006 issue, and focused on

vibrations in structures. In that article, I explained • the various waves that can propagate through structures,

and how bending waves are dispersive (their wavespeedsincrease with frequency);

• the modes of vibration of finite structures; • mobility and impedance, and how they are simply sum-

mations of individual modal responses;• structural damping;• how the mobility of an infinite structure is the mean

mobility of a finite one; and• modeling vibrations with finite ele-

ment (FE) analysis.Some of the feedback I received

from readers of the first article pointedout that it was more of a ‘cliffs notes’summary of structural acoustics than atutorial, but found the summary quiteuseful nevertheless. After thinking aboutit, I suppose that is true, and I thoughtabout renaming the second part of thisarticle ‘A Structural AcousticsCookbook,’ rather than a tutorial. Forcontinuity, though, I have retained theoriginal title, but hope the two articles will be a useful shortreference on the subject, where handy formulas and conceptscan be easily found.

In this article, I have added a co-author—Dr. JohnFahnline—who specializes in analyzing sound–structureinteraction using boundary element (BE) modeling tech-niques. John has already written one book1 on acoustic BEanalysis (with Gary Koopmann), and is working on a second.

In this article we will explain:• what structural vibrations do to neighboring acoustic flu-

ids, and• what sound fields do to neighboring structures.These problems are complementary (and reciprocal), and wewill use analytic, numerical, and experimental data todemonstrate their basic concepts. As with Part 1 of the arti-cle, we will supply plenty of useful terms and equations.

The overarching concept of linear sound–structure inter-action is simple: the normal particle velocity in the structureand fluid along the fluid–structure interaction boundary mustbe the same. This means that when a structure vibrates againsta fluid, the component of the vibration normal to the struc-

tural surface must be identical to the corresponding particlevelocity in the neighboring fluid. This simple equality allowsus to couple the equations that define structural and fluidmotion at the fluid–structure interface and solve for the totalsound–structure behavior. While the normal particle velocityis identical in the structure and fluid, the in-plane, or tangentparticle velocity is not. In fact, we allow a ‘slip condition’between the structure and fluid, so that a structure can slidealong a neighboring fluid without inducing any sound.

Of course, with any simple concept there are inevitablyseveral assumptions. Here are ours:• homogeneity (the fluid properties are the same every-

where),• isotropy (the fluid properties arethe same no matter what direction thewave propagates), and• linearity (the fluid properties donot depend on the fluctuating pressureamplitude or phase).With these simplifying assumptions, itis straightforward to couple the vibra-tions of structures with those inacoustic fluids. Incidentally, all of theinformation in Part 1 of this articlemade the same assumptions, but for thestructural materials!

We will start by explaining what astructure’s vibrations do to a fluid—they compress andexpand it. The spatial pattern of structural vibrations andtheir frequencies determines how much sound is radiated,and in what directions. It may be helpful to think of theacoustic fluid as an elastic blob surrounding the structure,being pushed and pulled over time by the motion of thestructural boundary (in the normal direction only, ofcourse). When the fluid’s mass density is comparable to thestructure’s, the fluid not only absorbs sound, but also mass-loads the structure. We will explain how two importantstructures—a circular baffled piston, and a flat rectangularflexible finite plate—radiate sound and are fluid-loaded bythe impedance of the surrounding acoustic fluid.

Next, we will consider the complementary problem—how acoustic waves induce vibration in a structure. Thesame physics are at work in this reciprocal problem, as weshall see later. We will conclude by discussing how to makemeasurements of sound–structure interaction, and how tomodel it using boundary elements.

A clarifying note: in this article we considersound–structure interaction, not fluid–structure interaction

Structural Acoustics Tutorial 2 9

“In this article we will

explain: what structural

vibrations do to neighboring

acoustic fluids, and what

sound fields do to

neighboring structures.”

STRUCTURAL ACOUSTICS TUTORIAL—PART 2:SOUND—STRUCTURE INTERACTION

Stephen A. Hambricand

John B. FahnlineApplied Research Laboratory, The Pennsylvania State University

State College, Pennsylvania 16804

10 Acoustics Today, April 2007

vibrations are time harmonic (eiωt), and consider pressures inthe far-field, or far away from the structure. The far-fieldpressure radiated by a baffled circular piston as a function ofangle θ and distance r is:

, ( 1)

where ρ0 is the fluid density, a is the piston radius, vn is thepiston velocity (assumed to be constant over the surface ofthe piston), and ko is the radial frequency ω divided by theacoustic sound speed co. We do not include the time-har-monic dependence in this, or in any future equations.

Figure 2 shows the far-field pressure at two frequencies,normalized to koa, where koa=π and 3π. For koa=π (a fullacoustic wavelength spanning the piston diameter), the pres-sure is in phase at all directivity angles (note that the direc-tivity angle is taken from a vector pointing normal to the pis-

(FSI). FSI usually refers to how moving fluids interact withsolid objects, such as the turbulent flow over a structure. Tolearn more about FSI, we recommend the references byBlevins2 and Naudascher and Rockwell.3

Piston vibrating against an acoustic fluidPerhaps no other structural-acoustic system has been

studied more than a circular baffled piston vibrating againstan acoustic fluid (see, for example, Fahy,4 pages 58–60 and118–121; Pierce,5 pages 220–225; and Junger and Feit,6 pages95–100 and 105–109). We will describe how a piston interactswith a surrounding acoustic fluid, and how the acoustic fluidaffects the piston’s vibrations. This simple problem allows usto introduce many structural-acoustic quantities of interest.

Let us start by considering a very slowly oscillating pis-ton in a rigid baffle, as shown in Fig. 1. As the piston movesoutward, fluid flows from the area of high pressure near thecenter of the piston to its outer edge, which is nearly at ambi-ent pressure. Conversely, as the piston moves inward, fluidflows toward the center of the piston, where the pressure isless than the ambient pressure. The level of sound radiationis a function of frequency because as the speed of the oscilla-tion of the piston increases, there is less and less time for thepressure to equalize, and eventually the surface vibrationsbecome much more efficient in compressing the adjacentfluid and causing sound radiation. Since the piston’s sizecontrols how far the pressure pulses have to travel beforethey can equalize with the surrounding fluid, it is importantin determining when it begins to radiate sound efficiently.

Thus, we can conclude purely from physical argumentsthat the main parameters which control the acoustic radiationfrom a vibrating structure are its speed of vibration (or fre-quency) and its size. It is common in acoustics to nondimen-sionalize the problem in terms of the quantity ka, where k is theacoustic wavenumber and a is the characteristic dimension ofthe vibrating structure. For a piston source, the transition froman inefficient, low frequency oscillation to an efficient, high fre-quency oscillation occurs at approximately ka = 1, where a isthe radius of the piston. At this frequency, the acoustic wave-length is of the same order as the size of the structure.

A source with zero average displacement is an evenmore inefficient radiator of sound at low frequencies, like anunbaffled piston, i.e., a loudspeaker without a cabinet. Whenthe diaphragm of the speaker moves, the fluid on one side ofthe speaker is compressed while the fluid on the other side isexpanded (see Fig. 1). There is a natural flow of fluid fromhigh pressure to low pressure, and thus the fluid flowsaround the edges of the speaker. The pressure equalization isvery nearly complete because the average displacement of thesurface is zero when both sides are taken into account, suchthat there is very little residual compression of the acousticmedium and sound radiation. As the frequency of vibrationincreases, the sound radiation also increases because there isnot as much time for the fluid to flow around the edges of thespeaker before it is compressed. At approximately ka = 1, asource with zero average displacement becomes as efficient aradiator of sound as a baffled piston.

Let us consider the pressure field radiated by an oscillat-ing piston at discrete frequencies. We assume the piston’s

Fig. 1. Low frequency vibration and acoustic fluid motion of a piston. Left – baf-fled piston; Right – unbaffled piston.

Fig. 2. Pressure distribution in the far-field of a radiating baffled piston. Top –ka=π; Bottom – ka=3π.


ton surface). However, at higher frequencies, like koa=3π, thepressure alternates its phase at varying angles. The directivi-ty of the sound radiated by piston sources is usually plottedas a pressure magnitude to remove the effects of alternatingphase.

It is hard to generalize how much sound a source makesby considering pressure at a specific location. Therefore, weusually measure or compute the total radiated sound powerinstead. The total radiated sound power is computed by inte-grating the active acoustic intensity over space.

The active, or propagating acoustic intensity in a fluctu-ating acoustic pressure field is:

(2)

where v�* is the conjugate of the acoustic fluctuating particlevelocity. Notice that both the particle velocity and intensityare vector quantities, and point in specific directions. Whenin the far-field, we consider only the radially propagatingcomponent of intensity. Also when in the far-field, the pres-sure and particle velocity are in phase with each other (this isnot the case close to the vibrating surface, or in the near-field) and v=p/ρoco. So, in the far-field, the radial componentof intensity simplifies to:

(3)

Figure 3 shows the far-field intensity for ka=3π on a dBscale. Notice how all the peak locations are positive, sinceintensity squares the pressure magnitude. The highest pres-sure is normal to the piston, with lower amplitude side lobesat various angles. As ka increases, more side lobes will appear.

The total radiated power is computed by integrating theintensity (which is just the localized power/area) over a far-field half-spherical surface surrounding the piston. Sphericalsurfaces are used to make the integration simple; the totalsound power may be integrated over any shaped surface,though.

The radiated sound power is related directly to the radi-ation resistance of the fluid, which acts over the surface of the

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Fig. 3. Sound intensity distribution in the far-field of a radiating baffled piston,ka=3π.


piston. In fact, power can be calculated knowing a structure’sradiation resistance and spatially- and time-averaged normalvelocity:

. (4)

The radiation resistance and reactance are the real andimaginary parts of the fluid loading on the structure’s sur-face. Imagine once again the fluid as an encompassing elasticblob surrounding the structure. As the structure pushesagainst this blob, it encounters an impedance, which resiststhe structure’s motion. The impedance, Zo, is complex, andequal to Ro + iXo, where Xo is the reactance. The resistanceand reactance are fluid-loading properties of the structure,and do not depend on how the structure vibrates. Equationsfor the resistance and reactance (shown without derivation)of a baffled circular piston are:

, and (5)

. (6)

Note once again that no piston structural properties appear inthe above equations; they are solely determined by the geome-try of the piston and the acoustic properties of the fluid.

We plot the resistance and reactance for a 16.6 mmradius piston in water at the top of Fig. 4. At low frequencies,the resistance (in red) looks like a parabola, while the reac-tance (in green) looks like a line with a constant slope.

Popular approximations for the low-frequency resistanceand reactance are:

(7)

At high frequencies, the resistance asymptotes to a con-stant value, while the reactance decreases inversely propor-tionally to frequency:

(8)

Now, notice how Ro and Xo in the example are plotted inabsolute terms, but also plotted normalized against ρocoA (seethe axis on right side of the plot) and against dimensionlessfrequency ka (see the axis on the bottom of the plot). Recallthat you can use ka to visualize how many acoustic wave-lengths fit over the characteristic dimension a. In our plot, themaximum ka is π, which corresponds to a half wavelengthover the piston radius, or a full wavelength across the pistondiameter. Ro /ρocoA converges to a value of 1 at high frequen-cies. In fact, Ro /ρocoA is called the normalized radiation resist-ance, or more commonly, the radiation efficiency:

. (9)

We will learn more about radiation efficiency when we dis-cuss how sound is radiated by plate modes.

At the bottom of Fig. 4, examples of how the fluid load-ing varies over the surface of a piston are shown at low, mid,and high frequencies. Below a ka of π/2, the fluid loading isprimarily reactive, or mass-like, weighing down the piston.Above ka of π/2, the fluid loading becomes more resistive,absorbing energy in the form of sound from the vibratingstructure.

Now, let us suppose that the piston is the mass elementof a simple harmonic oscillator, where the mass (m) rests ona grounded spring (k) and dashpot (b). We now consider theeffect of the complex fluid loading on the piston resonance,where the piston mobility in-vacuo is:

. (10)

The fluid loading (resistance Ro and reactance Xo) may beadded to the mobility equation to produce:

(11)

For a 1 gram piston of 16.6 mm radius, with spring con-stant k=1x105 N/m, and a damping constant b of 1, we com-pute the drive point mobility v/F in air (ignoring fluid load-

Fig. 4. Radiation impedance of baffled circular piston. Top–resistance and reac-tance as a function of frequency (and ka); Bottom–spatial variability of resistanceand reactance at three discrete frequencies.

.


ing, using Eq. 10), and in water (including fluid loading usingEq. 11). Next, we multiply Ro by the square of mobility to com-pute radiated sound power. Plots of the mobility magnitude,radiation resistance (in air and in water), and the radiatedsound power for a unit force input are shown in Fig. 5.

The effect of mass loading on the piston in water is pro-nounced, shifting the piston resonance frequency down-ward, and the overall mobility amplitude downward. Theradiation resistance (and reactance) of water is much higherthan that of air (be sure to note the multiple scales used onthe resistance comparison plot). Therefore, radiated soundpower is quite different in air and in water, and the pistonresonance peaks occur at different frequencies.

The radiated sound power curve in water illustrates thebasic principle of loudspeaker design: adjust the piston prop-erties to set the fundamental resonance as low in frequency aspossible, so that the piston response is controlled by the massterm in Eq. 11. Recall that the radiated sound power is theproduct of the radiation resistance Ro and the square of thepiston normal velocity. Since Ro at low ka is proportional to(ka)2, and the piston mobility above resonance is proportionalto 1/ω2, the frequency dependencies cancel, leaving a nearlyfrequency independent radiated sound power/F2 response.

In our example in Fig. 5, the in-air radiated sound powertransfer function is not flat above the piston resonance fre-quency because the radiation resistance is above the frequen-cy range where it is proportional to (ka)2. To resolve this, aspeaker designer would simply reduce the radius of the pis-ton, shifting the radiation resistance curve further out in fre-quency. This solution comes with a cost though—the radia-tion resistance amplitude reduces with surface area, whichwill in turn reduce the radiated sound power.

Structural waves vibrating against an acoustic fluidHow well do structural waves (rather than rigid oscilla-

tors) radiate sound? Since only structural motion normal toan object’s surface induces an equal motion in a neighboringfluid, we consider transversely vibrating, or flexural waves(we acknowledge, however, that longitudinal waves deform astructure transversely due to an elastic material’s Poissoneffect, but do not focus on the sound radiated by longitudi-nal waves here).

How well flexural waves in a structure radiate sounddepends on whether the waves, which essentially act as asource against the fluid, are subsonic (slower than thewavespeed in the fluid) or supersonic (faster than thewavespeed in the fluid). Supersonic waves radiate sound, andsubsonic waves do not.

Many structural acousticians like to consider the soundradiated by structural waves in wavenumber space, andexamine wavetypes on frequency-wavenumber plots.Consider the traveling flexural waves in an infinite plateshown in Fig. 6. The flexural and acoustic wavenumbers(computed by dividing radial frequency by the flexural andacoustic wavespeeds) are plotted against frequency in the topof the figure. Since the acoustic waves are non-dispersive, thewavenumber curve has constant slope. The flexural waves,however, are dispersive, causing a varying slope in thewavenumber curve.

At low frequencies, the structural wavenumbers arehigher than those in the acoustic fluid, corresponding to sub-sonic structural waves. These flexural waves radiate nosound at all (this is only true for infinite plates—we will dis-cuss the sound radiated by finite plates soon). This is becausethe particle velocity in the fluid normal to the structure’s sur-face must match that of the structure. At low frequencies,acoustic waves are faster than structural ones, so their wave-lengths are longer. This means the structure simply cannotinduce a propagating wave in the fluid.

The frequency at which the flexural and acoustic waveshave the same wavenumber (and wavespeed, and wave-length) is called the coincidence frequency, and the flexural

Fig. 5. Mobility magnitudes (top), radiation resistances (middle), and radiatedsound power transfer functions (bottom) of a 16.6 mm radius circular baffled pis-ton in air (black) and in water (blue).


waves now radiate sound, as shown in the image in the middleof Fig. 6. They do so in the plane of the plate, or grazing theplate. At frequencies above coincidence, the flexural wavescontinue to speed up—eventually becoming pure shear wavesat very high frequencies (see Part 1 of this article). At thesehigh frequencies, the sound radiated by the flexural wavespropagates in a preferred direction, which is computed bytrace matching the flexural wave to the shorter acoustic wave,as shown in the image on the bottom right of Fig. 6.

The angle of dominant radiation (taken from a vectornormal to the structure’s surface) is computed as:

, (12)

where cB is the flexural wavespeed. You can compute a plate’scoincidence frequency by setting the flexural and acousticwavespeeds (or wavenumbers) equal to each other. Forbending waves in thin plates at coincidence:

, so (13)

(14)

How does coincidence frequency vary with plate parame-ters? To find out, let us examine Fig. 7. Increasing a plate’s

elastic moduli (Young’s Modulus E and Shear Modulus G)speeds up flexural waves, and lowers the plate’s coincidencefrequency. Increasing a plate’s density increases its mass,slowing down flexural waves, and raises the plate’s coinci-dence frequency. Increasing thickness increases both stiff-ness and mass, but increases the stiffness at a greater rate, sothickening a plate will lower its coincidence frequency.

Therefore, stiffening a plate lowers its coincidence fre-quency, allowing it to radiate sound at lower frequencies.Conversely, mass-loading a plate raises its coincidence fre-quency, so that the plate does not radiate sound at low fre-quencies. It would seem that the answer to most noise con-trol problems would be to simply add mass to a plate whilereducing its stiffness! While this is a good way of reducingsound radiation, we have never had any of our sponsorsaccept it. As most of us have experienced, nearly all newstructures are lightweight and stiff, like carbon-fiber com-posites reinforced with ribbing. These sorts of structures typ-ically have low coincidence frequencies, and therefore radi-ate sound very well. Also, as we will see later, lightweight stiffstructures are very easy to excite by sound waves.

Since most practical structures are finite, we will nowexplain how well the mode shapes of a structure radiate (seePart 1 of this article for a discussion of structural reso-nances). The classic example studied by early structuralacousticians is our old friend—the simply supported rectan-gular plate.

Before we proceed with modal sound radiation, remem-ber a key concept: the standing waves in a mode shape arecomprised of multiple left and right (and forward and back)traveling waves, which propagate at the structure’s wavespeed. It is these traveling waves that radiate, or do not radi-ate sound. If our plate were infinite, such that there were noreflections from any of the plate boundaries, subsonic flexur-al waves would radiate no sound. However, the discontinu-ities at the boundaries ‘scatter’ the energy in subsonic flexur-al waves into many wavenumbers, some of them supersonic,so that a finite plate radiates sound below its coincidence fre-quency. The amount of sound radiation depends on the radi-ation efficiency of each of the plate’s modes.

Maidanik7 and Wallace8 computed how much sound afinite rectangular simply supported plate’s modes radiate. Inparticular, Wallace provides formulas for the frequency-

Fig. 6. Bending and acoustic wavenumber-frequency plot, with trace diagrams ofbending and acoustic waves at and above coincidence. Below coincidence, an infi-nite bending wave radiates no sound!

Fig. 7. Effects of stiffening and mass-loading on plate coincidence frequencies.Stiffening a plate reduces the coincidence frequency, and adding mass increases it.

.


dependent far-field pressure and acoustic intensity fields radi-ated by a given plate mode, along with an integral for comput-ing the sound power radiation efficiency for each mode. Weshow the far-field intensity below, near, and above the platecoincidence frequency for the first three low-order modes of asquare 1m x 1m plate in Fig. 8. Recall from the first article thatwe define mode orders as (m, n) pairs, where m and n corre-spond to the number of antinodes (regions of maximum defor-mation) in the plate’s x and y directions, respectively.

In the figure, the mode shapes of the plate are shown,along with the corresponding far-field intensity patterns. Atlow frequencies, the fundamental (1,1) mode radiates soundomnidirectionally, like a baffled circular piston. Also at lowfrequencies, the (1,2) mode radiates sound like a dipole, andthe (2,2) mode radiates like a quadrupole.

The figure also shows radiation efficiencies as a functionof acoustic wavenumber (frequency/acoustic wavespeed),with a line shown to indicate the frequencies of the directiv-ity plots. Notice how they resemble the normalized radiationresistance of the baffled circular piston (Fig. 4). Below coin-cidence, the efficiencies increase rapidly with increasing fre-quency. The efficiencies peak at coincidence (exceeding 1,showing that they are not true efficiencies!), and then asymp-tote to a value of one above coincidence.

Above coincidence, the far-field sound directivitychanges, with ‘lobes’ of sound radiated from the structure atcritical angles. These critical angles may be computed usingthe trace matching procedure described above for the infiniteplate. The critical angles exist for all plate modes except thefundamental (1,1) mode, which radiates sound normal to theplate at all frequencies, but with a spatial ‘beamwidth’ thatnarrows with increasing frequency.

Let us re-examine the radiation efficiencies below coin-cidence for the different mode orders. We see that the (1,1)mode radiates sound most efficiently, followed by the (1,2)mode, and then the (2,2) mode, which radiates sound leastefficiently. This trend shows an important result: modes withodd m and n orders radiate sound much better than thosewith mixed orders (odd-even or even-odd), which radiatebetter than those with purely even orders (even-even).

Using the analytical radiation efficiencies, we can com-pute how much sound a rectangular plate radiates whendriven by a point force. To do so, we combine the mobilityformula from Part 1 of this article with radiation efficienciescomputed using the formulas in Wallace.8 Ignoring any fluidloading effects on the structure (which we know will mass-load and radiation damp the structure from our exerciseswith the baffled circular piston), we show the mobility, radi-ation efficiency, and radiated sound power for a unit forcedrive in Fig. 9.

Starting with the mobility (top of the figure), we see howthe contributions from the individual modes compare to thetotal mobility. The mobility is dominated by resonantresponse at the resonance frequencies, and a mix of non-res-onant responses away from resonance.

Next, examining the radiation efficiencies of the modesshows the same trends we observed in Fig. 8—that odd-oddmodes radiate sound very well, but modes with mixed andeven orders radiate sound poorly. The fundamental (1,1)

mode radiates sound very much like a baffled piston, and theradiated sound power transfer function (power normalizedby the square of the input force) is dominated by the soundradiated by that mode. Peaks in the sound power occur forthe other modes, primarily the other odd-odd (1,3) mode,but the non-resonant sound radiated by the (1,1) mode isnearly flat with increasing frequency.

Fig. 8. Far-field sound intensity in air for various low-order simply supported 1 msquare flat plate modes. Top–below coincidence, Middle–near coincidence,Bottom–above coincidence.


This example brings up one of the key points of this arti-cle—non-resonant sound radiation can be higher than resonantsound radiation. The sound radiated by the (2,2) mode at its res-onance frequency just below 100 Hz is a good example—it islower than the non-resonant sound radiated by the (1,1) mode!

When we coupled the radiation resistance of a baffledpiston to a simple harmonic oscillator (where the pistonhead was the mass), we added it to the resistance of the oscil-lator. For finite structures, most analysts use mechanical lossfactors to model structural damping. We can compute a radi-ation loss factor which can be used in a structural analysis torepresent the radiation damping of an acoustic fluid. A use-ful equation for the radiation loss factor of a uniform thick-ness, homogenous plate or shell structure (flat or curved) is

(15a)

The equation shows that the radiation loss damping isdirectly proportional to fluid density, meaning that the heav-ier the fluid, the more sound power is radiated, and the moreheavily damped the structure. An example would be a vibrat-ing bell dunked in water. While in the air, the bell’s vibrationswould ‘ring’ for a long time. When immersed in water, thevibrations decay quickly. An extension of the above equation,well known in the Statistical Energy Analysis (SEA) commu-nity, relates the radiation loss factor to the radiation resistance:

(15b)

where M is the mass of the plate or shell.Table 1 provides a useful list of the various sound radia-

tion quantities and how they inter-relate.

The complementary problem—Structural vibrationsinduced by acoustic pressure waves

Whereas the sound radiated by vibrating objects is oftenjust an annoyance, the vibrations induced in structures byimpinging acoustic waves can be so high that the structures Fig. 9. Mobility magnitude (top), radiation efficiency (middle), and radiated sound

power transfer function (bottom) of a simply supported 1m square 5 mm thick flatsteel plate in water driven at its quarter point (x=0.25 m, y=0.25 m).Table 1. List of sound power radiation quantities and their interrelationships.

.

,


crack and fail. This is clearly a more serious problem, and hasbeen studied intently by those in the aerospace and nuclearcommunities, particularly in the 1960’s and 1970’s.

Try holding your hand or fingers lightly on any struc-tural surface, like a window when there is a lot of noise out-side. You will feel the window vibrating. The physics thatexplain this phenomenon are the same as those that explainhow a vibrating structure radiates sound. As we will see,there are several quantities in sound-induced vibration thatare complementary to those in vibration-induced sound.

Let us start by considering the acoustic field adjacent toa flat infinite plate that is struck by an incoming acousticwave. There are many pressure waves next to the structuralsurface—the incident wave, a reflected wave, and a wave re-radiated by the structure, which has been forced into vibra-tion by the incident and reflected waves. The sum of the inci-dent and reflected waves forms a ‘blocked’ pressure on thesurface, and if the surface is rigid, the blocked pressure fieldis the total pressure. If the structure is flexible and vibrates, itradiates a third pressure wave, which sums with the blockedpair to form the overall pressure field.

Figure 10 shows the incident and reflected waves for a 30degree angle of incidence (the angle is taken from the direc-tion normal to the plate), along with the total blocked pres-sure field acting on a rigid surface. Notice how the two wavescombine to form a standing wave pattern in the directionnormal to the surface. The standing wave pattern propagatesin the direction parallel to the surface at a speed cosin(φ),where φ is the angle of incidence.

Infinite plate theory (discussed in Part 1 of this article)can be used to estimate the plate vibration caused by theblocked pressure field:

(16)

where zfluid and zplate are the impedance (pressure/velocity inthis case) of the acoustic space and the plate.

The fluid impedance depends on the pressure wave’s

angle of incidence and the fluid’s characteristic impedance:

(17)

and the infinite plate’s impedance is

(18)

where η is the structural loss factor and h is the plate thickness.Examining the equation, we can see that the plate will

vibrate most when its structural impedance is minimized.This will occur when the stiffness and mass terms in theplate’s impedance cancel each other, or when D(ko sinφincident)4 =ρhω2 Let us find this frequency. To do so, we replace ko withω/co, and we find that:

(19)

ωco looks similar to a plate’s critical frequency, and to its crit-ical radiation angle for frequencies above coincidence. Infact, when φincident is 90 degrees (the acoustic waves propagatein the plane of the plate, or ‘graze’ the plate), the coincidencefrequency is the critical frequency. The critical frequency, infact, is sometimes called the lowest coincidence frequency.This means that there is no single frequency where the platevibrates most—there are several of them, each of which cor-respond to a different angle of incidence.

Now let us consider how finite structures respond toincident acoustic waves. However, we will do so not forwaves that are incident from specific angles, but for groupsof waves that are statistically incident from all angles, morecommonly known as a diffuse acoustic field. Smith9 consid-ered how a simple harmonic oscillator responds to diffuseacoustic fields in 1962, where the mean square velocityresponse of the oscillator is

(20)

where M is the oscillator mass, Rradiation and Rstructure are theradiation and structural resistances, and Gpp is the autospec-trum of the incident pressure.

The randomly excited simple harmonic oscillator prob-lem is one of the origins of Statistical Energy Analysis (SEA),a popular structural-acoustic analysis technique. For multi-resonant structures, like plates, the plate energy is computedrather than velocity:

(21)

Here, the radiation and structural resistances are averagedover a frequency band spanned by ωmin and ωmax. Also, weinclude the modal density nplate, which defines the number ofplate modes in the frequency band. As we can see, SEA is sta-

Fig. 10. Blocked pressure field acting on a plate (right side of images) at 30 degreeangle of incidence. ‘+’ and ‘–’ signs indicate phase variations in the waves.

,

,

,

.

,


tistical, averaging over bands of frequencies. SEA also com-putes averages of response over space, since <E>plate is aver-aged over the plate.

A full discussion of SEA is outside the scope of this tuto-rial, but you can find more information on SEA in the reviewpaper by Burroughs10 and a more thorough treatment of howstructures respond to diffuse fields in the article by Shorterand Langley.11

You can use Eq. 21 without using SEA, though. You justneed to know the incident pressure spectrum of the diffuseacoustic field, and the radiation and structural resistance. Youalso need an estimate of the plate’s modal density, which is

(22)

Note that the plate’s modal density depends on frequency (asdo all modal densities).

Examining the equation, we see that the higher the radi-ation resistance, the higher the plate energy (and thereforethe higher the plate’s vibration). So, the better a plate radi-ates, the easier it is to excite with incident pressure fields!

Now, what happens when there is also fluid on the otherside of the plate? How much of the incident sound getsthrough the plate to the other side? This is the classic soundtransmission loss problem, and may be solved easily for aninfinite plate, and not so easily for a finite one.

Fahy4 provides a derivation of the sound powertransmission coefficient through an infinite plate in his text-book, and we repeat it here (assuming the fluids on bothsides of the plate are the same):

(23)

The red, green, and blue terms in the equation represent thedamping, mass, and stiffness of the plate, respectively. Theamount of sound transmitted depends on the fluid proper-ties, the structural properties, frequency, and the angle of theincident pressure wave with respect to the plate.

Some typical transmission loss plots, computed as10log10(1/τ) are shown in Fig. 11. For acoustic waves not nor-mally incident to the plate, sharp dips appear in the transmis-sion loss. These dips correspond to sharp peaks in the trans-mission coefficient (transmission loss is the inverse of thetransmission coefficient), and act as strong pass-bands of inci-dent sound. The dips are at the coincidence frequencies of theplate. Recall that the coincidence frequencies depend not onlyon the plate, but on the angle of incidence of the sound waves.As the angle of incidence changes, the coincidence frequencyand the frequency of the transmission loss dip changes as well.

At low frequencies, the mass term in Eq. 23 determinesthe transmission loss, which increases with the square of fre-quency (6 dB/octave, or 6 dB for each doubling of frequen-

cy). At high frequencies, above the coincidence dip, platestiffness is dominant, and the transmission loss increaseswith the 6th power of frequency, or 18 dB/octave. Figure 11shows what most people already know from experience—it ishard to keep low-frequency sounds from propagatingthrough barriers. Consider this the next time you close adoor to block out sound from a hallway or another room.You stop hearing mid-high frequency sounds, but still hear‘muffled’ low-frequency noise.

To visualize the sound field incident on and transmittedby an infinite plate, Fig. 12 compares pressure and displace-ment of a plate at two conditions: well below, and near coin-cidence. In the example, we have set the plate loss factorequal to 0. Try setting loss factor to zero and computing thetransmission coefficient in Eq. 23 at coincidence (remember,this is where the acoustic wavenumber in the plane of theplate matches the free bending wavenumber in the plate, orkosinφ = kb). You should compute a transmission coefficientof 1, which is perfect sound transmission!

The strength, or depth of the coincidence dip dependsstrongly on the plate’s loss factor η. Designers of noise barri-ers (windows, doors) try hard to minimize the depth andbreadth of the coincidence dips. The most commonapproach for mitigating coincidence dips is using con-strained layer damping, or CLD (we learned about this inPart 1 of this article). Automotive glass in luxury vehicles,and glass in high-end office buildings usually have a thinlayer of clear vinyl sandwiched between two panes of glass.

For zero, or normal angle of incidence (sound wavesnormal to the plate’s surface), the transmission coefficient isnot indeterminant (you might think it would be, since thereare several terms in Eq. 23 that divide by sin(φ)). The trans-mission coefficient actually simplifies to:

, (24)

which corresponds to the well-known ‘mass law.’ The masslaw transmission loss is shown in green in Fig. 11, and

Fig. 11. Typical transmission loss plot for variable angles of incidence. Grazing inci-dence refers to acoustic waves that are nearly in the plane of the plate.

.

,


increases with the square of frequency over all frequencies,never showing a coincidence dip.

To improve transmission loss, many noise control engi-neers use two panels, particularly in windows. Called ‘doubleglazing,’ the panels are separated by an air (or other gas) gap.Since transmission loss is additive, there is a substantialimprovement, much more so than simply increasing thethickness of a single panel. However, most double panel sys-tems have different panel thicknesses so that the coincidentdips of the two panels occur at different frequencies. Whilethis leads to two pass-bands for incident sound, it is general-ly preferable to have two weak pass bands rather than onevery strong one.

In most practical situations, acoustic waves impingingon a panel do not arrive from the same angle. Consider awindow in an office building or a hotel. Sound waves arrivefrom all angles, with a random distribution of the angles overtime (imagine incident sound from passing airplanes andautomobiles, reflecting off of adjacent buildings). Therefore,statistical methods have been used to estimate a randomangle of incidence transmission loss for low frequencies:

(25)

where TL0 is 10log10(1/τφ=0), or the mass-law normal inci-dence transmission loss. Since transmission loss is maxi-mized (transmission coefficient is minimized) at normalincidence, the random angle of incidence transmission loss isa fraction of that at normal incidence.

MICROPHONE ARRAY MODULES

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Depending on the application and the nature of the emitted sound field, the microphones are available as pressure- or free-field types, all with a built-in Transducer-Electronic-Data-Sheet (TEDS).

For multi-channel systems without TEDS options, we offer uni-gain versions with pre-adjusted gain.

The array modules for mounting the microphones are available in various spacing configurations depending on frequency requirements. The amount of microphones and modules in the array is only limited by the channels of the analysis system.

As always, we are ready to act on request, if our standardsolutions do not fit in with your requirements.

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Fig. 12. Incident and transmitted sound fields around an infinite 25 mm thick steelplate, 30 degree angle of incidence. Top–at 50% of coincidence, Bottom–at coinci-dence. Dark blue and red indicate high pressures, and green indicates low pressures.


Be careful using the above expression at high frequen-cies! It does not include the coincidence dip, or the high-fre-quency stiffness effects. To consider these effects, and thoseof finite panel boundaries, other techniques are used, likeSEA. In fact, one of the early SEA applications was for singleand double panel transmission loss calculations. Price andCrocker’s famous papers12, 13 clearly show the coincidencedips in transmission loss, and the importance of varyingpanel thicknesses in double panel systems.

Measurements of sound—structure interactionThe sound radiated by vibrating structures, and conversely,

the vibrations induced in structures by sound fields, are com-monly measured in two types of chambers: reverberation rooms,and anechoic rooms. Interconnected rooms are used to measurethe transmission loss of barriers, like windows and doors.

Accelerometers mounted to structures can provide aspatially averaged normal velocity. Also, non-contact veloci-ty measurements are often made using laser vibrometry,eliminating the mass-loading effects of the accelerometers onthe test structure, and reducing test times.

Measuring sound power is more difficult than measur-ing vibration. In air, arrays of microphones may be used tomeasure the spatial variability of pressure. In water,hydrophones are used. Sound pressures are measured quitedifferently in reverberant and anechoic rooms, though.

In a reverberant room, the acoustic modes of the air inthe room are excited by a vibrating structure. When theacoustic cavity modal density is high, the room’s pressurefield becomes nearly diffuse, such that pressures measured atjust a few locations randomly spaced throughout the roommay be used to estimate the total sound power radiated by avibrating object. The sound power is computed using themeasured pressures, along with a few of the reverberationroom’s parameters, such as the volume and the reverberationtime. The reverberation time is inversely proportional to theroom’s loss factor—the longer a pressure impulse takes todecay, the lower the room’s loss factor.

The accuracy of a reverberation chamber sound powermeasurement depends on the room’s modal density and rever-beration time (among other things). The sound power is actual-ly a statistical mean, which is bracketed by a standard deviationat each measurement frequency. In general, the higher a room’smodal density and the wider the frequency bandwidth, thesmaller the standard deviation. Most reverberation room soundpower measurements are therefore made over wide frequencybands that contain several acoustic modes. One-third octavefrequency bands are commonly used.

There are standards14, 15 available to help an experimen-talist quantify a reverberation chamber’s characteristics, andconduct a sound power measurement. The standards, how-ever, are specific to rooms filled with air. To make measure-ments in a reverberant water tank, some modifications to theapproaches in the standards are necessary. Conlon describesthese modifications, and applies them to ARL/Penn State’sreverberant water tank in his NoiseCon 2004 article.16

Photographs of a metal pressure vessel being tested inARL/Penn State’s tank are shown in Fig. 13. The vessel is

struck by an impact hammer at several points, and the pres-sures measured at five hydrophone locations are used to esti-mate the sound power transfer function (Prad/F2).

The sound power transfer functions for several drive loca-tions on an elbowed pipe (the same pipe described in Part 1 ofthis article) are shown in Fig. 14, adapted from anotherNoiseCon article,17 this one from 2005. The measurementswere made in one-third octave bands, and show that thesound power transfer functions vary with drive location (justas mobility functions do). Annotations on the graph showwhere various shell modes cut on (again, see Part 1 of this arti-cle to refresh your memory on what shell modes are).

It is often useful to compare sound power transfer func-tions to that of an ideal dipole source:

(26)

where |F|2 represents the square of the r.m.s. force amplitude.This would be the sound made by an oscillating point drivein space. At some frequencies, most notably those of struc-tural resonance, the sound power transfer function for theelbowed pipe is higher than that of a point dipole, showing

Fig. 13. Impact hammer measurements on cylindrical shell structure submerged inARL/Penn State Reverberant Water Tank. Top–shell submerged in tank with diver;bottom, shell suspended over tank.

,


that the structure amplifies the force drive at those frequen-cies. At other frequencies, usually between structural reso-nances, the structure attenuates the drive.

The sound power transfer functions can also be com-bined with a surface-averaged mobility measurement tocompute radiation efficiency:

(27)

Pressure, intensity, and sound power are measured very dif-ferently in anechoic rooms. The walls of a typical anechoicchamber are coated with sound absorption materials. Foamwedges are common, as shown in Fig. 15. The absorbingmaterial nearly eliminates the reflection of sound by thewalls. The pressure measured around a sound source placedin an anechoic room is due only to acoustic waves propagat-ing away from the source.

Anechoic environments allow for more refined meas-urements of sound fields, both over space and frequency.Directivity plots like those in Fig. 3 and Fig. 8 may be meas-ured. Also, narrow-band frequency spectra may be comput-

ed. However, whereas only a few pressure measurements arerequired in reverberant rooms to compute radiated soundpower, many more pressure measurements are required todo so in an anechoic room.

Total sound power must be integrated over a closed sur-face around the sound source. Also, acoustic intensity mustbe measured, sometimes in the near-field of the structure.This means many intensity measurements are made overseveral points around the structure. The points may bedefined over a hemisphere or a box-like shape. Standards areavailable18 to guide your measurements, and acoustic intensi-ty probes, like the one shown in Fig. 16, are widely available.

Fig. 16. Hand-held acoustic intensity probe. Two microphones are separated by aknown distance ∆x so that particle velocity may be computed using a finite differ-ence approximation. The mean pressure and estimated particle velocity are com-bined to compute the active, or real part of acoustic intensity in the direction alongthe x-axis.

Two measurement chambers are required to determinea barrier’s transmission loss. Often, a thick, nearly rigid wallis built between the two rooms. The wall includes an open-ing, into which a barrier is mounted. One of the rooms isconsidered the ‘source’ room, and is ensonified with soundwaves, which strike the barrier. The sound power in the‘receiver’ room is measured, and a transmission coefficient iscalculated. Usually, the source room is reverberant, and thereceiver room is anechoic, with measurements made accord-ing to the procedures we described above.

Boundary element modeling of sound fieldsBoundary element methods have been used to predict

acoustic fields for about 45 years, and there is a vast amountof literature on the subject. Although these methods can beused to compute both interior and exterior acoustic fields,the focus here is on computing exterior acoustic fields whereboundary elements have distinct advantages. Since this arti-cle is at the level of a tutorial, our main goal will be to give thereader a brief synopsis of the basic analysis techniques andthe current state of the art. For the sake of simplicity, thespecified boundary condition and radiated acoustic field areassumed to be time-harmonic.

Fig. 14. Sound power radiated by Schedule 10 Steel 3 inch pipe with elbow, pipelength/diameter~12.

Fig. 15. ARL/Penn State's Anechoic Chamber.

.


A completely general solution for the pressure field of avibrating structure can be derived in the form of an integralequation, which is an equation with an unknown functionunder an integral sign. Even though we cannot actually solvethe integral equation for the unknown function explicitly forany but the simplest problems, it is still useful to know theform of the solution in terms of physical variables. Also, thesolution provides great insight into how the problem mightbe solved numerically, and, as such, the Kirchhoff-Helmholtz integral equation (KHIE) commonly provides thestarting point for many numerical formulations:

(28)

In the KHIE, r represents a point in the acoustic fluidand q is a point on the boundary of the fluid S, or boundarysurface. In the context of sound radiation from structuralvibrations, the boundary surface is the outer surface of thestructure in contact with the acoustic medium. The functionG stands for Green’s function, and G and dG/dn in the KHIErepresent the acoustic fields of simple (monopole) and dipolesources, respectively, distributed along the boundary surface.The dipole sources are aligned in the direction perpendicularto the boundary surface. The monopole and dipole sourcesare functions of the point q where they are located and thefield point r. The KHIE shows that the acoustic field dependsonly on what happens at the boundaries since the surfacevibrations cause the radiated acoustic field.

By studying the solution for the acoustic field given bythe KHIE, much can be learned about how to derive anumerical solution. Unfortunately, its simplicity is deceiv-ing. The acoustic fields of simple and dipole sources are sin-gular functions of the source and field point locations, suchthat they become infinite when the two points coincide(when r � q ). However, in the KHIE, these singular func-tions add together to yield nonsingular pressure and velocityfields. Taking R to be the distance between r and q, the sin-gularities for the different sources can be categorized as:

(1) pressure of a simple source G ~1/R, (2) velocity of a simple source dG/dn0 ~ 1/R2, (3) pressure of a dipole source dG/dn ~ 1/R2, (4) velocity of a dipole source d(dG/dn0)/dn ~1/R3. The process of taking a derivative makes a singular func-

tion more singular. For example, taking the derivative of r-1

with respect to r gives -r-2. When r is less than unity, the func-tion r-1 is smaller than r-2, so that r-2 goes to infinity faster asr � 0. A function that depends on 1/R is called weakly sin-gular, one that depends on 1/R2 is called strongly singular,and one that depends on 1/R3 is called hyper-singular.

How is it possible, then, for singular functions to add upto give finite pressure and velocity fields? First, integration isa “smoothing” operation. Thus, it tends to reduce the level ofsingularity, the same way differentiation increases it. Becausethe functions are integrated over a surface, this processessentially reduces the level of singularity by two orders.Thus, it makes sense that weakly and strongly singular func-

tions should yield finite values. However, we would notexpect a hyper-singular function integrated over a surface toyield a finite value. Indeed, the only reason the velocity fieldof a dipole source yields a finite value is because the surfacepressure and normal velocity, which are weighting functionsfor simple and dipole sources, are related through a gradientoperation. To illustrate, consider a right-angled corner,where the surface normal is discontinuous. As we travelalong the surface towards the corner, the pressure field mustchange such that the gradient operation produces the correctvelocity on the other side of the corner.19 The level of conti-nuity in the pressure field is also important. In the exact solu-tion, the surface pressure is an absolutely continuous func-tion, such that the function itself, as well as all of its deriva-tives, are continuous. This level of continuity is impossible toduplicate with simple polynomial approximations. However,the pressure and surface velocity, taken together, mustenforce - iωρ v = - p as we travel along the boundary. Thus,it is impossible to duplicate the level of continuity in theactual surface pressure distribution with simple interpola-tion functions and it is also impossible to exactly enforce thespecified boundary conditions!

From a practical point of view, how is all this relevant?First, many of the research papers written about boundaryelement methods in the last ten years are concerned withsorting out the mathematical details of the integrals. Thesepapers are written by, and for, people writing their ownboundary element codes. A novice would find it very difficultto understand all the mathematical complexities.(Admittedly, even after years of dedicated effort, we findmany of the papers incomprehensible.) A reader interestedin a simple explanation of boundary element methods willfind the earliest papers on the subject, written in the 60’s,much easier to understand.20, 21, 22 Also, for the casual userswho are not trying to write their own boundary elementcode, it is only important to understand that current bound-ary element codes are not perfect. They probably do not takecare of the singularities in the KHIE such that the acousticfield is strictly non-singular. Nonetheless, it has been wellshown over the years that even simple approximations forthe pressure and normal surface velocity weighting functionsyield adequate numerical solutions for many problems.22

After constructing and solving the matrix system, thepressures and normal velocities are known for each elementof the boundary surface. The KHIE can then be used todirectly compute the pressure at each desired field pointlocation. The overall sound power output can be computedby setting up a grid of field point locations on a surfaceenclosing the structure and numerically integrating theacoustic intensity over the surface. In theory, the power out-put can also be computed directly using the pressure andnormal velocity on the boundary surface, but this is prob-lematic because this is where the largest errors tend to occurin a boundary element solution.

Aside from the mathematical details in evaluating theintegrals, most recent innovations in boundary elementmethods have concerned ways to increase the speed of thecomputations and reduce storage requirements. As with all

∆


numerical computations, boundary element methods havebenefited enormously from the rapid increase in computerprocessor speed and memory. Since boundary element com-putations are repeated many times at different frequencies, itis easy to split jobs up and assign them to different comput-ers or processors.

In practice, boundary element methods are very com-petitive for all but the very largest engineering problems,where the matrix inversion and multiplication dominate thesolution times. This is especially true if the time required tocreate models is factored into the analysis. In this respect,boundary element methods are relatively quick and easy.Simple rules of thumb can be used to determine if boundaryelement methods will be applicable for a specific problem.Generally, six elements are required per acoustic wavelengthto accurately compute an acoustic field. Knowing the surfacearea S of the boundary and the maximum frequency of inter-est, the required number of acoustic elements is given asS/(π/3fmax)2. Three matrices of this size will be required tocompute the matrix multiplication, and it is usually best todo the computations in double-precision, thus requiring 16bytes for each element of the matrices.

Modeling sound—structure couplingRecall that a moving structure compresses and contracts

neighboring fluids, which act as a continuous elastic blobaround the structure. When the fluid is heavy, its pressureloading affects the structural vibrations. To compute theseeffects, the boundary element model must be coupled to arepresentation of the structure.

The previous analysis has shown that an acoustic imped-ance matrix, relating the pressures to the normal componentof velocity on the boundary surface, can be computed as afunction of frequency using boundary element methods. Wenow want to combine this result with a finite element analy-sis of the structural vibrations to include the effects of thepressure field. In a finite element analysis, the displacementsare written in the form

, (29)

where M, B, and K are the mass, damping, and stiffnessmatrices, d is the displacement field of the structure, and F isthe vector of forces applied to the structure.

To include the pressure field in the finite element analy-sis, the impedance matrix must be transformed from itsdependence on normal surface velocity to nodal displace-ments. For time-harmonic problems, the displacement fieldon the outer surface of the structure can be used to computethe normal component of the surface velocity as a simpledot-product: vn = v . n = -iω d . n. Thus, a matrix relationshipcan be derived to convert the complete displacement fieldinto a normal surface velocity vector. This matrix will havezeros for all interior nodes not in contact with the surround-ing fluid. Similarly, the matrix will also be zero for displace-ment degrees of freedom on the boundary surface tangentialto the outward surface normal. Post-multiplying the acousticimpedance matrix by the transformation matrix then yieldsa matrix relationship between the pressure field on the

boundary and the finite element displacement vector.Including the result for the pressure field on the boundarysurface in the finite element equations of motion yields

. (30)

Given an input force vector, it is theoretically possible tosolve this equation for the displacement vector, which nowincludes fluid coupling. In a finite element analysis, the equa-tion system is highly-banded because each element onlyinteracts with other elements through the nodes. However,in a boundary element analysis, every node interacts withevery other node, so that the acoustic impedance matrix isgenerally fully-populated. It then becomes very time-con-suming to solve the matrix system in its present form.Various alternative strategies are possible where the matricesare subdivided into degrees of freedom with and withoutfluid coupling. It is also possible to treat the acoustic pressurefield on the boundary surface as the primary variable.21

Ultimately, many researchers instead reformulate the prob-lem in terms of a modal frequency response analysis (whichis the approach we use at Penn State). In a modal frequencyresponse, the mass, damping, and stiffness matrices are pre-and post-multiplied by mode shapes, and the applied forcevector is pre-multiplied by the mode shapes. The resultingsystem, which is usually much smaller in size than the systemdefined in physical coordinates, is solved to compute modalcoefficients. The modal coefficients tell us how much eachmode contributes to the overall response, and are multipliedby the mode shapes to compute that physical response.

Using the coupled FE/BE formulation, the time requiredto compute and store the acoustic impedance matrix as afunction of frequency typically dominates the overall solu-tion times. To speed up this part of the process, it is commonto condense the structural displacement variables into acoarser set of acoustic variables. To illustrate, Fig. 17 showstwo structural and acoustic meshes for a loudspeaker.

Since both the structural and acoustic fields requireapproximately six elements per wavelength, this processassumes that the structural waves vary more rapidly than theacoustic waves. This is true below the coincidence frequency,and is valid up to a relatively high frequency for structuressubmerged in water.

To demonstrate the typical results of a coupled FE/BEanalysis, two example problems will be considered. The first

Fig. 17. A structural mesh for a speaker with two different acoustic meshes.


is the familiar example of a thin circular cylindrical shell inan infinite circular baffle. For simply-supported boundaryconditions, the structural displacements can be determinedanalytically. Similarly, the acoustic Green’s function isknown analytically for a simple source outside of a rigid cir-cular cylinder, and thus for this problem the acoustic fieldcan also be determined analytically, although it can only beevaluated numerically. Solutions for the in vacuo and fluid-coupled resonance frequencies are given by Berot andPeseux.23 For this problem, the cylinder has radius 0.4 m,length 1.2 m, and is 3 mm thick. Its material properties aregiven as E = 200 GPa, ρ = 7850 kg/m3, v =0.3, and the sur-rounding water has properties co = 1500 m/s, and ρo = 1000kg/m3. We note that the thickness–to-radius ratio is lessthan 0.01 for this example, and thus thin shell theory isapplicable and we expect a relatively large shift in the reso-nance frequencies due to fluid-loading. Our goal will be toreproduce Berot and Peseux’s results. In the numericalcomputations, the structural analysis proceeds as usual, butwe cannot simply model the boundary surface in theacoustic analysis because the circular baffle extends toinfinity in both directions.

Following the procedure used by Berot and Peseux intheir numerical computations, we will truncate the baffleon both sides of the cylinder approximately half its lengthbeyond the vibrating portion and close off the ends with flatendcaps. Figure 18 shows the first two computed in vacuomode shapes and resonance frequencies for the shell.

In the finite element analysis, the displacement is set tozero at the ends of the cylinder in the radial and torsionaldirections, but the axial displacement is unconstrained. Aswe will show, a fairly fine mesh is required in the circum-ferential direction to properly represent the acousticimpedance of the higher order modes. For example, to cor-rectly represent a mode with 8 circumferential wavelengths,we need (8)(6)=48 acoustic elements around the circumfer-ence. The acoustic mesh does not have to be nearly asrefined in the axial direction because the structural stiffnessis much higher. With this in mind, the structural mesh has24 elements in the axial direction and 48 elements aroundthe circumference. The acoustic meshes have 8, 12, and 12

elements in the axial direction, and 16, 24, and 48 elementsaround the circumference. Table 2 lists the resonance fre-quency ratios given by Berot and Peseux along with numer-ical predictions for the various acoustic element meshes.

In the table, NA is the number of acoustic elements foreach of the boundary element meshes. Clearly, the numeri-cal and analytical predictions match each other closely,although a relatively fine acoustic mesh is necessary in thecircumferential direction to achieve convergence for thehigher order modes, as was expected. Overall, for structuressubmerged in water, the fluid-loading analyses has beenshown to yield excellent predictions for the added mass.

We consider a cavity-backed plate as a second exampleproblem with structural-acoustic coupling. In the 1970’s,Guy and Bhattacharya24 studied transmission loss through acavity-backed finite plate and their results have been usedsubsequently by several authors to validate numerical pre-dictions. We will similarly use the problem to illustrate howdipole sources can be used to model interior and exterioracoustic fields simultaneously in a scattering problem.Figure 19 shows the geometry of the cavity and plate.

The plate is 0.914 mm thick and is made of brass withproperties E = 106 GPa, ρ = 8500 kg/m3, v = 0.3, and thesurrounding air has properties co = 340 m/s, and ρo = 1.2kg/m3. The plate is simply-supported and the backing cavi-ty has rigid walls. We want to compute the transmissionloss through the plate to a field point location at the centerof the back wall of the cavity. Guy and Bhattacharya’s trans-mission loss is actually just a ratio of the incident pressureand the pressure near the back wall of the box:

. (31)

For the numerical analysis, we can perform the compu-tations in one of two ways. We could simply apply mechan-ical forces to produce a uniform pressure to the top surfaceof the plate, as in the studies by S. Suzuki, et al.25 and M.Guerich and M. A. Hamdi,26 or we could simulate theexperiment using an acoustic source as the excitation. Wewill use the latter method. The surface of the cube is divid-ed evenly into 144 quadrilateral elements, yielding aboundary surface mesh with 864 structural elements.

The incident pressure is computed knowing the sourcelocation and the distance to the center of the plate. Theresults in Fig. 20 show very good agreement between ournumerical predictions and Guy and Bhattacharya’s experi-

Fig. 18. Finite element predictions for the first two mode shapes and resonance fre-quencies of a thin shell. The rigid ends on both sides of the shell simulate semi-infi-nite rigid baffles.

Table 2. Analytical and numerical predictions for the resonance frequency ratios(in fluid/in vacuo) of the circular shell. Mode orders are indicated as (m,n) pairs,where m is the order along the axis, and n is the circumferential harmonic.


mental measurements, thus demonstrating that dipolesources can be used to model both the interior and exterioracoustic fields simultaneously.

SummaryIn Part 2 of this tutorial on structural acoustics, we

have learned about how acoustic fluids interact with struc-tures, both as an acceptor and a cause of vibrational energy.We have presented some simple and difficult concepts in arelatively short article (entire textbooks are devoted to thesubjects we have discussed), and hope the information isuseful as a handy reference. For those of you who are inter-ested in learning more about these topics, please lookthrough the references we have provided. You are also wel-come to enroll in the Sound—Structure Interaction courseoffered by the Penn State Graduate Program in Acousticsfor a more thorough treatment of this subject.

Of course, there is much we have not explained, but wecan refer you to other strong references on those subjects.For example, we have focused almost entirely on the inter-action of structures with exterior fluids. Acoustic cavitiescontain resonances which can interact with the walls thatbound the interior space, particularly at low frequencies.Some classic papers which introduce this topic are those byPretlove27 and Dowell.28 During the Active Noise Control(ANC) boom of the 1980’s and 1990’s, many people inves-tigated how to control the sound within acoustic cavitymodes by driving the enclosure boundaries with tunedforces. Nelson and Elliott’s textbook29 is a good reference onANC.

Another relatively modern structural–acoustic topic isNearfield Acoustic Holography (NAH), which is an inversetechnique for inferring a structure’s surface vibrations froma complex pressure hologram measured near the surface.Once the surface velocities are known, numerical boundaryvalue techniques can be used to compute the far-fieldsound radiation. We recommend the textbook byWilliams30 to those interested in NAH.

AcknowledgmentsWe are grateful for our daily interaction with the

ARL/Penn State Structural Acoustics Department (AndrewBarnard, Robert Campbell, Stephen Conlon, David Jenkins,and Tim McDevitt), along with some of our students in PennState’s Graduate Program in Acoustics, in particular, BenDoty, who measured the response of the elbowed pipe.AT

References for further reading1 G. H. Koopmann and J. B. Fahnline, Designing Quiet Structures

(Academic Press, San Diego, 1997).2 R. D. Blevins, Flow-Induced Vibration, 2nd Edition (Krieger

Publishing Company, Malbar, FL, 2001).3 E. Naudascher and D. Rockwell, Flow-Induced Vibrations–An

Engineering Guide (Dover Publications, Mineola, NY 2005).

Fig. 19. Dimensions for the cavity-backed plate example problem.

Fig. 20. Transmission loss as a function of frequency for a cavity-backed plate.


4 F. J. Fahy, Sound and Structural Vibration: Radiation,Transmission, and Response (Academic Press, San Diego,1987).

5 A. D. Pierce, Acoustics–An Introduction to its Physical Principlesand Applications (Acoustical Society of America, Melville, NY,1991).

6 M. S. Junger and D. Feit, Sound, Structures, and Their Interaction(Acoustical Society of America, Melville, NY, 1993).

7 G. Maidanik, “Response of Ribbed Panels to ReverberantAcoustic Fields,” J. Acoust. Soc. Am. 34, 809–826 (1962).

8 C. E. Wallace, “Radiation Resistance of a Rectangular Panel,” J.Acoust. Soc. Am. 51, 946–952 (1972).

9 P. W. Smith, Jr., “Response and Radiation of Structural ModesExcited by Sound,” J. Acoust. Soc. Am. 34, 640–647 (1962).

10 C. B. Burroughs, R. W. Fischer, and F. R. Kern, “AnIntroduction to Statistical Energy Analysis,” J. Acoust. Soc. Am.101, 1779–1789 (1997).

11 P. J. Shorter and R. S. Langley, “On the Reciprocity RelationshipBetween Direct field Radiation and Diffuse ReverberantLoading,” J. Acoust. Soc. Am. 117, 85–95 (2005).

12 M. J. Crocker and A. J. Price, “Sound Transmission UsingStatistical Energy Analysis,” J. Sound Vibration 9, 469–486(1969).

13 A. J. Price and M. J. Crocker, “Sound Transmission ThroughDouble Panels Using Statistical Energy Analysis,” J. Acoust. Soc.Am. 47, 683– 693 (1970).

14 ISO 3741/ANSI S12.51, Determination of Sound Power Levels ofNoise Sources using Sound Pressure – Precision Methods forReverberation Rooms (Acoustical Society of America, Melville,NY, 1999).

15 ISO 354, Measurement of Sound Absorption in a ReverberationRoom (International Organization for Standardization,Geneva, 2003).

16 S. C. Conlon, S. A. Hambric, and W. K. Bonness, “Evaluationof a reverberant water tank for radiated power measurements,”Proceedings of NoiseCon 2004, Baltimore, MD, 12–14 July2004. (Institute of Noise Control Engineering, Ames, IA,2004).

17 B. J. Doty, S. A. Hambric, S. C. Conlon, and J. B. Fahnline,“Structural-Acoustic Measurements of Pipes with Ninety-

Degree Elbows, Under Water Loading,” Proceedings of NoiseCon2005, Minneapolis, MN, 17–19 October 2005 (Institute of NoiseControl Engineering, Ames, IA, 2005).

18 ANSI S12.35–1990 (R1996), Determination of Sound PowerLevels of Noise Sources in Anechoic and Hemi-Anechoic Rooms,(Acoustical Society of America, Melville, NY, 1996).

19 L. J. Gray and E. Lutz, “On the Treatment of Corners in theBoundary Element Method,” J. Comp. Appl. Math. 32, 369–386(1990).

20 L. G. Copley, “Integral Equation Method for Radiation fromVibrating Bodies,” J. Acoust. Soc. Am. 41, 807–816 (1967).

21 H. A. Schenck, “Improved Integral Formulation for AcousticRadiation Problems,” J. Acoust. Soc. Am. 44, 41–58 (1968).

22 L. H. Chen and D. G. Schweikert, “Sound Radiation from anArbitrary Body,” J. Acoust. Soc. Am. 35, 1626–1632 (1963).

23 F. Berot and B. Peseux, “Vibro-acoustic behavior of submergedcylindrical shells: Analytical formulation and numerical model,”J. Fluids and Struct. 12(8), 959–1004 (1998).

24 R. W. Guy and M. C. Bhattacharya,“The transmission of soundthrough a cavity-backed finite plate,” J. Sound Vib. 27(2),207–223 (1973).

25 S. Suzuki, S. Maruyama, and H. Ido, “Boundary element analy-sis of cavity noise problems with complicated boundary condi-tions,” J. Sound and Vib. 130(1), 79–91 (1989).

26 M. Guerich and H. A. Hamdi, “A numerical method for vibro-acoustic problems with incompatible finite element meshesusing B-spline functions,” J. Acoust. Soc. Am. 105(3),1682–1694 (1999).

27 A. J. Pretlove, “Forced Vibrations of a Rectangular Panel Backedby a Closed Rectangular Cavity,” J. Sound Vib. 3, 252–261(1966).

28 E. H. Dowell, G. F. Gorman, and D. A. Smith,“Acoustoelasticity: General Theory, Acoustic Natural Modesand Forced Response to Sinusoidal Excitation, IncludingComparisons with Experiment,” J. Sound Vib. 52, 519–542(1977).

29 P. A. Nelson and S. J. Elliott, Active Control of Sound (AcademicPress, San Diego, 1993).

30 E. G. Williams, Fourier Acoustics–Sound Radiation and NearfieldAcoustic Holography (Academic Press, San Diego, 1999).

Stephen A. Hambricis head of theStructural Acous-tics Department atthe Applied ResearchLab at Penn Stateand Associate Prof-essor in the Grad-uate Program inAcoustics. Prior tojoining Penn Statein 1996, Dr. Hambricworked for nineyears in the Comp-u t a t i o n a l

Mechanics Office at the Naval Surface Warfare Center,Carderock Division. Dr. Hambric has directed many numeri-cal and experimental flow and structural acoustics research anddevelopment programs for the Navy, U.S. industry, and theU.S. Nuclear Regulatory Commission. He has authored over60 conference and journal articles and advised many graduatestudents at Penn State. He teaches courses in StructuralAcoustics, and Writing for Acousticians on campus at PennState, and also to off-campus students working in industry andgovernment. He currently serves on the board of directors ofthe Institute for Noise Control Engineering (INCE), on theExecutive Committee of the ASME’s Noise Control and


Acoustics Division, and as an associate editor of ASME’sJournal of Vibration and Acoustics.

John B. Fahnline isa research associ-ate at the AppliedResearch Labora-tory and an assis-tant professor ofacoustics at PennState. He holds aPh.D. in Acousticsfrom the Pennsylv-ania State University.As a post-doctoralr e s e a r c h e r , D r .Fahnline designedand implementeda practical, effi-

cient, and accurate method for numerically solving general,three dimensional acoustic radiation problems which can beapplied to problems in design optimization and active noisecontrol. The code, named POWER, is currently being usedat the ARL Penn State to predict noise radiation from thestructural resonances of compound propulsors. Dr.Fahnline coauthored a book on design optimization with Dr.Gary Koopmann.

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