Acoustic signals of underwater explosions near surfaces · 2006-05-18 · Acoustic signals of underwater explosions near surfaces John R. Krieger and Georges L. Chahine Dynaﬂow,

Acoustic signals of underwater explosions near surfacesJohn R. Krieger and Georges L. ChahineDynaflow, Inc., 10621-J Iron Bridge Road, Jessup, Maryland 20794

�Received 12 November 2004; revised 13 July 2005; accepted 4 August 2005�

Underwater explosions are conventionally identified and characterized by their seismic and/oracoustic signature based on spherical models of explosion bubbles. These models can be misleadingin cases where the bubble is distorted by proximity to the free surface, the bottom, or to a solidobject. An experimental and numerical study of the effects of various nearby surfaces on thebubble’s acoustic signature is presented. Measurements from high-speed movie visualizations andacoustic signals are presented which show that the effect of proximity to a rigid surface is to increasethe first period, weaken the first bubble pulse, and affect significantly the second period, resultingin a peak value at standoffs of the same order as the maximum bubble radius. These results arecompared to results under a free surface, over a bed of sand, and over a cavity in a rigid surface. Inall cases, the first period is increasingly lengthened or shortened as the motion of fluid around thebubble is increasingly or decreasingly hindered. The effect of bubble distortion is to weaken the firstbubble pulse and increase the bubble size and the duration of the second cycle. © 2005 AcousticalSociety of America. �DOI: 10.1121/1.2047147�

PACS number�s�: 43.30.�k, 43.30.Lz �WMC� Pages: 2961–2974

I. INTRODUCTION

Underwater explosions produce bubbles of gas and va-por that alternately expand and collapse, generating a train ofacoustic pulses that propagates acoustically through the wa-ter, and seismically through the seafloor. The timing of thesepulses can be used to determine the yield of the explosion,provided its depth can be estimated. In some cases, rever-berations from the sea surface and ocean bottom provideinformation about the explosion depth �Baumgardt and Der,1998�. If the explosion is far from the free surface and thebottom and is otherwise unconstrained, the classic formula,as presented for example by Cole �1948�, serves to define the“bubble pulse” period �T in seconds� in terms of the explo-sive yield �W in pounds�, its depth �Z in feet�, and a param-eter �K=4.25 for TNT� characterizing the explosive

T = KW1/3

�Z + 33�5/6 . �1�

The ability to assess the character of an underwater explo-sion �UNDEX� and estimate its yield and depth is importantfor a variety of reasons. For example, the use of explosivesas underwater acoustic sources has a long and rich history�Weston, 1960�. A common method of surveying oceangeoacoustic properties is to detonate calibrated charges andmeasure the propagated signal from acoustic sensors or ar-rays at various ranges �Potty et al., 2004, 2003; Spiersberger,2003; Godin et al., 1999; Popov and Simakina, 1998�. Inaddition, the bubble pulse signature is also vital in the con-text of nuclear test-ban verification �Weinstein, 1968�, wherea key issue is to distinguish an UNDEX from an underwaterearthquake �Baumgardt, 1999� and in tracking dynamite fish-ing practices �Woodman et al., 2003�. Exploiting dynamitefishing explosions as sound sources of opportunity providesanother motivation to this study �Lin et al., 2004�.

In all cases, erroneous conclusions result from the use of
Eq. �1� if the explosion is near an obstacle, the bottom, or if
J. Acoust. Soc. Am. 118 �5�, November 2005 0001-4966/2005/118�5

it is partly contained. In particular, the bubble generated byan UNDEX can be severely distorted from the presumedspherical geometry by a nearby object or surface. This couldresult in “bubble pulse periods” that correspond to a yieldsubstantially different from that predicted by Eq. �1�.

Considerable efforts have been devoted to experimentaland theoretical understanding of the physics of underwaterexplosions as sound sources. These attempts have typicallyused ad hoc models of bubble migration �Buratti et al., 1999�or have been based on tests with actual full-scale explosives�Chapman, 1985, 1988; Hannay and Chapman, 1999�. Thispaper presents controlled scaled laboratory tests to define themulticycle motion and geometry of the explosion bubble,and compares with a sophisticated numerical simulationbased on the dynamics of bubbles near boundaries. Quanti-tative results �in particular, depths and time of acoustic ra-diations� that are most important for defining impulse pointsources of acoustic propagation are presented. Section II de-scribes the basic physics, Sec. III describes the experimentaland numerical conditions of the study, Sec. IV presents theresults of the study, and Sec. V provides a discussion of theseresults in terms of characterization of explosions.

II. BACKGROUND

The detonation of an explosive charge underwater re-sults in the emission of a shock wave and the creation of ahigh-pressure gas and vapor bubble. The shock wave consti-tutes the first of a series of acoustic pulses detectable at longranges, and the remaining bubble pulses are generated by thesubsequent motion of the bubble. The pressure of the bubbleinitially causes it to expand, but inertia of the liquid carriesthe bubble past its equilibrium point, causing it to overex-pand, and then collapse again. The final stages of the col-lapse are quite violent, and return the bubble nearly to its
original state, generating large pressures and another acoustic
© 2005 Acoustical Society of America 2961�/2961/14/$22.50

pulse. The bubble then begins the cycle again and continuesuntil it has broken up and its energy has been dissipated.

The shock and subsequent bubble pulses radiate soundas point- or compact impulsive sources at different depthsand times. While boundaries affect the sound propagationfrom a point source in well-understood ways, the problem ofhow the boundaries affect the multicycle motion of thebubble acoustic source is the focus of this paper.

A. Numerical model

Following an underwater explosion detonation, and aftera very short period of time where compressible effects areimportant, the characteristic liquid velocities become smallcompared to the velocity of sound, enabling one to neglectthe liquid compressibility. Similarly, viscous effects can beneglected and one can consider the water flow field as po-tential, with u=��, where u is the liquid velocity vectorand � is the velocity potential. � satisfies Laplace’s equa-tion in the fluid domain, D

�2��x,t� = 0, x � D , �2�

where x is the spatial variable and t is time �Chahine andPerdue, 1989�. On the bubble surface Sb, of local normal, n,� satisfies the kinematic boundary condition, which ex-presses equality between the fluid and the free-surface nor-mal velocities

dx/dt · n = �� · n, x � Sb, �3�

and the dynamic boundary condition

��

�t= −

1

2��2 +

P� − Pb

�− �gz�x�Sb

, �4�

where P� is the hydrostatic pressure at the point of initiationof the explosion, Pb�x , t� is the local pressure in the liquid atthe bubble interface, and z is the vertical coordinate atpoint x.

The pressure inside the bubble is assumed spatially ho-mogeneous, and the bubble content is assumed to be com-posed of noncondensable gas arising from the explosionproducts, which follow a polytropic compression law of con-stant k. These assumptions have been shown, for a widerange of free-field bubbles, to lead to an excellent fit betweenexperimental data and numerical results �Chahine et al.,1995�. This leads to the following form of the normal stressboundary condition at the bubble interface:

Pb�x,t� = Pg0�V�t�V0

�k

+ Pv − �C , �5�

where Pg0 is the initial pressure of the noncondensable gasinside the bubble and Pv is the water vapor pressure. V0 isthe initial volume of the bubble, � is the surface tensioncoefficient, and C�xs , t� is twice the local mean curvature atxs given by

C = � · n . �6�

The local normal to the surface, n, is defined as

2962 J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005

n = ±�f

��f �, �7�

where f is the bubble surface equation. The appropriate signis chosen so that the normals point towards the liquid.

In addition to conditions �3� and �4�, the boundary con-dition on any nearby solid body St is given by

��

�n= un, x � St, �8�

where un is the local normal velocity of the solid body inresponse to the bubble loading. This includes body motionand deformation �Kalumuck et al., 1995; Chahine and Kalu-muck, 1998�. At infinity, the fluid velocities due to thebubble dynamics vanish, and the boundary condition is

�lim�x→�� = 0. �9�

To complete the description of the problem, appropriate ini-tial conditions are used. At the initiation of the computation,and only at that instant, the bubble dynamics is assumed tobe that due to a spherical explosion bubble in a free field.The Rayleigh–Plesset equation �Plesset, 1948� governingspherical bubble dynamics is used to obtain these initial con-ditions �i.e., a relationship between the initial bubble radiusand the initial bubble wall velocity or gas pressure�. Thisresults in the following relationship between the initialbubble radius, R0, and the initial gas pressure inside thebubble, Pg0, for R0=0:

Pg0 =3�1 − k�1 − �0

3k−3� p� − pv

3�1 − �0

−3� +�

R0�1 − �0

−2�� ; �0

=R0

Rmax. �10�

Thus, a closed system of equations is obtained to solve for �using Green’s identity

��x� = �S

ny · �y��y�G�x,y�dsy

− �S

ny · �yG�x,y��y�dsy , �11�

where x is selected on the boundary, y is an integration vari-able on the surface S ,ny is the normal to S, and

G�x,y� =1

�x − y��12�

is the Green’s function. � is the solid angle subtended in thefluid at the point x.

Equation �11� is an integral equation which relates thepotential at any point x to the values of � and its normalderivatives on the boundary of the liquid domain, S. We canuse Eq. �11� to determine at a given computation time theunknown quantities: � on solid surfaces and normal deriva-tive of the potential, �� /�n, on the free surface, knowing thenormal velocities on the solid surfaces and the velocity po-
tential on the free surface. To start the computations, we
J. R. Krieger and G. L. Chahine: Explosion signals near surfaces

know the potential on the initial bubble surface through ��=R0R0�, and we assume a zero body velocity.

Once the flow variables are known at a given time step,values at subsequent steps can be obtained by integratingEqs. �3� and �4�, and using an appropriate time-steppingtechnique. The material derivative, D� /Dt, is then computedusing the Bernoulli equation �4�, and noting that u=��

D�

Dt=

��

�t+ �� · �� =

��

�t+ �u�2. �13�

This time stepping proceeds throughout the bubble growthand collapse, resulting at each time step in the knowledge ofall flow-field quantities and the shape of the bubble.

This is the basis of the numerical method used in thispaper and which has been extensively used and validated forboth axisymmetric �Zhang et al., 1993; Choi and Chahine,2003� and three-dimensional interactions between UNDEXbubbles and nearby structures �e.g., Chahine and Perdue�1989�; Chahine �1996�; Chahine, Kalumuck, and Hsiao,�2003��.

B. Discussion and definitions for analysis

The acoustic bubble pulses, generated by the bubble vol-ume and shape changes, may be treated theoretically prima-rily as a monopole radiation in the far field with strengthdependent on the volume acceleration of the bubble, plus adipole with strength dependent on the bubble migration, andhigher-order poles dependent on its deformation.

In the absence of gravity effects or nearby objects, thebubble remains spherical in shape, and the period of its vol-ume oscillations can be predicted analytically. Rayleigh�1917� analyzed the collapse of empty spherical cavities andfound the time of collapse to be

Tcollapse = 0.915Rmax �

P�

, �14�

where Rmax is the initial �maximum� radius of the cavity,and � and P� are the density and pressure of the ambientfluid, respectively. This dependence of the period on theambient pressure and the bubble size is another formula-tion of Eq. �1�. The effects of vapor pressure, noncondens-able gas, surface tension, viscosity, and a time-varyingambient pressure could also be included, to produce theRayleigh–Plesset equation �Plesset, 1948�, the equation ofwall motion for a collapsing and rebounding sphericalbubble. For given conditions, the period of the bubbleoscillation can be predicted from the solutions of thisequation.

If the explosion occurs near an object or at shallowdepths where the effects of gravity are strong, the bubblecollapses asymmetrically, resulting in reentrant jet and toroi-dal bubble formation, or if both effects are strong and inopposite directions, a pear shape and bubble splitting �Cha-hine, 1996, 1997�, and the bubble does not produce the sameseries of pulses as it would in deep submergence and awayfrom boundaries.

The effect of a nearby surface is quantified in terms of
the normalized standoff distance
J. Acoust. Soc. Am., Vol. 118, No. 5, November 2005 J. R

X =x

Rmax, �15�

where x is the standoff distance, i.e., the shortest distancefrom the initial explosion center to the surface, and Rmax isthe maximum radius the bubble would achieve if the ex-plosive were detonated in an infinite medium at the samedepth.

For large values of the standoff parameter, the bubble isessentially spherical and unaffected by the boundary. As thestandoff distance is reduced, the bubble becomes more andmore affected by the boundary. At values near unity, thebubble nearly touches the surface on its first expansion, andat values near zero, the first bubble pulsation is quasihemi-spherical. Examples of this behavior for small-scale explo-sions generated by a spark in a water tank �Chahine et al.,1995� are shown in Fig. 1.

The effect of buoyancy is quantified in terms of theFroude number �Chahine, 1997�

F =P� − Pv

2�gRmax. �16�

This parameter is the ratio of the difference between theambient pressure at the charge location and the vapor pres-sure to the difference in pressure between the top and bottomof the bubble. It expresses the relative importance of thecompressional force tending to collapse the bubble and thegravity force tending to deform and migrate the bubble up-ward. For large values of the Froude number, the effect ofgravity is relatively weak, and the bubble remains spherical.For small Froude numbers, upward motion and bubble dis-tortion due to gravity are important. This parameter isstrongly responsible for differences between UNDEX bubblebehaviors at different geometric scales. The Froude number�or the gravity acceleration term� does not appear explicitlyin the equations, but the gravity term can be seen in the

FIG. 1. Spark-generated bubbles over a rigid surface. Horizontally, fourdifferent times are shown—the time of the first volume maximum, just priorto the first collapse, the time of the first collapse, and the time of the secondvolume maximum—and vertically three different standoff distances areshown, with standoff distance increasing from bottom to top. The ambientpressure was approximately 60 mbar, and Rmax was approximately 15 mm.

boundary condition �4� on the bubble and free-surface walls.

. Krieger and G. L. Chahine: Explosion signals near surfaces 2963

For problems of interaction between an UNDEX bubble andthe ocean bottom, or an obstacle located below the bubble,gravity acts in an opposite direction than the bottom, i.e., ittends to force the bubble to have a reentrant jet moving up-ward, while the bottom induces a downward motion. Thisresults, at small Froude numbers, in large differences in thebubble behavior between two different Froude number con-ditions for otherwise similar UNDEX conditions.

III. EXPERIMENTAL AND NUMERICAL ANALYSIS

A. Experimental methods

Bubbles generated by the underwater discharge of ahigh-voltage charge between two coaxial electrodes, exposedto each other at their ends and contained within a transparentvacuum cell, were used as laboratory-scale models of under-water explosions �Chahine et al., 1995�. The cell is 3�3�3 ft.3 made of 1-in.-thick Plexiglas wall, which enables itto withstand reduced pressure down to 0.5 psi. By reducingthe pressure in the cell, relatively large and slow-collapsingbubbles were generated, and the transparent walls allowedrecording of the bubble dynamics by a high-speed digitalcamera �Redlake Imaging model PCI8000S�. The acousticsignals of the bubbles were measured with a piezoelectrictransducer having a 1-s rise time �PCB Piezotronics model102A03�. A photograph of the experimental arrangement isshown in Fig. 2. Concerning the physics of the problem, thearrangement electrodes/nearby boundary is quite simple andis further illustrated in the bubble pictures in this paper, suchas in Fig. 1. Note that the acoustic properties of the acrylicplatform are close to those of water, making the reflectioncoefficient near zero over the relevant range of reflectionangles. Furthermore, reflections from walls and other distantobjects resulted in path lengths much longer than thestraight-line path from bubble to transducer. Because of this,only the pulses propagating directly from the bubble to thetransducer were strong enough to be detectable.

The energy of the bubble was controlled by adjusting thevoltage supplied to the capacitor of the spark generator. Theeffects of the tank pressure and of the variations in the dis-charge output were removed from the data by measuring

FIG. 2. The experimental arrangement for recording bubble dynamics neara rigid surface.

Rmax and normalizing periods using a characteristic time


�equal to the Rayleigh collapse time omitting the factor0.915� as in Eq. �15�. The bubble maximum radius is used tonormalize the standoff distance, and is obtained by directmeasurement from the video pictures corresponding to eachsignal. When the bubble was significantly distant from thebottom, it retained a spherical shape at maximum volumeand a radius could easily be measured. In cases where thebubble was distorted by the presence of the nearby boundary,an equivalent radius, equal to the radius of a sphere of equalvolume, was used. Simulations of identical initial bubbles atvarious distances show that the bubble achieves the samevolume, within ±1%, at the time of its maximum growthregardless of the standoff distance, so that the equivalentRmax is a consistent measure of bubble size. The volumes ofdistorted experimental bubbles were calculated from the ob-served bubble outline at the time of maximum growth, as thevolume of a solid of revolution around its vertical axis. Thevalidity of the assumption of axisymmetry is supported bytop views, which show circular profiles, and by the mirrorsymmetry seen in side views.

The ambient pressure was calculated based on the abso-lute pressure in the vacuum cell and the depth of the elec-trodes, and the vapor pressure was calculated based on themeasured ambient water temperature. All times, t, and dis-tances, r, were normalized using the quantities �, Rmax, andPamb− Pv

t t

Rmax �

Pamb − Pv

, �17�

r r

Rmax. �18�

Although spark-generated bubbles are considerably smallerthan typical explosion bubbles, surface tension and viscosityeffects remain negligible, and the bubble dynamics is an ac-curate scaled representation of a full-scale case having thesame normalized standoff and Froude number �Chahine etal., 1995�. Conversion between scales is accomplished usingthe characteristic length lcharac=Rmax and the characteristictime Tcharac=Rmax�� /P�1/2, as above. Differences betweenvarious scaled results obviously exist if the Froude num-bers, i.e., the relative influence of gravity, are not thesame.

B. Error estimates

One of the principal sources of experimental uncertain-ties lies in the determination of Rmax for each case, whichaffects both the normalized standoff distance and the normal-ized period. Repeated measurements for several differentcases give a standard deviation of about 1% of the mean. Incases where an equivalent Rmax was measured for the bubbleat its second maximum, the standard deviation was about4%, since the axisymmetric assumption probably introducesmore significant errors in these cases. Vapor pressure is afunction of temperature, and uncertainty in the temperaturemeasurement led to errors of about 2% or less. The least
significant digit on the pressure gauge is 1 mbar, or about

2% for low-pressure tests. For high-pressure tests, in whichthe bubbles are small, the dominant source of uncertainty layin the measurement of standoff, which could have been asmuch as 10%. In the graphs presented here, it was decided torely upon the scatter among a large number of data points asan indication of repeatability. One series of tests was con-ducted with each test repeated five times, which producedstandard deviations in normalized period and normalizedstandoff of approximately 5%.

C. Numerical approach

DYNAFLOW’s boundary element method code2DYNAFS© �Choi and Chahine, 2003� was used to conductsimulations of bubbles under conditions matching those ofthe spark tests. 2DYNAFS© is based on the boundary elementmethod described in Sec. II for solving potential flows witharbitrary boundaries, and calculates the motion of one ormore interfaces in an axisymmetric geometry. Each interfaceis discretized in a meridional plane, each node is advanced ateach time step using the velocities and the unsteady Ber-noulli equation, and the new velocity potentials and normalvelocities are calculated using a discretized version ofGreen’s equation �Zhang et al., 1993�. In cases where a re-entrant jet impacts the opposite wall of the bubble, the dy-namic cut relocation algorithm of Best �1994� was used tocontinue computations with a new toroidal bubble. Forlaboratory-scale simulations, the initial conditions of thebubble were specified as the same ambient pressure as thecorresponding experimental test, and an initially sphericalexplosion bubble with a radius and pressure such as to pro-duce an Rmax value of 1.3 cm, which is a typical value for thespark tests in the pressure range tested.

IV. RESULTS

A. Bubble dynamics near a rigid surface

Figure 3 shows an example series of pressure signalsfrom bubbles generated by spark discharges at varying dis-tances above a flat, rigid plate, with each signal offset verti-cally in the figure by an amount proportional to its corre-sponding standoff distance. The time has been normalized asin Eq. �15�. Each signal contains initial strong fluctuationscomposed of a combination of the initial shock wave andelectrical noise associated with the spark discharge, followedby two and sometimes three detectable peaks generated bythe bubble collapses. Two principal differences among thesignals can be observed. First, the timing of the first collapsepulses appears in general delayed for bubbles near the plate.Second, the first bubble pulse decreases in strength dramati-cally as the bubble approaches the plate.

For bubbles at moderate distances from a rigid surface,Chahine and Bovis �1983� have demonstrated, using
matched asymptotic expansions for large X, that

T* � 1.83�1 +1

4X�

R0

Rmax

R�t�dt � 1.83�1 +0.79

4X ,

�19�

where T* is the normalized period of the bubble in the pres-ence of the surface. This first-order approximation provides auseful theoretical comparison for moderate values of stand-off distance, although it is not valid for small values.

In order to investigate the effects of proximity to a rigidboundary over the entire range of standoff distances, severalseries of spark tests were conducted using approximatelyconstant energy but varying standoff distances, including aconcentrated series of measurements for values of X less than1, and these results were compared to the results from aseries of simulations using 2DYNAFS©. The measured andcalculated first bubble periods are shown in Fig. 4 along witha curve calculated from Eq. �19�, and a correlation calculatedfrom Eq. �20�, presented below.

Both the measurements and the simulations agree withthe predictions of the first-order asymptotic expansion �19�

FIG. 3. Pressure signals from a spark-generated bubble with varying stand-off distance. Signals were measured using a quartz pressure transducer sev-eral inches to one side of the bubble. The ambient pressure was1 atmosphere and the maximum bubble radius about 6 mm. All signals havebeen offset vertically by an amount proportional to the respective standoffdistance.

FIG. 4. First bubble period as a function of standoff distance. Experimentaldata were collected for several cell pressures, and simulations were con-ducted using an ambient pressure of 50 mbar plus a correction for depth,
and an Rmax value of 1.3 cm.

for values of X larger than 1, where the period is approxi-mately twice the characteristic time, as expected. The contri-bution of this work is the description of a nonasymptoticregion where larger bubble deformations occur. As thebubble nears the bottom, the period shows a steady gradualincrease, up to a value at the bottom approximately 20%–25% greater than the “free” value. This increase is consistenttheoretically with the additional energy due to an imagebubble on the opposite side of the surface. If a hemisphericalbubble is generated exactly at the surface, it will act with itsmirror image as a single spherical bubble having twice theenergy as the actual bubble, and according to the period–energy relation will have a period greater than the actualbubble if it were spherical by a factor of 21/3=1.26. Thus, thetheory predicts a normalized period of 2�0.915=1.83 forfree bubbles, and with the image a period of 2�0.915�21/3=2.31 for bubbles on a rigid surface. Both experimen-tal measurements and numerical simulations using2DYNAFS© agree with these extrema and show a steady pro-gression in between, while, as expected, the asymptotic ex-pansion �and the classical corrections� fails at the lower val-ues of X.

A general analytic prediction of the bubble first periodover all standoff distances is lacking. However, the empiricalexpression

T* � 1.83�1 +0.1975

0.5776 + X2 , �20�

does a reasonably good job as a correlation, as shown in Fig.4. This expression matches the theoretical values for X=0and X→�, and agrees with the asymptotic expansion for-mula for large X. It may be used to predict the period of anexplosion bubble of any size near a large, flat, rigid surface,as long as the effects of buoyancy are not too strong, i.e., aslong as the Froude number is above approximately 10. Simi-lar expression could be derived for other values of F butwere not attempted here.

The period of the second cycle, shown in Fig. 5, dependson the history of the bubble through the first cycle, and ex-hibits more variation with standoff than does the first period.The second period is normalized using the same characteris-tic time as the first period. At large standoff distances, thesecond period is nearly half that of the first cycle, whichaccording to the period–energy relation for spherical

FIG. 5. Normalized second period as a function of standoff. Experimentaldata are shown from the same set of tests as in Fig. 4, normalized using Eqs.�17� and �18�.

bubbles, and for negligible migration, corresponds to an en-


ergy of � 12

�3 or 13% of the original energy, or an energy lossat the first collapse of about 87%. This differs slightly fromperiod ratios and energy losses for free-explosion bubbles�Cole, 1948�, but is consistent with reported values for laser-generated bubbles �Vogel et al., 1989; Vogel and Lauterborn,1988�. As the standoff distance decreases to a value of X�1, the second period increases to a large fraction of itsoriginal value, suggesting a reduction in the bubble energyloss during the first collapse. A reduction in energy loss isalso suggested by a reduction in emitted acoustic energy, asis indicated in Fig. 6, which shows a pronounced minimumin the peak pulse pressure near X=1. However, a change inperiod is also to be expected for these standoff distancessimply due to the formation of a vortex ring bubble, whoseperiod is known to be larger than that of a spherical bubbleof equal energy �Chahine and Genoux, 1983�. As the standoffdistance decreases below X�1, the period again decreases.This behavior of exhibiting an extremum near X�1 is alsoshown in the measurements of Lindau and Lauterborn�2003�, who measured maximum and minimum volumes oflaser-generated bubbles near a plate and found that the ratioof maximum to minimum volume near X�1 was reduced byroughly 2 orders of magnitude from its value very near andvery far from the plate. In other words, the strength of com-pression of the bubble contents is greatly weakened when thestandoff distance is approximately 1 bubble radius from thesurface.

Note that the value of peak pressure shown in Fig. 6 isprobably lower than the actual value due to the extremebrevity of the pulse �laser-induced bubbles produce pulseswith a duration of tens of nanoseconds; see Vogel and Lau-terborn, 1988�, and the relatively large size and slow re-sponse �rise time of 1 s� of the quartz transducer. The pres-sures in all cases were measured several inches to the side,and were normalized to correspond to a 3-in. separation be-tween electrodes and transducer, using an assumed 1/r de-pendence of pressure with distance.

Since standoff values much less and much greater thanthe bubble size correspond, respectively, to hemisphericaland spherical symmetry, while standoff values nearly equalto the bubble size result in highly nonspherical bubbles andin the development of strong linear and rotational motion, itseems reasonable to explain the extrema in all of these prop-

FIG. 6. First bubble pulse strength as a function of standoff. The maximumpressure recorded by the transducer 3 in. from the bubble during the firstbubble pulse is shown from the same tests as in Fig. 4 and Fig. 5.

erties in terms of bubble distortion during the first cycle and


energy loss during the first collapse. For hemispherically andspherically symmetric bubble collapse, all of the energy ofthe fluid is directed into compression of the gaseous contentsof the bubble, producing small minimum volumes, largemaximum pressures, and large energy losses due to acousticradiation. For values of X�1, much of the fluid energy goesinto the formation of a jet and a toroidal bubble, and theasymmetric collapse deprives the gas of some of its compres-sion energy, producing larger minimum volumes, smallerpressures, and weaker acoustic emission.

Larger second-cycle bubble volumes, as well as largersecond periods, are also observed near X=1. Figure 7 showsapproximate measurements of the bubble equivalent radius atthe second volume maximum �Rmax 2�, as a fraction of thefirst Rmax value. Similar to the second period, the secondmaximum radius displays a maximum near X=1. Further-more, the similarity between the radius ratio and the ratio ofsecond to first period, shown in Fig. 8, suggests that theperiod scales approximately as the equivalent maximum ra-dius for nonspherical bubbles as for spherical bubbles, andboth are decreased in nearly the same proportion by the lossof energy during the first collapse. Note that Cole �1948�reports a period ratio of approximately 0.77 for free-explosion bubbles, and a decrease in the ratio as the explo-sion approaches the bottom, while a period ratio between 0.5and 0.6 is consistently obtained for free spark-generated

FIG. 7. Ratio of second maximum radius to first maximum radius as afunction of standoff distance. Data are shown from the same set as shown inprevious figures.

FIG. 8. Ratio of second period to first period as a function of standoffdistance. Data are shown from the same experimental tests as shown in
previous figures.

bubbles in the ambient pressure conditions considered here,and this ratio increases as the bubbles approach the bottom.The energy losses for spark-generated bubbles are consistent,however, with reported energy losses for laser-producedbubbles �Vogel et al., 1989�. Also, the influence of gravity inthe two cases is very different, since the Froude number forthe conditions reported by Cole is approximately half of thevalue for a typical spark test reported here, and the opposingeffects of gravity and a nearby bottom produce significantlydifferent effects from the effects of a surface alone, such asbubble lengthening and splitting �Chahine, 1996�. In addi-tion, energy losses between one bubble cycle and the follow-ing for free-field bubble have been shown experimentally,over a very large range of explosion conditions, to depend ona parameter equivalent to the Froude number, i.e., Rmax/Z,where Z is the hydrostatic head at the explosion center �Snay,1962; Chahine and Harris, 1997�.

The ratio of the peak pressures between the second andthe first bubble cycle is also very instructive, and shows avery distinctive peak at a normalized standoff distance ofapproximately X=0.9. This is clearly illustrated in Fig. 9.

B. Bubble dynamics near a free surface

It is well known that proximity to a free surface reducesthe first period of an explosion bubble. An analysis using theimage theory can be applied to a free surface, similar to thatfor a rigid surface but with an opposite sign, resulting in ananalogous formula �Chahine and Bovis, 1983�

T* � 1.83�1 −0.79

4X . �21�

In this case, X is the normalized depth, or standoff distancefrom the free surface. This formula is essentially the same asthat given by Cole �1948�, and except for depths shallowerthan 1 bubble radius, agrees quite well with both laboratory-scale experimental data, and with simulations, as shown inFig. 10. Note that the Froude number for the experimentalconditions is approximately 8, indicating a mild influence ofbuoyancy, while Eq. �21� assumes negligible influence ofbuoyancy �F�1�. Also note that data are unavailable below

FIG. 9. Ratio of second peak pressure to first peak pressure as a function ofstandoff distance. Data are shown from the same experimental tests asshown in previous figures.

approximately X=0.5, as the spark-generated bubbles vent


their contents to the atmosphere in this region, lose theiridentity, and fail to emit bubble pulses. In the range 0.5�X�1.0, the top of the bubble does rise above the equi-librium free-surface level, but the surface is notbreached—it maintains its integrity and exhibits behaviorresembling an elastic membrane, and a water layer re-mains between the bubble and the free surface �Chahine,1977�.

As with bubbles near a rigid surface, the first bubblepulse and the second cycle behavior of bubbles near a freesurface show more complex variations than the period of thefirst cycle. The first bubble pulse strength, and the ratios ofsecond to first period and second cycle bubble size to firstcycle bubble size, are shown in Fig. 11. When the bubble isclose to the surface, the surface has a repulsive effect whichforces the bubble downward during and after the collapse.When the bubble is sufficiently deep, the effect of buoyancydominates over the effect of the free surface, and the bubbleis distorted and lifts upward. At some intermediate depth, thetwo opposing effects cancel each other, and the bubble isobserved to remain nearly spherical and to pulsate withoutrising or descending. Under the laboratory conditions shownhere, this occurs at a normalized depth of approximately 2.Unfortunately, when the bubble remains spherical and sta-tionary it collapses at the tip of the electrode, and this tendsto make the bubble pulse magnitudes somewhat erratic inthis region.

Figure 11 shows that, near a normalized depth of 2, thestrength of the first bubble pulse is relatively strong, and thatthe second bubble cycle is relatively short with a small maxi-

FIG. 10. Normalized first period versus normalized depth for bubbles nearthe free surface.

FIG. 11. Size and duration of the second bubble cycle, and magnitude of the
first bubble pulse for bubbles near a free surface.

mum bubble radius. Thus, just as for bubbles near a rigidsurface, when the collapse is symmetric the compression ofthe bubble contents is relatively strong, resulting in relativelylarge energy losses and weaker second cycles. When thebubble is distorted during collapse, the compression is weak-ened, resulting in smaller energy losses and stronger secondcycles.

C. Bubble dynamics near a nonflat rigid surface

In addition to being nonrigid, surfaces near explosionbubbles are often not perfectly flat. Explosions near the sea-floor may result in a crater. Explosions near or on an objectmay interact with a cavity in the object. As a simple model ofa bubble interaction with a nonflat surface, a series of simu-lations and experimental tests of a bubble near a flat surfacecontaining a hemispherical cavity was conducted. These testsare characterized not only by the normalized standoff dis-tance x /Rmax, but also by the ratio of cavity size to maximumbubble size, Rc /Rmax. The standoff distance for these tests isdefined relative to the bottom of the cavity, as shown in Fig.12 in order to avoid ambiguities in the limit as the wallcavity radius becomes very large.

If the explosion bubble is very much larger or verymuch smaller than the cavity, the behavior of the bubbleresembles that near a flat surface, with the bubble interactingwith a large region of the surface around the cavity, or withthe bottom of the cavity, respectively. If the bubble size is ofthe same order as the cavity size, however, the behavior isdifferent. If the bubble is initiated at the bottom of the cavity,it splits during collapse, as shown in Fig. 13. The mostprominent effect of the cavity on the acoustic signals is thesignificant increase in first period for bubbles partially en-closed within the cavity. Figure 14 shows the normalized firstperiod as a function of normalized standoff for various ratiosof cavity size to bubble size, with Fig. 14�a� showing experi-mental data, and Fig. 14�b� showing the results of simula-tions. In both cases, a curve similar to the normalized periodcurve for flat surfaces is produced when the cavity to bubble

FIG. 12. Schematic of spark-generated bubbles over a hemispherical cavityin a rigid surface.

FIG. 13. Collapse, splitting, and rebound of an explosion bubble initiated at
the bottom of a moderately sized hemispherical cavity.

size ratio tends towards very small or very large values,since, in the limit of very small cavities, the bubble shouldinteract with a flat surface with negligible influence of thecavity, and in the limit of very large cavities, the bubbleshould interact with the bottom of the cavity as if it were avery large flat surface. Note that curves for large values ofRc /Rmax maintain higher values than curves for small valuesof Rc /Rmax at large standoff distances, since smaller bubblesimply that larger normalized standoff distances are requiredto emerge from and to escape the effects of the wall cavity. Ifthe wall cavity to bubble size ratio is not extremely large orsmall, i.e., if the wall cavity and maximum bubble radii areof the same order, then there is a marked increase in thebubble period when the explosion is initiated near or withinthe wall cavity, demonstrating the effect of increased inertiaof the surrounding water due to increasing confinement ofthe explosion.

Figure 15 shows more clearly the effect of the relativecavity and bubble sizes. The normalized first period is shownfrom experiments and simulations of explosion bubbles ini-tiated at the bottom of the crater, for various values of
Rc /Rmax. As expected, the period approaches the theoretical

value for an infinite flat surface for large and small values ofRc /Rmax, and increases by a factor of almost 2 in the rangeRc /Rmax�2–3, indicating a maximum restriction of bubblemotion when the bubble size is a large fraction of the size ofthe containing cavity.

Figure 16 and Fig. 17 show the experimental results forthe second bubble oscillation period, and the amplitude ofthe first bubble pressure pulse, for bubbles near a cavity.Overall behavior is similar to that near a flat surface, exceptthat the curves are shifted towards larger standoff values asthe bubbles become smaller, due to the fact that smallerbubbles must be further away from the bottom of the cavityin order to escape its effects.

D. Bubble dynamics near a bed of sand

In order to investigate the effect of nonrigid surfaces, aseries of experimental tests over a bed of sand was con-ducted, for comparison to the rigid surface. Commercialplayground-quality sand �grain size approximately

FIG. 14. Normalized first period ofbubbles over a cavity in a rigid sur-face.

0.1–1.0 mm� was thoroughly rinsed and poured into a bed


2 cm deep. Each bubble growth and collapse disturbed thesurface of the sand to a certain extent, so the surface wassmoothed flat before each test.

When the bubble is within approximately 2 radii fromthe surface, the bubble tends to lift the sand underneath itinto a mound during its collapse, as shown in Fig. 18. Thedynamics of the first cycle appears otherwise unaffectedcompared to behavior over a rigid surface, and shows similardistortions and similar jetting behavior at similar standoffdistances. Furthermore, the first period of the bubble oversand is within experimental variability of the first period overa rigid surface, as shown in Fig. 19, indicating that the abilityof a solid surface to deform slightly or fluidize within a shal-low region is not sufficient to significantly alter the hydro-dynamics around a bubble near the surface. Greater elastic-ity, such as that of a free surface, is required to produce largevariations in the bubble behavior.

The most pronounced difference between the effect of asandy bottom and of a rigid bottom lies in the behavior fol-lowing the impact of the bubble with the sand. For a rigidsurface, the bubble and remnants of the bubble may be ob-served in motion for a long time, but when a bubble strikesthe sandy surface, the sand appears to immediately swallowthe bubble, with fragments occasionally reemerging only af-

FIG. 16. Normalized second period of the bubble oscillations.


ter a long time. A second bubble pulse is normally detect-able, indicating a second collapse, but the strength of thispulse is much diminished for bubbles very near to the sand,as shown in Fig. 20. �As in Fig. 3, each signal has been offsetvertically by an amount proportional to the standoff distance.Note that several signals are overlaid at certain standoff dis-tances, representing multiple tests at those distances.� Thesecond period, shown as the ratio of second to first period inFig. 21, shows a similar trend to the period ratio for bubblesover a rigid surface, except for somewhat diminished valuesfor small standoff distances. The reduced period of the sec-ond cycle shown in Fig. 20 and Fig. 21 for standoff distancesless than about 1, which is the region where the bubble col-lapses directly against the sandy surface, indicates a weak-ened rebound and second cycle, presumably due to dissipa-tion of the bubble’s energy as it rebounds within and movesthrough the sand. The weakened second bubble pulse is con-sistent with greater energy losses as the pulse propagatesfrom deeper within the sand bed.

E. Spectra of the first bubble pulse

In cases where the explosion occurs near a rigid surface,the bubble pulse is often not a single peak, but has a smaller,

FIG. 15. Normalized first period ofexplosion bubbles initiated at the bot-tom of a cavity, as a function of thecavity size relative to the bubble size.

FIG. 17. Amplitude of the first bubble pulse peak pressure.


initial peak preceding the primary collapse peak, as shown inFig. 22, which shows typical peak profiles for normalizedstandoff distances of 0.06, 0.30, 0.45, and 0.69. The initial,or precursor, peak is evident for X=0.06 and quite pro-nounced at X=0.3. Also evident is the decrease in peak am-plitude for standoff distances near X=1. A multitude ofpeaks, such as Fig. 22�c�, is often observed for standoff val-ues in the neighborhood of X=1, and sometimes for bubblesstrongly influenced by gravity as well, and is probably aresult of partial or complete bubble breakup during collapse.The precursor peak is known to occur as a result of impact ofthe reentrant jet on the surface �Chahine and Duraiswami,1994; Thrun et al., 1992�, while the main peak is produced atthe bubble’s volume minimum. This precursor peak, which ismost pronounced at a normalized standoff value of approxi-mately 0.3, may also hold some explanation for the mini-mum in the second bubble period which occurs at this stand-off value, due to increased energy loss accompanying the jetimpact and pulse emission. Bubble collapse is an unstableprocess, and any perturbations, such as disruption of flowaround an electrode, or any other object present in a nonidealflow field, will promote breakup of the bubble. This breakuphas sometimes been observed in overhead views to producedaughter bubble fragments which collapse separately.

In order to identify the effect of the changing features ofthe bubble pulse on spectra of the acoustic signal, a series ofspectra of the first bubble pulse has been compared. Figure23 shows profiles of the first bubble pulse and precursorpeak, with the primary peak centered at t=0, while Fig. 24shows spectra of the first bubble pulse for standoff distancesless than approximately unity. In Fig. 23, the signals fromfive repeated tests at each standoff distance have been over-laid, as an indication of variability in the signals. As is evi-dent in Fig. 23, the jet impact peak becomes pronounced atX�0.3, although with variable delay between it and the pri-mary peak, and both the precursor peak and the collapsepeak become more variable before nearly vanishing as the

FIG. 18. Growth and collapse of an explosion bubble over a bed of sand.

FIG. 19. First period of a bubble over a bed of sand.


standoff distance approaches X�1. Similar behavior may beobserved in the spectra. The full width at half-maximum ofthe bubble pulse is typically approximately 16 s, the loga-rithm of the reciprocal of which is 4.8. A distinct peak in theaverage spectra can be discerned at a value of 4.8, at least forthe lower standoff distances where the pulse is strong. Atstandoff distances of approximately 0.5, where the bubblepulses become more erratic, the spectra do as well. At stand-off distances of approximately 0.3, where the jet impact peakis most distinct and the interval is approximately 80 s, thespectra show a faint rise at 104.1 Hz=1/80 s.

V. CONCLUSIONS

We have presented measurements of the acoustic signalsof spark-generated bubbles �laboratory-scale underwater ex-plosion bubbles�, as well as simulations for similar condi-tions, which demonstrate the effects of bubble deformationson the acoustic signals. The influence of a nearby rigid sur-face increases the first period up to a value of approximately25% greater than the free-field value when the explosion isright at the wall, and increases the second period signifi-cantly near a normalized standoff value of X=1, due to jet

FIG. 20. The first and second bubble pulses of bubbles over a bed of sand.Signals were measured using a quartz pressure transducer several inches toone side of the bubble. The ambient pressure was 1 atmosphere and themaximum bubble radius about 6 mm. All signals have been offset verticallyby an amount proportional to the respective standoff distance.

FIG. 21. Ratio of second period to first period for explosion bubbles over
sand.

and toroidal bubble formation during the first collapse. Thisincrease in the second period corresponds to reductions inthe first bubble pulse pressure and increases in the secondcycle bubble size near X=1.

The presence of a free surface produces similar but op-posite effects for moderate distances, causing a decrease inthe bubble period for bubbles close to the surface. In addi-tion, the distortion caused by the free surface has similareffects on the bubble pulse and second period as the distor-tion caused by a rigid surface, namely a weakening of thebubble pulse and a lengthening of the second period of the

FIG. 22. Experimentally measured acoustic pressure profiles taken aroundthe time of the first bubble pulse. At the lower standoffs, a precursor pulsecan be seen �noted 1�, and is attributed to reentrant jet impact.


bubble. The influence of a sandy surface produces similareffects to a rigid surface for moderate distances, but the sandtends to engulf the bubble and dissipate its energy for smallstandoff distances, greatly reducing the strength of laterbubble pulses and the period of later cycles. An explosionbubble near or within a hemispherical cavity in a rigid sur-face exhibits behaviors similar to a bubble near a flat rigid

FIG. 23. Precursor peaks for bubbleswithin approximately 1 maximumbubble radius of a rigid surface. Eachsignal has been offset vertically by anamount proportional to the standoffdistance. Five signals from repeatedtests have been overlaid over eachother for each standoff distance.

FIG. 24. Averaged spectra of the first bubble pulses from bubbles near arigid surface. Each spectrum has been offset vertically by an amount pro-portional to the standoff distance.


wall if the bubble is very much larger or smaller than the sizeof the cavity, but if the bubble size is of the same order asthat of the cavity, the restriction of motion caused by thecavity surrounding the bubble causes significant increases inthe duration of the bubble cycle and the period betweenpulses.

These results are consistent with general rules summa-rized as follows: Any factors which tend to confine thebubble, i.e., which restrict the motion of the fluid around thebubble, will tend to increase the first period. Any factorswhich tend to distort the bubble, e.g., large buoyancy forcesor the presence of nearby objects, will tend to increase theminimum volume of the bubble during its collapse and thustend to weaken the compression during collapse, resulting inweaker bubble pulses and longer periods in subsequentcycles.

The significance of these results to seismo-acoustic sig-nals can be demonstrated by considering a standard 1.8-pound signal charge, which at a depth of 56 ft. produces amaximum bubble radius of about 1 m, and a first bubbleperiod of about 0.12 s. If the explosion is within 1 m of thebottom at this depth, the period will be larger than this, andcould be as much as 0.15 s, according to the data presentedhere. The period is weakly dependent on the yield, as indi-cated by Eq. �1�, so that this difference in the periods willproduce an even greater difference in estimates of the yield.Given the same depth, a period of 0.15 s would correspondin the inverse problem to an overestimate of the yield of3.3 pounds, if Eq. �1� is used. This factor of 2 excess couldalso be predicted from the observation that a hemisphericalbubble on the bottom will have a period corresponding to aspherical bubble of twice the energy. The results for explo-sions over a hemispherical cavity indicate that, if the explo-sion is near a partial enclosure, the first bubble period couldbe increased by as much as a factor of 2, depending on theproximity and size of the enclosure. A factor of 2 increase inthe period would result in an overprediction of the yield by afactor of 8.

Finally, we remark that, given this new ability to makeaccurate predictions of bubble pulse, depths, and times, onecan numerically model the multiple impulsive point sources�shock and bubble pulses� to simulate a pressure time seriesthat defines more realistically the acoustic radiation from anunderwater explosion. This capability could be useful forrefined geoacoustic measurements using calibrated explo-sives, since each impulse occurs at a different depth andtime. It should also be possible, by inverting bubble pulsemeasurements to estimate yield and depth of an explosion�Gumerov and Chahine, 2000�, to calibrate dynamite fishingand other “explosions of opportunity.”

ACKNOWLEDGMENTS

This work was sponsored by the DCI Postdoctoral Fel-lowship Program and was conducted at DYNAFLOW, INC.,�www.dynaflow-inc.com�. The authors wish to thank manycolleagues at DYNAFLOW, who have contributed to one as-pect or another of this study, and most particularly Dr. Jin-
Keun Choi, Dr. Chao-Tsung Hsiao, and Mr. Gary Frederick.

We are also very grateful to Dr. John Cyranski of the Depart-ment of the Navy for his critical guidance and his support.

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