M. Ichihara et al- Airwaves generated by an underwater explosion: Implications for volcanic...

8/3/2019 M. Ichihara et al- Airwaves generated by an underwater explosion: Implications for volcanic infrasound

1/15

Airwaves generated by an underwater explosion:

Implications for volcanic infrasound

M. Ichihara,1 M. Ripepe,2 A. Goto,3 H. Oshima,4 H. Aoyama,4 M. Iguchi,5

K. Tanaka,6 and H. Taniguchi3

Received 10 May 2008; revised 9 December 2008; accepted 29 December 2008; published 26 March 2009.

[1] A shallow explosion in a fluid is one of the fundamental processes of airwavegeneration in volcanic eruptions. To better understand the mechanism of the airwavegeneration, underwater explosion experiments were conducted. Although the underwaterexplosions have been intensely studied over the last century, airwaves have received littleattention. In this study, pressure waves were measured in air and under water, and thecorresponding motion of the water surface was captured by a high-speed video camera.Airwaves and associated styles of the surface motion show similarities determined by thescaled depth, defined as the depth divided by the cubic root of the explosion energy. Theair waveforms are quite different from the pressure waves measured in the water and

cannot be completely explained by linear transmission of pressure waves from water to air.The mechanism of airwave generation is a combination of wave transmission anddynamics of the interface boundary. The first mechanism directly reflects the explosionsource, and the same source is also observed as the underwater pressure waves. On theother hand, the second is not necessarily visible in the underwater pressure waves but will provide information on the mechanical properties and behavior of the material above theexplosion source. Each mechanism is analyzed in the experiments.

Citation: Ichihara, M., M. Ripepe, A. Goto, H. Oshima, H. Aoyama, M. Iguchi, K. Tanaka, and H. Taniguchi (2009), Airwaves

generated by an underwater explosion: Implications for volcanic infrasound, J. Geophys. Res., 114, B03210, doi:10.1029/

2008JB005792.

1. Introduction

[2] Acoustic airwaves from an active volcano areexpected to provide useful information of the activities atthe vent [Ripepe et al., 1996; Sakai et al., 1996; Vergniolleet al., 1996; Morrissey and Chouet, 1997; Garces et al.,1998; Johnson et al., 2003]. However, methods to decodethe airborne information have not been established yet, andit is usually difficult to uniquely determine the source modelfor a single event.

[3] At the Stromboli Volcano, for example, visual obser-vations have demonstrated how an acoustic wave, whichhas a characteristic waveform, is generally produced by the

bursting of a bubble or bubbles at the magma surface[ Ripepe et al., 1996]. Vergniolle and Brandeis [1996]

proposed the acoustic signal is generated by the vibration

of a large bubble at the magma surface before it bursts.

Buckingham and Garcs [1996] presented a model in whichsound is generated as a wave transmitted to the air from anexplosive source embedded within the magma column, andits waveform is due to the resonance of the magma column.Since then, these models have been respectively applied toother Strombolian eruptions on different volcanoes [e.g.,

Johnson et al., 2004; Vergniolle et al., 2004; Hagerty et al.,2000].

[4] For Vulcanian eruptions, another model considers themagma surface to be covered by a solid plug before theexplosion. Gas overpressure increases and the sudden dis-ruption of the plug may generate strong airwaves [Johnsonand Lees, 2000]. When the degassing is not a single largeexplosion event, but rather a sequence of regular events, the

plug will act like a valve on a pressure cooker, where the

motion of the plug is the source of the observed acoustictremor in the air [Lees and Bolton, 1998].[5] To properly analyze eruption behavior using airwave

signals, a better understanding of the link between thewaveform and source behaviors is needed. In the case ofseismic waves, both inversion methods and forward mod-eling have been developed on the basis of linear elasticwave theory to analyze the waveform, and have become

powerful tools for determining source behaviors [e.g.,Chouet, 1996; Ohminato et al., 1998; Jousset et al., 2003;

Nakano et al., 2007]. In the case of airwaves, the nonlinearand complex nature of the wave transmission processesrequires quite difficult mathematical calculation. Numerical

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B03210, doi:10.1029/2008JB005792, 2009ClickHere

for

FullArticle

1Earthquake Research Institute, University of Tokyo, Tokyo, Japan.2Dipartimento Scienze della Terra, Universita di Firenze, Florence,

Italy.3CNEAS, Tohoku University, Sendai, Japan.4Institute of Volcanology and Seismology, Graduate School of Science,

Hokkaido University, Sapporo, Japan.5Sakurajima Volcano Research Center, DPRI, Kyoto University,

Kagoshima, Japan.6Research Institute for Computational Sciences, AIST, Tsukuba, Japan.

Copyright 2009 by the American Geophysical Union.0148-0227/09/2008JB005792$09.00

B03210 1 of 15
http://dx.doi.org/10.1029/2008JB005792http://dx.doi.org/10.1029/2008JB005792http://dx.doi.org/10.1029/2008JB005792http://dx.doi.org/10.1029/2008JB005792http://dx.doi.org/10.1029/2008JB005792http://dx.doi.org/10.1029/2008JB005792


2/15

methods have not been fully developed to treat the wavetransfer through a magma-air interface with a large imped-ance contrast, the large distortion and rupture of the magmasurface, and the jets of gas and magma fragments. It isimportant to clarify the significance of these effects and thelimit of linear wave theory in the transmission of airwavesfrom magma to the air.

[6] A group of researchers carried out underground

explosion experiments using dynamite to experimentallydetermine the relationship between the explosion source andthe associated surface phenomena [Goto et al., 2001; Ohbaet al., 2002]. They presented similarity laws using the cubicroot of the explosion energy (E1/3) in scaling the amplitudeand waveform of the airwave, the crater diameter, and thematerial ejection. They then characterized the observedsurface phenomena in terms of the scaled depth: Ds = d/E

1/3,where dis the depth of the explosion [Goto et al., 2001; Ohbaet al., 2002]. The scaling laws were applied to phreaticexplosions at Usu Volcano in 2000 to connect the surface

phenomena (airwaves and ejection styles) to the explosionsources (energy and depth) [Yokoo et al., 2002].

[7] The underwater explosion experiments presented in

the present paper were initially motivated by questionsabout the applicability of the underground explosion resultsto the phreatic explosions, and were conducted to determinedifferences caused by the different material properties

between the soil and water. However, the purpose of thispaper is not to present empirical similarity laws for under-water explosions to compare with phreatic eruptions, but tounderstand the mechanisms that generate features of theairwaves in relation to the scaled depth, underwater pressurewaves, and motion of the surface. Mechanisms and simi-larity laws for the surface motion and underwater pressurewaves have already been given in the literatures [Cole,1948; Kedrinskii, 2005], but only a few studies discussedairwaves [ Kuwabara et al., 1987; Adushkin et al., 2004].Therefore, experimental results and analyses presented inthis paper are new in the research of underwater explosionsas well as in the volcanology, and knowledge obtained bythis study will be useful in developing tools for bothvolcanological and engineering applications.

[8] In section 2, a brief review of underwater explosionphenomena is presented. Experimental methods and resultsare presented in sections 3 and 4, respectively. The observedwaveforms in the air are compared with those predicted bylinear-acoustic theory in section 5, and their relationshipwith surface motion is discussed in section 6. The applica-

bility of linear theory and processes that affect the wave-forms are discussed in section 7.

2. A Brief Review of Underwater ExplosionPhenomena

[9] The underwater explosion process occurs in a seriesof events: generation of high-pressure gas, pressure waveradiation, perturbation of the water surface, and motion ofthe gas bubble. All these events are described in detail in theliterature [Cole, 1948; Kedrinskii, 2005]. Here we brieflyreview the sequence of the phenomena and the background

physics of the E1/3 scaling law.[10] An underwater explosion is generated by the release

of gas at very high pressure and, in most cases, at high

temperature. In the case of a chemical explosion, the gas isgenerated by an extremely rapid chemical reaction. In thecase of a vapor explosion including an explosion by water-magma interaction, the gas is generated by an extremelyrapid evaporation of superheated water [Shepherd andSturtevant, 1982; Wohletz, 2002]. The pressure of the gasat the source begins to be released by the emission of anintense pressure wave into the water driving the outward

motion of the water. The pressure wave is characterized by aso-called shock wavefront with a roughly exponential decayin the tail, and it expands outward spherically. The only

parameter characterizing the length scale of the outgoingwavefield is the linear dimension of the explosion source,which is denoted by a. If all material properties are constant,the hydrodynamic equations suggest the solutions for dif-ferent explosion sizes are comparable when the distance (r)and time (t) are rescaled as r/a and t/a, respectively. This iswhat is called the similarity law of the explosions [ Cole,1948]. Because the linear dimension of the explosive isapproximately proportional to E1/3, where E is the totalenergy of the explosive, the similarity of the underwaterwavefield has frequently been discussed in terms of the

scaling of E1/3. (The weight of the explosive, W, can alsobe used instead of E [Cole, 1948].) It has been found the peak pressure of the shock wave is well approximated bya function of r/E1/3 for the range of 0.006 (r/E1/3) 0.12 m/J1/3 [Cole, 1948]. The concept of the scaled depththat is used in the present and preceding studies [Goto et al.,2001; Ohba et al., 2002] is based on this similarity nature ofthe wave field [Cole, 1948].

[11] The compressive wave is reflected as a rarefactionwave at the water surface. The interaction between thereflected wave and the expansion phase in the tail of theincident wave generates a negative pressure field. Whenthe negative pressure is strong enough to break the water,it generates cavitation, surface rupture, and the throwingout of water drops to form a structure called a spray dome[Cole, 1948]. If the similarity law of the underwatershock wave holds, the peak pressure of the shock wavethat hits the water surface directly above the source will bea function of the scaled depth, Ds = d/E

1/3, where d is thedistance between the explosion source and the watersurface.

[12] The gas generated by the explosion forms a bubble,and its dynamics covers another important aspect of under-water explosion phenomena. The pressure in the gas bubbleremains much higher than the equilibrium hydrostatic

pressure, though it decreases considerably after radiationof the main part of the shock wave. The water in theimmediate region around the bubble has a large outwardvelocity and the diameter of the bubble increases rapidly.The radial motion of the bubble turns into a strong oscil-lation, in which the gas pressure works as the spring andthe surrounding water works as the mass. The oscillationof the bubble is the secondary source of the pressure waveemitted into the water. Every time the bubble approachesthe point of having the smallest volume, a compressivewave is emitted, which is called the bubble pulse [Cole,1948]. The strong oscillation rapidly decays owing to theloss of energy by the wave transmission, heat generation,and turbulent flows. While the bubble oscillates, it alsomoves upward owing to buoyancy. After a while, depending

B03210 ICHIHARA ET AL.: AIRWAVES BY UNDERWATER EXPLOSION

2 of 15

B03210


3/15

on the depth of the explosion, the bubble reaches the watersurface and breaks into the atmosphere. The swelling of thewater surface by the bubble is called a plume. If the bubbleis sufficiently close to the surface, it deforms the watersurface in an early phase of its expansion. A strong vertical

jet of water then develops. It is noted the radial and verticalmotions of the bubble do not follow the same scaling law asthe wave field does because they are influenced by hydro-static pressure and gravity.

[13] All the above mentioned phenomena, the underwaterpressure waves, the behaviors of the bubble, and the motionof the water surface, have already been studied in detail[Cole, 1948; Kedrinskii, 2005]. In the present paper, weinvestigate the pressure waves induced by these phenomenain the atmosphere. We employ the E1/3 scaling for lengthand time as a reference, and compare the results in terms ofthe scaled depth.

3. Experimental Methods[14] The main parameters investigated in the experiments

are the depth and energy of the explosions, combined hereas the scaled depth (Table 1). We performed underwaterexplosion experiments with different scales, data acquisitionsystems, and topographies.

[15] The first experiment (E01) was conducted at LakeToya, Hokkaido, Japan (Figure 1a), in 2001 using anemulsion-type explosive (Ultex, Nihon Kayaku Co. Ltd.)with the explosion energy of 5.8 MJ/kg. We measured theunderwater pressure waves with piezoelectric sensors(PCB,138A01 and 138A05, 2.5 Hz to 700 kHz) using a

recording rate of 104 samples per second (10 kS/s) with a band-pass filter of 0.05 Hz to 1 kHz. The airwaves werem easu r ed o n t h e l and u si ng m i cro p ho n es ( A CO ,3348+7144, 0.1100 Hz 3 dB) with 1-kS/s recordingand a piezoelectric blast sensor (PCB, 137A23 5 Hz to250 kHz) with 1-MS/s recording. Most of the sensors wereset in a linear array (Figure 1a). The water surface motionwas monitored with a high-speed video camera (nac MEM-

RICAM Ci-4) at 1000 frames/s. The data were synchro-nized with an accuracy of 1 ms. Another campaign (E02)was conducted at the same site in 2002. The water surfacewas observed with three high-speed video cameras (nacMEMRICAM fxK3, nac MEMRICAM ci-Expo, VisionResearch, Phantom V4.1) from two different angles and,in addition to the ACO microphones, different sensors (e.g.,Bruel and Kjaer 4155+2639, 4 Hz to 16 kHz 2dB) wereused. Unfortunately, the underwater pressure sensors werenot installed because of poor weather conditions.

[16] The second experiment (E04) was conducted at anatural pool on a coastline in 2004 (Figure 1b). We madesmall explosions using an electrical detonator (NOF Corp.,IED No.6) with no explosive. According to the manufac-

turer, the explosion energy of the percussion cap was0.029 MJ. The time resolution of the measurement systemwas improved using a broad-band microphone (Bruel andKjaer 4193+2669L, 0.13 Hz to 20 kHz 3 dB) and a shockwave sensor (Kistler 701A, 5 Hz to 80 kHz) with highersampling rates of 20 kS/s and 200 kS/s, respectively. Toinvestigate the directional difference of the wavefield, thesensors were placed so as to have different directions andelevations (Figure 1b). The shock wave sensor was set overthe water so as to be free from the effects of the land. Theunderwater pressure waves were measured by the same

piezoelectric sensors as in the E01 experiment but at ahigher sampling rate of 200 kS/s.

4. Experimental Data

4.1. Pressure Waves

[17] The measured pressure waves for various scaleddepths and absolute energies are shown by the black linesin Figure 2, where the gray lines are the results of calcula-tion, as explained in section 5.1. The time axes are normal-ized assuming the cubic root scaling law, ts = t/E

1/3, where tis the real time and ts is the scaled time [Goto et al., 2001].The same features of the underwater pressure waves de-scribed in the literature [Cole, 1948] are observed in the

present data. The first pulse is the shock wave generated bythe explosion. The successive strong pulses observed for Ds! 0.028 m/J1/3 (Figures 2a, 2b, 2d, and 2e) are the so-called

bubble pulses, which are generated by the oscillation of a bubble as described in section 2. The expansion phasebehind each pulse indicated by arrows with Rs is dueto wave reflection from the water surface. Reflection fromthe bottom of the water is observed in the wave profiles fromthe E01 experiments approximately 30 ms after each pulse, asis indicated by an arrow with Rb in Figures 2a2c. Thistime delay is consistent with the mean depth of the lake, 35 m,around the explosion site. The reflection from the bottom isnot observed in the E04 experiments (Figures 2d2f).

[18] In contrast to the underwater signals, the pulsesmeasured in the air have the following features.

Table 1. Experimental Parameters

ShotWeightW (kg)

EnergyE (MJ)

Depthd (m)

Scaled DepthDs (m/J

1/3)

Experiments 2001 Using ExplosiveE01-01 0.10 0.58 0.47 0.0056E01-02 0.25 1.45 0.63 0.0056E01-03 0.25 1.45 3.15 0.028E01-04 0.25 1.45 6.30 0.056E01-05 0.50 2.9 0.79 0.0055

E01-06 0.50 2.9 3.97 0.028E01-07 0.50 2.9 7.94 0.056E01-08 1.00 5.8 10.0 0.056E01-09 1.00 5.8 5.00 0.028E01-12 0.50 2.9 0.50 0.0035

Experiments 2002 Using ExplosiveE02-01 0.50 2.9 1.82 0.013E02-02 2.00 11.6 2.89 0.013E02-05 2.00 11.6 1.00 0.0044E02-06 0.50 2.9 0.79 0.0055E02-07 1.00 5.8 5.00 0.028E02-08 7.80 45.2 9.93 0.028E02-12 2.00 11.6 2.89 0.013E02-14 1.00 5.8 10.0 0.056E02-16 4.0 23.2 0.99 0.0035

Experiments 2004 Using Percussion CapE04-14 - 0.029 1.00 0.030E04-15 - 0.029 0.87 0.028E04-16 - 0.029 1.73 0.056E04-17 - 0.029 0.87 0.028E04-18 - 0.029 0.17 0.0056


3 of 15

B03210


4/15

[19] 1. For shallow explosions (Figures 2c and 2f), the

first pulse has a distorted M shape. As the scaled depthincreases, the M shape disappears and the pulse becomes

broader (Figures 2a, 2b, 2e, and 2d).[20] 2. The relative strengths of the successive pulses

differ for the airwaves and for the underwater waves.[21] 3. The airwaves contain less energy in the high-

frequency range than the underwater waves do, but they aresimilar in the common low-frequency range, as is shown inFigure 7 and discussed in section 5.3.

[22] In the present paper, we focus on how the pressurewaves are transformed through the water-to-air boundary.We mainly use the first pulse because, in contrast to the casefor bubble pulses, the wavefield in the water is known tosatisfy the cubic root scaling law [Cole, 1948] and its

propagation is not affected by perturbation of the mediumgenerated by the explosion. The accuracy of the waveformmeasurements is discussed in Appendix A.

4.2. Water Surface Motions

[23] Selected frames of the high-speed video images fromthe E02 experiments are presented in Figure 3 for the scaleddepths Ds = 0.028 m/J

1/3 (Figure 3a) and Ds = 0.0055 m/J1/3

(Figure 3b). In the other cases with Ds being larger than 0.028or smaller than 0.0055, we observed phenomena at thesurface qualitatively similar to those seen in Figures 3aand 3b, respectively. Figure 3c shows a case from the E04

experiments for a scaled depth comparable with that in

Figure 3b, but with 100 times less explosion energy.Applying the scaling of E1/3 to both the spatial and timelengths, 1 m and 1 ms in Figure 3c are compared with 4.6 mand 4.6 ms in Figure 3b, respectively. The images arecorrected for comparable scaled time, and displayed nextto each other in Figures 3b and 3c. We note that thefollowing features are consistent with features discussed

by Cole [1948] and Kedrinskii [2005].[24] In the case of a relatively large scaled depth

(Figure 3a), the first fluctuation of the water surface isseen when the explosion shock wave reaches the watersurface. The white area expands on the surface (8 ms) andthe spray dome is generated (198 ms). The water spraysare produced by strong decompression in the water due to

reflection of the shock wave at the surface. The successive bubble pulses are visually observed two or three times. Inthe case of Figure 3a, the second pulse is observed between368 ms and 372 ms, but it is difficult to see in the still

pictures. After several periods of the bubble oscillation,the bubble appears at the surface (1,112 ms) and breaks(1,512 ms).

[25] In the case of Figure 3b with a smaller scaled depth,the water spray is generated within the first frame after theexplosion and is jetted vertically (14 ms). Large motion ofthe water surface associated with the bubble expansion andthe partial ejection of the gas from the bubble are observed

Figure 1. Experimental setups. The explosion point is indicated by a black star. The numbers aredistances in meters. (a) In the E01 and E02 experiments at Lake Toya, the height of the microphones atA1, A2, and A3 is 1 m, and the depth of the pressure sensors at P1 and P2 is 5 m. (b) In the E04experiments at a natural pool, the height of the shock wave sensor at S is 0.2 m, and the depth of the

pressure sensors at P1 and P2 are 1 m.


4 of 15

B03210


5/15

from the first expansion phase of the bubble oscillation(14130 ms).

[26] Comparing the results from the E01 and E02 experi-ments and the results from the E04 experiments, it isconfirmed the scaled depth-dependent similarity holds forthe surface motion, qualitatively (e.g., Figures 3b and 3c).However, the scaled time and length are not completelycomparable quantitatively in the entire sequence of theimages of the water surface movements, because they arecontrolled not only by the underwater pressure waves butalso by the bubble dynamics and the rupture process of

water. The latter processes do not follow the E1/3 scalinglaw as explained in section 2.

5. Comparison With Linear Theory

5.1. Waveform of the Explosion Pulse

[27] We calculate the acoustic field transmitted into theatmosphere using basic linear acoustic theory [Ziomek,1995]. We assumed the system consists of semi-infinite

bodies of water and air separated by a planar boundary. Thesource of the pressure waves is represented by a pointsource embedded in the water, because the size of the

Figure 2. Black lines are data from (a) E01-08, (b) E01-09, (c) E01-05, (d) E04-16, (e) E04-15, and (f)E04-18. In each case, the upper frame shows the underwater pressure wave measured at P1, and the lowerframe shows the airwave measured at A1 in E01 and B in E04. The time axis is scaled by the cubic rootof the explosion energy, and the actual time is displayed on the upper axis. The insert is an expanded viewof the gray area. The gray lines superposed on the airwave data are the results of calculation, as discussedin section 5.1. Tb indicated in Figure 2a is the period of the first cycle of the bubble pulse. The arrowswith Rs and Rb on the underwater pressure waves indicate the reflection phase from the surface and

bottom of water, respectively. The arrows with H indicate the short pulses mainly passing through thewater. These pulses are observed in the theoretical waveform (the gray lines), but are not recognized inthese data. The arrows with M indicate the characteristic airwave waveform with an M shape.


5 of 15

B03210


6/15

Figure 3


6 of 15

B03210


7/15

explosive is much smaller than the wavelength and thelength scale of the measurement system. The mathematical

procedure is detailed in Appendix B. Because of the pointsource assumption, this calculation itself does not explicitlyinclude the scaling effect of E1/3. However, the nature of thehydrodynamic equations asserts that the similarity of theunderwater pressure wave is observed when both length andtime of the measurements are rescaled by a common factor[Cole, 1948]. This factor is totally arbitrary and can be set

equal to E1/3

.[28] We assumed the source function po(t) of the under-water shock wave is well represented by the waveformmeasured in the water without the reflection phase. Sincethe underwater signal in the E01 experiments is stronglyaffected by the low-pass filter of the acquisition system, wealso considered a theoretical waveform for the underwatershock wave [Cole, 1948]. For both cases, the resultantwaveforms in the atmosphere were the same. Below, wecompare the calculation and the data in terms of waveforms.The amplitude is discussed in section 5.2.

[29] We found linear theory well explains the dependenceof the pulse width on the scaled depth. However, a strongshort pulse (indicated by arrows with H in Figures 2a, 2b,

2d, and 2e) is observed in the calculated waveform muchearlier than the arrival of the main pulse. On the basis of the

propagation time of the wave, we suggest this wave mainly propagates in the water and is then transmitted to the airnear the observation point. This pulse is not clearly visiblein the data (Figure 2), and we infer it has been lost in the

interaction with the ground before reaching the micro-phones. In the E04 experiments, the shock wave sensor (Sin Figure 1b) was installed above the water, and it detectedthe short pulse (indicated by arrows with H in Figure 4).Linear theory explains the arrival time of the short pulse aswell as the width of the subsequent main phase, though itfails to reproduce the waveform and amplitude of the short

pulse (gray lines in Figure 4).

[30] On the other hand, there are considerable differences between the theoretical waveform and the measured pres-sure waveform. Linear theory fails to reproduce the high-frequency oscillation observed for relatively large Ds(Figures 2a, 2b, 2d, and 2e) and the characteristic M shapeobserved for relatively small Ds (indicated by arrows with Min Figures 2c, 2f, and 4c,). The ground motion due to theseismic waves partially generates the high-frequency com-

ponents by local transmission of the airwave and by vibrationof the microphone, but there are certainly high-frequencycomponents in the airwave from the source (Appendix A).We consider high-frequency components and the M shape are

partially generated by interference with the water surface,which is not included in linear theory. These effects are

discussed in section 7.

5.2. Amplitude of the Explosion Pulse

[31] We also apply linear theory to the amplitude of themeasured pressure signals. In each calculation presented inFigures 2 and 4, the theoretical amplitude of the source

pressure po(t) calculated 1 m from the point source has beenadjusted to fit the amplitude of the measured airwave. Thevalues are compared with the signals measured by theunderwater pressure sensors in the following ways.

[32] The peak pressure of the shockwave at a distance rfrom the source is approximated by a power law:

Ppeak r K rW

1=3TNT

!a Kqa=3 r

E1=3

a; 1

according to Kirkwood-Bethe theory [Cole, 1948], whereWTNTis the TNTequivalent weight of the explosive and r/E

1/3

is the scaled distance. The empirical constants have beendetermined as K = (5.2 0.2) 107 in MKS units and a =1.14 0.01 for several types of commonly used explosives[Cole, 1948]. The weight, WTNT, is related to the explosionenergy, E, as WTNT = E/q, where q = 4.6 MJ/kg is theexplosive energy of TNT, and thus the right-hand side ofequation (1) is obtained. The coefficientKqa/3 = (1.41.7) 105 is given in MKS units. This empirical relation is fairly

accurate for 0.006 (r/E1/3

) 0.12 m/J1/3

[Cole, 1948].[33] We use equation (1) and the underwater pressure data

to estimate the amplitude (Ppeak) 1 m from the point source.The following two methods are applied. In the first method,we use the amplitude of the explosion shockwave measured

by the underwater sensor (P1 in Figure 1a). Figure 5

Figure 3. Selected frames from the high-speed motion pictures for (a) E02-07 (Ds = 0.028 m/J1/3), (b) E02-06 (Ds =

0.0055 m/J1/3), and (c) E04-18 (Ds = 0.0056 m/J1/3). The white number at the right bottom of each image is the time (ms)

from the explosion. According to the cubic root scaling law, the ratio in both the length and time is Figure 3a:Figure3b:Figure 3c = 5.8:4.6:1. Figures 3a and 3c compare the surface phenomena between different scaled depths. Figures 3band 3c compare the phenomena between the same scaled depth but different scales.

Figure 4. Black lines are airwave data measured by theshockwave sensor over the water (S in Figure 1b). (a) E04-16, (b) E04-15, and (c) E04-18. The superposed gray linesshow the results of calculation discussed in section 5.1. Thearrows with H indicate the wave mainly passing throughthe water. The arrow with M indicate the characteristicairwave waveform with an M shape.


7 of 15

B03210


8/15

compares the measured amplitudes with those given byequation (1). We use the peak amplitudes measured forrelatively deep explosions (Ds ! 0.028 m/J

1/3) (solidsymbols with white crosses in Figure 5). Strictly speaking,the peak pressure measured by the second sensor (P2 inFigure 1a) is beyond the range of the scaled distance forwhich the reliability of equation (1) is guaranteed [Cole,1948] (squares in Figure 5). However, we include these

points because they follow the same power law and change

the fitting parameter by less than 1%. Peak pressures fromshallower explosions (solid symbols without crosses inFigure 5) are excluded because they have been influenced

by the reflection wave from the water surface. The datafrom the E01 experiments follow the same power law asequation (1) does, but their absolute values are smaller thanthe empirical values. The absolute values from the E04experiments are also smaller. Although we cannot check thereliability of the power law for the E04 experiments becauseof the limited range of the scaled distance, we determined Kassuming the power law with a = 1.15. We estimate theamplitude of the shockwave at r = 1 m from the sourceusing the power law equation (Figure 5) and compare thevalue with the theoretical source pressure po.

[34] In the second method, we use the time intervalbetween the first and second pulses, Tb, which correspondsto the first cycle of bubble oscillation (Figure 2a). Since Tbrepresents the energy of the bubble oscillation, it can beused as a measure of the explosion energy. Tanaka et al.[1981] investigated the relation between Tb and both theexplosion energy and the amplitude of the underwatershockwave. They obtained

TbP5=6

so CE1=3; 2

where Pso is the static pressure at the explosion point and Cis an empirical constant. Using equation (2), we caneliminate E from equation (1) to obtain

Ppeak r Ksr

TbP5=6

so

!a; 3

where Ks = Kqa/3

Ca

. They showed that for various typesof explosives, the experimental data are fitted well byequation (3) with a = 1.15 and Ks = 3.6 10

3 for Ppeak inunits of pascals (Appendix C). The results of the secondmethod are independent of those of the first method becausethe second method does not use the nominal explosionenergy and peak pressure measured in the water. The secondmethod has the great advantage of being unaffected by theuncertainty of the actual explosion energy or errors in the

peak pressure measurement. We apply this method toestimate the peak pressures of relatively deep explosions(Ds ! 0.028 m/J

1/3) for which the bubble pulses are clearlyseen.

[35] Figure 6 compares the amplitudes ofpo(t) used to fit

the measured airwave and the values of Ppeak at r = 1 mestimated by the above two methods. If both the theory andthe measurement are correct, the ratio is expected to beunity. When the ratio is less than unity, the measuredunderwater airwave is weaker than that predicted by lineartheory. Possible causes of the misfit are (1) the amplitudeand/or waveform of the underwater wave has not been

properly measured, (2) the amplitude of the airwave ampli-tude has not been correctly recorded, and (3) linear theory isnot appropriate for describing the observed phenomena. Toreduce the errors from cause 1, we used the two independentmethods to estimate Ppeak. We also tested with a theoreticalwaveform for the underwater shock wave [Cole, 1948], butno significant difference was observed. To reduce the effects

of cause 2, we fitted the airwave data recorded by stationsA1 and A2 in the E01 experiments (Figure 1a) and stationsB and S in the E04 experiments (Figure 1b), and the resultsare plotted in Figure 6. As a result, we conclude lineartheory does not allow one to accurately calculate theamplitude of the airwave. In particular, the error is moresignificant for the E01 experiments than for the E04 experi-ments, and the measured airwave for the E01 experiments issmaller than the theoretical prediction. Although verticalchanges in temperature and pressure have a large influenceon the propagation of wave energy in large distances, we donot believe this effect is significant on the scale of the E01experiments ($100 m), because the changes are in the orderof 1 degree and 1000 Pa. We infer in a large explosion,

some of the wave energy is lost at the water surface owingto fluctuation, fragmentation, and evaporation of water.These effects are not included in linear theory and arediscussed in section 7. For the E01 experiments, the ratioof the measured amplitude to the theoretical amplitude

becomes even smaller as the scaled depth increases (Figure6). We have not yet identified the mechanism to cause thisdependence on the scaled depth.

5.3. Comparison at Low Frequency

[36] The waveforms shown in Figure 7 are from the E0108 experiments (Figure 2a) using a band-pass filter from

Figure 5. Peak pressures of the underwater shockwave,Ppeak (MPa), are compared with the empirical power law,Ppeak = K(r/E

1/3)1.15, given in the literature [Cole, 1948].The fitting constant K is determined by the solid symbolswith white crosses as K= 0.098 (9.8 104 forPpeak in Pa).The solid symbols without white crosses are data fromshallow explosions and are affected by the reflection fromthe water surface. These data are excluded from the fitting.Data from the E04 experiments (open triangles) determine

K = 0.057 (5.7 104 for Ppeak in Pa). All of the datameasured in the present experiments are smaller than theempirical values (black line).


8 of 15

B03210


9/15

2.5 Hz to 10 Hz. The lower cutoff frequency is chosen accord-ing to the response of the piezoelectric sensor used tomeasure the underwater pressure wave. The upper cutofffrequency is chosen so that the features of the bubbleoscillation remain, while the reflections from the surfaceand bottom of the lake become obscure. In this frequencyrange, the waveforms in the water and air are similar.Comparing this low-frequency component of the pressurewaves with that predicted by linear theory is useful in testingthe applicability of the theory.

[37] We use the pressure wave in the water shown inFigure 2a as the source time function po(t) with correctionof the amplitude for the distance. The theoretical wave inthe air is calculated in the same way as described in section5.2. The same band-pass filter of 2.510 Hz is then appliedto the solution. Figure 7b shows the calculated waveform(gray line) represents the low-frequency features of the

bubble pulses (black line). The amplitude has been reducedby a multiplication factor of 0.56 to fit the data, though thisreduction is less than the factor of $0.1 used to fit the first

pulse in Figure 2a. The waveform of the first pulse isdeformed by the filter and is not compared in this analysis.

[38] Our comparative analysis demonstrates how lineartheory explains the general features of the airwave ade-quately. Nevertheless, as presented in section 5.1, there arefeatures that are not explained by the theory; namely theamplitude of the airwaves, the high-frequency oscillationfor relatively large Ds, and the characteristic M shape forrelatively small Ds. We consider these features to represent

processes that are not included in linear theory nor visible inthe pressure wave under water.

6. Comparison With the Water Surface Motion

[39] The M shape is a dominant feature of the airwave fora shallow explosion, and is not explained by linear wave

theory. Comparing Figures 2c and 2f, we can see a similar Mshape for the same scaled depth of the explosion regardlessof the respective values of depth and energy. This evidencesuggests the mechanism generating the waveform is relatedto the E1/3 scaling law, and the motion of the water surface isa good candidate (Figures 3b and 3c) for explaining theobserved M shaped acoustic wave.

[40] However, in the high-speed video images, correlation

between the waveform and the movement of the watersurface is not obvious. Figure 8 compares the images withthe airwave data. The time delay for the propagation of theairwave to the sensor is corrected. To emphasize the

perturbation of the water surface, each frame in the left-hand column is a differential image between the frame at theindicated time and the frame right before the explosion, andeach in the right-hand column is a differential image

between two sequential frames. In the left-hand column,the underwater shock wave is clearly visible as an expand-ing white area on the water surface, while the shock wave

propagating in the air is highlighted by the distortion of thebackground scenery, indicated by arrows in Figures 8a8c.The water around the jet becomes darker during the expan-

sion phase of the airwave (Figures 8e8g). Although weobserved the images carefully, we noticed no clear evidencethat can explain the onset of the second compression phase(Figures 8h and 8i).

[41] Theoretically, an expanding volume, V(t), on thewater surface generates an acoustic wave, pa(t), as [Lighthill,1978; Vergniolle and Brandeis, 1996]

pa r; t ra

2pr

d2

dt2V t r=ca : 4

Inversely, V(t) is estimated from the acoustic signal as

V t 2prra

Ztto

dt0Zt0

to

pa r; t00 r=ca dt00: 5

[42] As a test, we assume the observed acoustic waveshown in Figure 8 is generated by the volume expansion

Figure 6. Ratio of the underwater shockwave amplitudeused to fit the airwave data by the linear wave theory, po,and that estimated from the underwater pressure data, Ppeak,

both of which are estimated at a distance of 1 m from theexplosion source. The values of Ppeak(1) are obtained bytwo methods. Method 1 used the peak pressure of the dataand assumed the empirical power law (1), whereas method2 used the period of the first bubble pulse and assumed itsempirical relation to the shockwave amplitude (3). In theE01 experiments, the ratio is significantly smaller than

unity, indicating that the measured airwave is by an ordersmaller than what is expected from the underwatershockwave.

Figure 7. A band-pass filter from 2.5 Hz to 10 Hz isapplied to the data shown in Figure 2a. (a) The underwater

pressure wave and (b) the airwave are similar in thisfrequency range. The gray line is obtained by the lineartheory with the amplitude adjusted to fit the data.


9 of 15

B03210


10/15

Figure 8. Comparison between the airwave and the surface motion for E02-16. (left) A differential fromthe image before the explosion. (middle) Sequential differences. The airwave front is shown by thearrows. (right) The times corresponding to Figures 8a, 8b, 8c, 8d, 8e, 8f, 8g, 8h, and 8i are indicated onthe pressure wave profile with correction for the time delay of airwave propagation.


10 of 15

B03210


11/15

directly above the explosion source and calculate V usingequation (5), even though the observed wave includes the

explosion signals passing through the larger area of thewater surface. The result is shown in Figure 9. We alsoestimated the volume expansion of water surface bytracking the edge of the jet, h(x), in the high-speed videoimages (Figure 9d). Here, x is the horizontal distance fromthe center of the jet and h is the height of the jet at x.Assuming a semicylindrical symmetry, the volume, Vimage, iscalculated by

Vimage

Zpjxjh x dx: 6

The result forVimage is presented in Figure 9c by points withthe axis on the right. The magnitude of Vimage issignificantly larger than the magnitude of Vestimated fromthe measured airwave. We consider the above methodoverestimates the volume expansion because the volume ofthe jet does not represent the pure bulge expansion of thewater surface into the atmosphere, but is rather a mixture ofwater drops and the entrained atmospheric air. The volume

acceleration (d2V/dt2), which is directly connected with theairwave by equation (4), is not large enough to be clearlyvisible in the plot of V (Figure 9c), and the accuracy ofVimage is insufficient to compare d

2Vimage/dt2 with the

observation of the airwave. Therefore, it is yet to beconfirmed whether this motion of the water surface canexplain the observed M shape acoustic wave.

7. Discussion

[43] The airwaves generated by underwater explosionshave two features that are not explained by linear theory.The first feature is the amplitudes of the airwaves beingmuch smaller than those expected from the theory. This

difference was observed only in the experiments performedin the lake. The second feature is that the air waveformshave high-frequency components for explosions with rela-tively large Ds and a distinct M shape for explosions withrelatively small Ds.

[44] We infer that the small amplitudes are due to ineffi-cient wave transmission from the water to the air. Someenergy is converted into different energies (e.g., thermalenergy, kinetic energy of the water drops, and higher-frequency wave energy) by processes such as the fragmen-tation of water, evaporation, and large deformation of thesurface. For example, the velocity of the water jet in theE01 05 experiments is 40 m/s, obtained from the first10 frames (10 ms) of the high-speed video. The accelerated

area is approximately 1 m in radius. If the thickness of theaccelerated water layer is assumed to be 0.1 m, the kineticenergy is 0.7 MJ, equivalent to one quarter of the energy ofthe explosive charge (Table 1). On the other hand, theenergy transferred into the shockwave by an underwaterexplosion is usually from one fifth to one third of the totalenergy [Tanaka et al., 1981]. This means a considerablefraction of the shockwave energy is converted into thekinetic energy of the water drops. These energy losses areconsidered more significant in a larger explosion, in which alarger area of water experiences tensile stress above thefracture strength [Kedrinskii, 2005]. Therefore, the airwaveweakening was observed in the E01 experiments but not inthe E04 experiments.

[45] The other unexplained feature observed in theexperiments is the high-frequency component or the Mshape of the airwaves. Adushkin et al. [2004] recognizedfeatures of the first pulse similar to those shown in Figure 2b.

Adushkin et al. [2004, p. 712] stated the acoustic signal ofthis explosion has the first positive phase characterized by asmooth increase in the leading front and a comparativelysharp decrease in the rear front and explained the characteras a result of cavitation. Although how the cavitationgenerates the sharp decrease in the rear front has not

been determined, we consider it can certainly generate high-frequency perturbations in the airwaves; the oscillation of

Figure 9. (a) Plot of pressure wave in air versus time.(b) Plot of dV/dt versus time. (c) Plot of V (line) and Vimage(colored diamonds) versus time. (d) Trace of the edge ofthe expanding jet. The pressure wave in the air (Figure 9a)is converted into volume expansion rate (Figure 9b) and

then to the volume expansion (Figure 9c) at the source,assuming Lighthills [1978] equation (4). The volumeexpansion is also estimated from the high-speed videoimage by tracking the edge of the jet (Figure 9d). Then thevolume is calculated by equation (6), and is shown inFigure 9c by diamonds with axis on the right. Eachdiamond in Figure 9c has the same color as the edge ofthe jet in the corresponding time in Figure 9d. The data arefrom E02-16.


11 of 15

B03210


12/15

the small individual bubbles in the cavitation zone generateshigh-frequency pressure waves, and these are efficientlytransmitted into the air because impedance of the water with

bubbles is much smaller than that of pure water and matchesbetter with that of the air.

[46] The M shape may be explained as a continuation ofthe waveform described above, that is the sharp decrease inthe rear front, and it may also be generated by the cavitation.However, there is another possible explanation. In shallowexplosions that generate the M shape, we observe upwardhigh-velocity flows of water that form a characteristic shapewith a thin projection at the center (Figures 3b and 8). Thevertical water column is called a sultan in Russian

publications and is associated with the reflection of a shockwave and the dynamics of the bubble generated by theexplosion [Kedrinskii, 2005]. The formation of a sultan isexplained as the rapid expansion of the bubble with theinteraction with the water surface [Kedrinskii, 2005]; nu-merical calculation demonstrates the following sequence ofthe bubble expansion and the motion of the water surface.At first, both the bubble expansion and the surface motionare strongly accelerated by the explosion. Since the pressurein the bubble decreases, the bubble expansion soon decel-erates. The motions of liquid particles on the free surfacefirst follow the boundary of the bubble; that is theyaccelerate and then decelerate. The upper part of theexpanding bubble then begins to approach the surface and

expands upward, while the bottom half continues to decel-erate. At this point, the water surface is accelerated for asecond time. This second upward acceleration may generatea second compression phase for the M shaped airwave, assuggested by equation (4). The M shaped waveform is theninterpreted as the combined result of wave transmission (1)from a source embedded in the liquid and (2) from the rapidexpansion of a bubble at the surface. It is interesting to pointout the two models have been separately proposed toexplain the origin of volcanic airwaves [ Buckingham andGarcs, 1996; Vergniolle and Brandeis, 1996]. Here we

propose the possibility that a waveform can be generated

not only by a single mechanism but by a sequence involvingboth processes.

[47] We have found the airwave generated by underwaterexplosions follows the E1/3 scaling law. That is, the wave-form of the explosion pulse depends on the scaled depth ofthe explosion (d/E1/3), and for the same scaled depth, itswavelength is scaled by E1/3. The waveforms of the bubble

pulses are also similar for the same scaled depth, even

though the period of the bubble pulse is not scaled by E1/3because of the influence of hydrostatic pressure. The scaleddepth similarity of the airwave waveforms has also beenreported for previous underground explosion experiments[Goto et al., 2001; Taniguchi et al., 1999]. Some examplesare shown in Figure 10 and some have been published[Goto et al., 2001]. Waveforms produced by explosions inthe ground and in water are quite different, even for thesame scaled depth. As the explosion depth becomes shal-lower in the underground explosions, the waveform doesnot have the M shape but approaches an N shape [ Goto etal., 2001], which is a typical waveform generated by anexplosion in air [Kinney and Graham, 1985]. This differ-ence indicates the waveform similarity does not hold for

different media. This may sound like a negative result forapplying results of underground or underwater explosions tovolcanic explosions. We agree the application is notstraightforward. However, the knowledge obtained in thisstudy is important and useful in analyzing airwaves gener-ated by volcanic explosions; it is important to understandthe mechanism of airwave generation is not just a linearwave transmission through the interface boundary, but acombination of wave transmission and dynamics at the

boundary. The former mechanism directly reflects theexplosion source, and the same source is also reflected inthe underwater pressure waves. On the other hand, the latteris not necessarily visible in the underwater pressure waves.

[48] For an airwave generated by an explosion in magma,we may be able to separate the two mechanisms bycomparing seismic waves and airwaves. The latter processwill provide information on the mechanical properties andsurface dynamics of magma in the crater. Inversely, if weknow the physical properties of the material in the crater, wemay be able to infer the depth and energy of the explosionusing the airwave waveform. Further studies, however, arerequired to understand the relationship between the airwave,the material properties, and the dynamics of the free surface,and the relationship between these three factors and the realsource.

8. Conclusion

[49] We have investigated airwaves generated by under-water explosions. The airwave consists of a primary pulse,that is directly related to the explosion process, and succes-sive pulses associated with the oscillation of the gas bubble

produced by the explosion. In the present study, the primarypulse in the air is analyzed by comparing with results fromlinear wave theory and by analysis of movement of thewater surface. We conclude the following:

[50] 1. The waveform depends on the scaled depth of theexplosion. For the same scaled depth, the time width of the

pulse is scaled by E1/3.

Figure 10. Airwaves generated by underground explosions[Taniguchi et al., 1999] with scaled depths comparable withthe underwater explosions of this study at (a) 0.021 m/J1/3,(b) 0.0084 m/J1/3, and (c) 0.0044 m/J1/3.


12 of 15

B03210


13/15

[51] 2. The width of the first compression phase of thepulse is reproduced by linear wave theory assuming a pointsource in the water. The same theory explains the low-frequency component of the airwave associated with the

bubble oscillation as well.[52] 3. Explosions with small scaled depths generate an

M shape waveform in the primary pulse, and this is notexplained by linear wave theory. We assume the M shape isgenerated by the combined effect of linear wave transmis-sion from the explosion source and deformation and rupturedynamics of the surface.

[53] 4. In explosions with relatively large scaled depths,the airwave pulse contains high-frequency components notexplained by linear wave theory. We infer the oscillation isgenerated by the perturbation of the water surface, includingcavitation and water fragmentation.

[54] 5. Linear wave theory fails to explain the amplituderatio between the airwave and the underwater shockwave inthe E01 experiments. In this case, the measured airwavesare an order of magnitude smaller than those predicted bytheory, indicating a significant loss of wave energy occurs atthe interface.

[55] 6. The mechanism of airwave generation is a combination of wave transmission from the explosion sourceand dynamics of the boundary. The latter mechanismsuggests that the air waveform contains information onthe mechanical properties and surface dynamics of thematerial above the explosion source in addition to the

physical process inherent in the source. To fully appreciatethis duality, further studies are required to understand therelationship between the airwave, the material properties,

and the dynamics of the free surface, and the interaction ofthese three factors with the explosion source.

Appendix A: Sensor Calibrations

[56] In each set of experiments, the sensitivities andcharacteristics of the sensors were tested by comparing datataken at the same position. Some examples are presented in

Figure A1. Although we used different acquisition systemsand microphones for the E01 and E02 experiments, themeasured amplitude and waveform are almost the same forthe same explosion conditions (Figure A1a). The shockwave sensor used in the E04 experiments on the water(Kistler 701A) is sensitive to the acceleration (100 Pa/G).The strong waves detected before the airwave in Figure A1bare due to the motion of the sensor with the ground.However, the oscillation of the sensor does not influencethe measurement of the pressure wave, and the signalsrecorded by the shockwave sensor and by the microphone(BK) are quite comparable (Figure A1b). Figure A1ccompares the measurements of the underwater pressure

Figure A1. Comparison of the signals recorded by twodifferent sensors at the same position to confirm theaccuracy of the measurements. (a) Airwaves measured atA1 (Figure 1a) by ACO (3348+7144) for E01-05 (blackline) and by Bruel and Kjaer (4155+2639) for E02-06 (grayline). (b) Airwaves measured by the shock wave sensor(Kistler 701A) and Bruel and Kjaer (4193+2669L) at thesame position. (c) Underwater pressure waves measured bythe two sensors at the same position.

Figure A2. Comparison of (a) the vertical component ofseismic data, (b) the microphone signal, and (c) the samemicrophone signal with a 30 Hz high-pass filter. The solidand dashed gray lines with AC and S indicate

propagation of acoustic and seismic waves, respectively,demonstrating the existence of high-frequency components

propagating as airwaves. Data are taken in the E01-09explosion at stations A1 and A2 shown in Figure 1a.


13 of 15

B03210


14/15

wave taken by the two sensors at the same position. The testprovides evidence that the amplitudes and waveforms are allcomparable and that no errors are introduced by the use ofdifferent sensors.

[57] The explosion generates seismic waves, which vibratethe microphones and generate noise in the signals. Theground motion is measured by a short-period seismometer(Mark, L3C-3D) at each location of the microphones.Figure A2 compares the vertical velocity of the groundwith the original and high-pass filtered airwaves at twoobservation points. The broken gray lines with S indicatethe propagation of the seismic waves. The solid gray lines

with AC characters indicate the propagation of the airwaves.We see strong high-frequency signals are detected by themicrophone after the arrival of the airwaves and the signals

propagate at the speed of the airwave.

Appendix B: Linear Acoustic Theory

[58] We consider a system consisting of semi-infinite bodies of water and air separated by a planar boundary.The acoustic wavefield, Ptw, transmitted to the air from atime harmonic point source, eiwt/r, in the water is given as[Ziomek, 1995]

Ptw

ieiwt

4pZ1

0

rakrJ0 krR eikzajzjikzwd

rakzw rwkza dkr; B1

where t is time, w is the angular frequency, z is the verticalaxis taken downward from the water surface, d is the depthof the explosion source, R is the horizontal distance fromthe source, J0 is the zeroth-order Bessel function of the firstkind, and subscripts w and a indicate the values of the waterand the air, respectively. The densities are rw = 1000 kg/m

3

and ra = 1.2 kg/m3. The wave numbers are defined as kw,a =

w/cw,a with speeds of sound of cw = 1450 m/s and ca =346 m/s, respectively, which are calculated from the travel

time between the stations in the E01 experiments. The zcomponents of the propagation vectors are

kzw

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2w k

2r

p; k2r k

2w;

iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2r k2w

p; k2r > k

2w;

8 k

2a :

8

M. Ichihara et al- Airwaves generated by an underwater explosion: Implications for volcanic...

Documents

Transcript of M. Ichihara et al- Airwaves generated by an underwater explosion: Implications for volcanic...