Experimental Laboratory Study on the Formation of Multiple 1

Shock Waves observed during Volcanic Eruptions 2

E. F. Médici1, and G. P. Waite2

1Department of Mechanical Engineering - Engineering Mechanics

Michigan Technological University

Houghton, Michigan, USA.

2Department of Geological and Mining Engineering and Sciences

Michigan Technological University

Houghton, Michigan, USA.

Abstract

Sequences of multiple pressure waves have recently been observed in different explosive 3

volcanic eruptions, but their origin is not clearly understood. Although a pulsating type of 4

eruption with many individual discharges can produce this effect, the formation of multiple 5

shock waves could also result from a single discharge through: 1) the dynamics of the shock 6

wave itself; 2) the coupled shock wave-supersonic jet dynamics; or 3) a combination of these 7

factors. Shock tube experiments were performed to evaluate these potential mechanisms under 8

controlled laboratory conditions. The amount of energy released was varied to achieve different 9

weak shock wave conditions and to promote formation of single or multiple shock waves 10

through both shock wave dynamics and coupled shock wave-supersonic jet interactions. Our 11

experimental results indicate that both mechanisms may play a role in the formation of multiple 12

pressure waves in low intensity eruptions. 13

1. Introduction

Explosive volcanic eruptions are characterized by a sudden release of gases, liquid and 14

solid particles. During the first instants of eruptions, the multiphase mixture expands into the 15

atmosphere and may commonly reach supersonic speeds. The supersonic jet accelerates the 16

gasses in front of it, potentially triggering a shock wave. The jet may continue ascending and 17

expanding behind the initial shock wave and persist while there is sufficient energy in the 18

conduit to sustain it. The shock wave, on the other hand, can travel many kilometers from the 19

vent of the volcano but will eventually decay into a sound wave [Morrissey and Mastin, 2000]. 20

The shock wave, and the eventual sonic pressure wave, carries important information about the 21

energy and dynamics of the eruption. A better understanding of supersonic jet dynamics and 22

shock wave propagation may lead to new methods for assessing vent conditions in real-time 23

using existing infrasound and video monitoring technologies. 24

Supersonic jets, weak shock waves, and pressure waves have been observed and 25

measured in volcanic eruptions for decades [Nairn 1976; Morrissey and Chouet, 1997; Morrissey 26

and Mastin, 2000]. Still, there are many fundamental aspects regarding the formation and 27

expansion of supersonic jets and shock waves that require further study. One intriguing 28

observation is the formation of multiple consecutive shock waves during a single eruption pulse 29

[e.g., Taddeucci, et al., 2014]. Typically, in shock tube testing or controlled detonation 30

experiments, a single shock wave is observed for a single discharge or detonation event. The 31

Friedlander wave profile [Freidlander, 1946], and its approximation equation, provides a simple 32

model of a sharp pressure rise followed by a smooth rarefaction, typical of a detonation test. 33

Although the phenomenon described by the Friedlander equation can accurately describe the 34

initial high-pressure front, more complex gas dynamics can take place behind this initial front. 35

For example, it is well known that a second shock wave might develop as consequence of the 36

rarefaction pressure drop behind the leading shock wave [Landau, 1945]. The pressure in the 37

rarefaction phase can drop far enough below atmospheric pressure to trigger a second shock 38

wave. The early work by Landau [1945] discussed the physical mechanism triggering the second 39

shock wave. This phenomenon has since been observed in experimental tests [Biss, and 40

McNesby, 2014; Gitterman and Hofstetter, 2014] and the theoretical analysis generalized for 41

supersonic aerodynamic and sonic booms [Whitham, 1956]. The idealized shock wave pair 42

consists of a linear decrease from the initial shock discontinuity to a value below atmospheric 43

pressure before the second shock is triggered. Because the pressure time history resembles the 44

letter “N”, this type of wave has historically being referred as “N-wave” [Whitham, 1956; 45

Blackstock, 2000]. Rarefaction pressure decay observed in a Friedlander type of pressure profile 46

is not considered as N wave because it does not result in a second shock. 47

The theory describing the N wave profile can explain the formation of two, and 48

sometimes more shock waves. However, it may not sufficiently explain the multiple consecutive 49

shock waves observed in some volcanic eruptions, such as those captured by Taddeucci, et al. 50

[2014] at Yasur in 2011. If the rarefaction between subsequent shocks is not far enough below 51

atmospheric pressure to trigger a shock, they must be explained by another mechanism. The 52

complex gas flow dynamics taking place between the leading shock wave and perhaps, the 53

expanding jet head might produce these pressure waves or weak shock waves. Supersonic jets 54

are well known to produce lateral shock waves known as Mach wave radiation and jet crackle, 55

the cause of which is unclear [e.g., Nichols et al., 2014]. Mach wave radiation [Tam, 1995] and 56

jet crackle [Williams, et al., 1975] have been historically described in association with heated jet 57

engine exhaust, but recently, have been suggested to occur during volcanic eruptions as well 58

[Fee, et al., 2013]. The field measurements presented by Taddeucci, et al. [2014] show images of 59

both lateral and upward sequence of pressure/weak shock waves generated after an eruption 60

pulse. In this letter, the described sequences of pressure waves and model are ahead of the jet, so 61

are distinct from crackle. 62

While a number of studies have been conducted with detonations and shock tubes as 63

explosive eruption analogs, there is an important difference between these two types of analog 64

models. In detonations, the event consists only of the initial shock wave (Friedlander profile) and 65

perhaps, if the energy and observation time are long enough, a second shock wave (N wave). On 66

the other hand, in shock tube experiments, the initial shock waves will be followed by a 67

relatively long duration supersonic jet. Although the flow system generated in shock tube 68

experiments is more complex when compared to detonations, they tend to more closely resemble 69

the dynamics occurring during an explosive volcanic eruption. For instance, the multiple 70

pressure waves associated with the supersonic jet that are intended to reproduce experimentally 71

in this article are unattainable in a detonation experiment. 72

Shock tube experiments allow for study, not only of the formation and propagation of the 73

initial shock wave, but also the dynamics of jet expansion and the possible interaction of the jet 74

with the leading shock wave. The interaction of shock waves and the supersonic jet may be at 75

least partially responsible for the formation of the multiple shock waves observed in some 76

volcanic eruptions. In fact, the expansion and deceleration of the jet from supersonic to subsonic 77

speed alone has also been shown to be responsible for acoustic emission in actual volcanic 78

eruptions [Genco et al., 2014; Taddeucci, et al., 2014]. In this paper, shock tube experiments 79

designed to provide insight into the generation of multiple shock waves occurring ahead of an 80

expanding supersonic jet are described. To the best of the authors’ knowledge, the sequence of 81

pressure waves occurring between a leading shock wave and a supersonic jet have not been 82

previously studied. 83

2. Experimental Testing

A series of shock tube tests were performed to demonstrate the formation of multiple shock 84

waves in the open space under a single discharge of compressed dry nitrogen. The number of 85

shock waves and their intensity was measured with video imaging and pressure records. The 86

shock tube is made of a driver section with annular cylindrical shape of volume 8.51 10-3 m3, 87

followed by a 1.2 m long driven section, followed by open space [Médici et al., 2014]. The 88

driven and driver section are separated by a piston which is activated by a rapid decompression 89

of the rear chamber. After the piston opens the driver section, nitrogen expands through the 90

driven section, generating a shock wave, and then out through the open end. 91

92

93

94

95

96

Figure 1:

tests: 517

the first f

the first s

M

The disch

shock tub

respectiv

nitrogen

: Image sequ

70 kPa in a a

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shock wave

Multiple tests

harge energy

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uence of the

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ble shock wa

and the jet fr

s at two diffe

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essures, also

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multiple pre

450 kPa in c

aves are labe

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rmodynamic

essure waves

c and d. For t

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he driver sec

c analysis ass

s observed at

the 3450 kPa

ditional shoc

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were perfor

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ction when c

suming an is

t different di

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ately 15.2 x1

rmed to ensu

the driver se

re set at 3450

ompletely fi

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ischarge pre

pressure, on

e present betw

13.0 cm.

ure repeatabi

ection of the

0 and 5170 k

illed with dry

rocess:

essure

nly

ween

ility.

e

kPa

y

= (1) 97

where is the initial pressure (atmospheric pressure), is the loading, or discharge pressure, 98

and is the volume of the driver section. Using the discharge pressures of 3450 and 5170 kPa, 99

the discharge energies become 1.04 105 and 1.74 105 J, respectively. Video images were 100

recorded at 20,000 fps using Photron FASTCAM-APX RS high-speed camera mounted on a 101

shadowgraph array illuminating the open space at the front of the shock tube. Figure 1 shows 102

image sequences of the expanding spherical shock waves for the two discharge pressures. 103

At the lower energy test (1.04 105 J), a series of shock waves was observed in the open 104

space following the leading shock front (Figure 1c and d). After the shock wave travels through 105

the driven section, it starts expanding spherically in the open space. This first shock wave is 106

weak enough that only a fraction of the spherical shape develops. Behind this shock wave, the jet 107

head decelerates as it expands into the open space and a series of shock waves are generated in 108

its wake. Only the first four clearly visible shock waves are marked, although more were 109

generated. 110

At the higher discharge energy test of 1.74 105 J, the shock front is stronger and a shock 111

wave leading the discharge was observed to fully expand into a half spherical shape (Figure 1a 112

and b). As with the lower energy test, a sequence of weaker shock waves following the main 113

shock was observed. 114

115

116

117

118

Figure 2:

for three

panels a,

indicates

The notic

repeatabi

P

imaging

placed on

The wedg

: Pressure sig

tests at each

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test pressure

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g sequence c

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were made in

on of the sho

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rom the end

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(3450 kPa)

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under the sa

n separate tes

ock waves. T

n, centered 5

al shock wav

of the driven

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in the right p

ith the video

ame testing c

sts to avoid i

The pressure

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test set (5170

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the end of th

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high

deo

was

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e

recording on the center line for two reasons. First, the shock wave can be described as a 119

hyperbolic problem, which guarantees that the shock wave will remain unaffected by anything 120

located in front such as the wedge [Hoffmann and Chiang, 2000]. Second, the spherical 121

symmetry of the shock means splitting the wave in half does not lead to losing generality. 122

However, since the jet is not hyperbolic problem, using a wedge in this way is not valid for 123

studying jet dynamics. 124

The signal from the pressure transducer was recorded at a 250,000 samples per second. 125

Figure 2 shows the pressure traces for both discharge experiments. Figure 3 show the shock 126

waves as they passed over pressure sensor. The instantaneous dimensionless shock wave speed, 127

Mach number, was obtained for every test from the video images (Figure 4). While in this 128

manuscript these pressure waves are referred to as shock waves, their Mach numbers are close to 129

1, meaning they are near the transition between supersonic and subsonic wave speed. This can 130

explain the relatively slow rise time of the shock waves observed in Figure 2, compared with 131

strong shock waves. 132

133

134

135

136

137

Figure 3:

pressure

d. The fir

approxim

R

multiple

passes ov

labeled in

first shoc

: Image sequ

sensor at dif

rst eight noti

mately 15.2x

Repeated test

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ver the senso

n the pressur

ck wave sepa

uence of the

fferent disch

iceable wave

13.0 cm.

ts with each

plicated. Fig

or for the low

re plot of Fig

arates from t

multiple pre

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es are labele

of the discha

gure 2b show

wer discharg

gure 2b and

the jet head d

essure waves

re tests: 5170

ed for the 345

arge pressur

ws the pressu

ge energy tes

the correspo

due the incre

s observed a

0 kPa in a an

50 kPa test.

res demonstr

ure fluctuatio

st. The first e

onding imag

easing relati

s they passe

nd b; and 34

Field of view

rate that the

ons as every

eight shock w

ge in Figure 3

ive velocity d

d over the

450 kPa in c

w is

formation of

y shock wave

waves are

3c and d. As

difference, t

and

f

e

s the

the

pressure behind the shock decreases. If the pressure decreases to a value far enough below the 138

atmospheric pressure, it will initiate a second shock wave [Landau, 1945]. This process may 139

repeat, generating additional shock waves. Between the second shock wave, labeled as 2a, and 140

the third shock wave there are two more shock waves, labeled as 2b and 2c that can be traced in 141

the pressure profile and the video imaging. Because they are very weak they are considered as 142

part of the second shock wave, generated by the rarefaction that followed the leading shock. The 143

pressure drop behind the second shock wave may have been large enough to generate a third 144

shock wave in the same way. However, five more shock waves were observed and the pressure 145

drop between shock wave pairs 3-4, 4-5, 5-6, 6-7, and 7-8 are not as prominent as between 1-2 or 146

2-3. This leads to the notion that these later shock waves could have been generated by a 147

different mechanism than the rarefaction of the N-wave, perhaps through the deceleration of the 148

jet. 149

Behind the third shock, pressure decreased as it separated from the jet head, which was 150

still moving at supersonic speed. As the jet head decelerated, the pressure ahead of it decreased 151

and it emitted a new shock wave. This newly-generated shock wave then separated from the jet 152

head, as the jet continued losing speed. This process repeated until the jet lost most of its 153

supersonic speed and was no longer able to sustain the formation of pressure waves. The region 154

in space at which the jet no longer generated pressure waves is out of the field view in the video 155

imaging. 156

The test performed at higher discharge energy, (Figure 3a and b), has similar flow 157

characteristics, but the formation of the multiple shock waves can all be attributed to the 158

dynamics associated with the N-wave pattern. Between every pair of shock waves the pressure 159

decreased below atmospheric pressure (Figure 2a). However weaker pressure waves similar to 160

those observed in the lower discharge energy test may have formed outside the field view. 161

Figure 4: Non dimensional shock wave speed, Mach number, versus position from the end the

shock tube for both experiments at 3450 and 5170 kPa. For the 3450 kPa experiment, the Mach

numbers of the 1st, and 2nd shock waves shock waves were measured. The relative position of the

pressure transducer is indicated for comparison with the Mach number obtained from the

pressure measurements shown in Figure 2.

The pressure difference across a shock wave is correlated with the Mach number through 162

the Rankine-Hugoniot relationship [Shapiro, 1953]: 163

= (2) 164

where is the initial pressure (atmospheric pressure), is the discharge pressure, and is the 165

specific heat ratio (1.4 for air). At the lower discharge energy test, the measured pressure reached 166

by the first shock wave was 20 kPa, which gives 120 kPa absolute for an atmospheric pressure at 167

the time of the experiments of 100 kPa. This gives a pressure ratio of 1.2, which according to the 168

Rankine-Hugoniot relationship corresponds to a Mach number of 1.08. For the second shock 169

wave of the same test the peak pressure was 13 kPa gauge (113 kPa absolute) which corresponds 170

to a Mach number of 1.05. At the higher discharge energy test, the pressure reached by the first 171

shock was 27 kPa gauge (127 kPa absolute) which corresponds to a Mach number of 1.11. These 172

Mach numbers are in good agreement with shock-wave speed measurements obtained through 173

the video imaging despite the uncertainties in speed calculations using the images. The 174

uncertainty in the Mach number calculation due to ±1 pixel error estimation on the shock wave 175

position is 0.06. Figure 4 shows the Mach number of the first two shock waves for the low 176

energy test and the leading shock for the high energy test as a function of the position relative the 177

end of the shock tube. Figure 4 also indicates the relative position of the pressure where the 178

Mach number from the pressure recording and the video imaging are compared. 179

3. Discussion

Different mechanisms can trigger the formation of multiple shock waves for a single discharge 180

event, all of which could resemble potential scenarios of explosive volcanism. The particular 181

case of pairs of shock waves was explained as consequence of the natural expansion of weak 182

shock waves by Landau [1945] and later by Whitham [1956]. In this case, the formation of this 183

second shock wave is due to the shock wave dynamics and does not involve the jet. The 184

existence of the double shock wave can be identified wherever the pressure trace follows the 185

patterns described by Landau [1945] and is normally referred as N-wave. Both shock waves will 186

eventually dissipate into sound waves. For example, the experimental observations presented by 187

Biss and McNesby [2014] clearly show the N-wave pressure profile of the double shock wave, 188

which were also captured on the video imaging. This type of N-wave pressure profile has also 189

been observed in infrasound recordings from different volcanic eruptions [Marchetti et al., 2013; 190

Fee et al., 2014; Matoza et al., 2014]; from detonation tests at the surface [Gitterman and 191

Hofstetter, 2014] and shallow depth [Goto et al., 2001; Bowman et al., 2014]; as well as on both 192

tests reported here. 193

But in addition to the classic N-wave double shock wave, formation of multiple shock 194

waves during volcanic eruptions, could also be a consequence of a series of individual, closely-195

spaced pressure release events (discharges) that resemble a pulsating phenomenon. The multiple 196

shock waves formed under these conditions will be associated with a pulsating jet and not result 197

from the shock wave dynamics described in the previous discussion. Although this could 198

certainly be the case in many eruptions, such us the eruptions of Sakurajima described by Fee et 199

al. [2014], there are other cases that call for a mechanism such as those observed in the shock 200

tube experiments. In some eruptions where multiple shock waves were present, such us the 201

Eyjafjallajökull eruptions in August 2010, the periodicity of the events (eruptions) was slower 202

than the time observed between each shock wave of the couple. Ripepe et al. [2013] using 203

thermal imaging of the Eyjafjallajökull eruption plume, showed that it consisted of a series of 204

puffs at an average period of 20s. The infrasound measurements showed a wider range of 205

pulsating events in the order of 2 to 50 s. Although the spacing between many of the discharges 206

(puffs) was quite long, the close spacing of a subset of the discharges could be explained by 207

shock wave fluid dynamics in the upper conduit and above the conduit. This gives rise to the idea 208

that the multiple shock waves are the outcome of a single discharge rather than the result of 209

multiple discharges. Some of the video footage taken during the eruption shows the 210

coupled/multiple shock waves followed by a single jet/puff indicating that they are the result of a 211

single event. 212

While the scenarios described previously could be associated either with the shock wave 213

or the jet dynamics, at least some of the later multiple shock waves formed during the shock tube 214

tests are due to the coupled shock wave-jet interaction dynamics. Regarding the Eyjafjallajökull 215

eruptions in August 2010, a question still remains: are the observed of shock wave couples a 216

consequence of a pulsating jet or due to the coupled shock wave-jet interaction dynamics 217

observed in the experiments presented here. Clearly more work must be done before this can be 218

decisively answered. For example, an important aspect of multiple shock formation that has not 219

yet been explored is the influence of vent shape. Preexisting vent or crater shape is known to 220

cause changes in the jet characteristic and the associated pressure trace when compared with a 221

no-crater eruption [Taddeucci, et al., 2013]. 222

The vent for this particular eruption is particularly interesting. According to Magnússon 223

et al. [2012], the subglacial vent of Eyjafjallajökull melted cauldrons into the overlying ice that 224

acted as an elongation of the conduit. The shock wave propagation inside the constrained space 225

of the cauldron could be comparable to the shock wave propagation inside the driven section of 226

the experimental shock tube. For a relatively low energy discharge, the shock front could 227

dissipate some of its energy inside the cauldron while the jet was still advancing at supersonic 228

speed giving rise, as the in the experiment, to a shock wave couple in the open atmosphere. On 229

the other hand, for discharges with relatively higher energy, single shock wave events would 230

have been produced; single shock wave events were also observed during this eruption. The most 231

probable explanation of the observed events may be a combination of the factors described 232

herein. 233

In addition to the role of vent geometry, the influence of particles, and other complexities 234

related to the formation of shock waves, there are challenges associated with comparing 235

laboratory measurements with field data. Shock waves can be captured in volcanic eruption 236

pulses using normal and high-speed video imaging, but only under certain environmental 237

conditions [e.g., Nairn, 1976; Yokoo and Ishihara, 2007; Taddeucci et al., 2014]. The relatively 238

low energy, low Mach number, shock waves produced in laboratory experiments decay to sound 239

waves at a short distance from the source. Similarly, by the time comparable pressure waves 240

from a volcanic eruption are recorded at a microphone station in the field, they likewise will 241

have decayed into sound waves. Although estimates of the source pressure can still be made 242

[Medici et al., 2014], this aspect remains a challenge. 243

Clearly the multiple shock waves presented in this experimental work will not be 244

observed in every type of eruption. Based on our experimental evidence, they will only be 245

observable in low intensity eruptions. More intense eruptions will still show the traditional N-246

wave pattern discussed by Marchetti et al. [2013]. 247

4. Conclusion

A series of experimental tests were performed with an open-ended shock tube to model the 248

generation of multiple, coupled shock waves for single discharge volcanic eruptions by at least 249

two distinct mechanisms. The experimental observations at lower discharge energies, show that 250

the first weak shock wave was followed by a series of N-wave pattern shock waves produced by 251

shock wave dynamics. This first set of shock waves was then followed by additional shock 252

waves generated by the decelerating jet. The formation of multiple shock waves was also 253

observed in higher discharge energy tests, although they are more difficult to distinguish in the 254

available field of view. This implies that while moderate and large eruptions will exhibit the 255

traditional N-wave pressure pattern, multiple shock waves might be more common on low 256

intensity eruptions. A combination of different the factors such as vent shape and the amount of 257

energy being released in every puff may play a role in the formation of multiple shock waves. 258

5. Acknowledgments

Support was provided by the Michigan Tech’s Earth, Planetary, and Space Sciences Institute 259

(EPSSI). Support for the Shocktube Facility was provided by the National Science Foundation 260

(EAR-1250153). The authors thank Jeffrey Allen for the technical discussions regarding the 261

experiments; REL, Inc. of Calumet, Michigan, USA for design, fabrication and technical support 262

of the shock tube; and the thorough reviews of David Fee and an anonymous referee which 263

resulted in an improved manuscript. Data are archived at Michigan Tech’s HazHub archive: 264

http://www.mtu.edu/geo/research/hazard-data. 265

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1 2 3 12 3 4

150 µs 350 µs

150 µs 350 µs

a) b)

c) d)

1

2a

2c

3

4 5

6

78

2b

50 µs 350 µs

1 2 12 2c 3 4 5 6 7

8

50 µs 250 µs

PressureSensor

a) b)

c) d)

0 5 10 151

1.05

1.1

1.15

1.2

1.25

x [cm]

M

1st

2nd

5171 [kPa]

Pressure Sensor