ARL-TR-2203 May 2000

Army Research Laboratory

Adelphi, MD 20783-1197

Modification of the Acoustic

Spectrum of Detonation Tube

Shock Waves by Timed

Multiple-Pulse Addition

H. Edwin Boesch, Jr., Christian G. Reiff

Sensors and Electron Devices Directorate, ARL

Bruce T. Benwell

Directed Energy Technologies

Approved for public release; distribution unlimited.
Detonation tubes are simple devices capable of producing

substantial acoustic power that may be useful for the simulation of

high-level acoustic environments. We report results of an

investigation into the modification of the acoustic spectrum

produced by detonation tubes by timed addition of the shock-wave

outputs of six detonation tubes fired in sequence. We first examined

the output of a single detonation tube as a function of range and

found that it conformed to existing models for spherical blast waves

when appropriate initial conditions were derived. We found timing

schemes for the firing of the multiple tubes that (1) produce a

substantial shift of the acoustic energy to lower frequencies by

maximizing the duration of the positive pressure pulse, or

(2) maximize the acoustic energy output in a narrow frequency range

by matching the pulse-to-pulse delay to the total duration (positive

Abstract

and negative pressure phases) of a single detonation wave.

ii
iii

Contents

1. Introduction ....................................................................................................................................... 1

2. Addition of Ideal Blast Waves ........................................................................................................ 4

3. Experimental Setup .......................................................................................................................... 8

4. Single Tube Experiments ................................................................................................................. 9

4.1 Results.............................................................................................................................................. 9

4.2 Discussion ...................................................................................................................................... 12

5. Multiple Tube Experiments .......................................................................................................... 15

5.1 Simultaneous Firing of Six Tubes .................................................................................................. 15

5.2 Staggered Firing of Six Tubes ........................................................................................................ 16

6. Conclusions ...................................................................................................................................... 21

References ............................................................................................................................................... 22

Distribution ............................................................................................................................................ 23

Report Documentation Page ................................................................................................................ 25

Figures

1. Schematic of secondary shock-wave formation by jet from DT............................................... 1

2. Idealized blast wave for t+ = 1 ms, τ = 3.58 ms ........................................................................... 3

3. Linear superposition of five blast waves from figure 2 with delays of ∆t

i

 = 1 ms ................ 5

4. Linear superposition of five blast waves from figure 2 with delays of ∆t

i

 = 0.5 ms ............. 6

5. Linear superposition of five blast waves from figure 2 with delays ∆ti = 1.0, 0.65, 0.55,

and 0.47 ms ...................................................................................................................................... 7

6. Schematic of experimental arrangement ..................................................................................... 8

7. Time histories of SPL measured at various ranges r from a single DT ................................. 10

8. Pulse from a single DT measured at r = 15.24 m .......................................................................11

9. Range dependence of single DT overpressure characteristics ............................................... 13

10. Range dependence of single DT overpressure pulse duration .............................................. 13

11. Pulse measured at 15.24 m from six DTs fired with ∆ti = 0 ms .............................................. 15

12. Pulse measured at 15.24 m from six DTs fired with ∆ti = 8 ms .............................................. 17

13. Pulse measured at 15.24 m from six DTs fired with ∆ti

 = 2 ms .............................................. 18

14. Pulse measured at 15.24 m from six DTs fired with measured ∆ti

 = 2.2, 1.5, 1.6, 1.2,

and 0.5 ms ...................................................................................................................................... 19

Table

1. Dominant and mean power frequencies and percent power above 1 kHz from single

and multiple detonation tube firings measured at 15.24-m range......................................... 20

1

1. Introduction

Detonation tubes (DTs) are conceptually simple devices that can repetitively and reliably produce acoustic energy at high intensities. These

characteristics make detonation tubes attractive for applications that

require substantial acoustic power from a compact source. However, the

acoustic output of a single DT is not suitable for many applications, since

the output is by nature impulsive or broadband rather than continuous.

Also, the similarity characteristics of blast waves [1] ensure that the

frequency content of the acoustic output of a single detonation tube is

largely constrained by the energy of the shock-wave source and our range

from it. In this report, we present the results of our initial experiments

directed at exploring the means to generate impulsive acoustic energy

with a modified frequency spectrum by timed and repetitive firing of an

array of detonation tubes.

DTs produce pressure transients that have a shock profile that is typical of

those produced by a wide range of shock-wave sources, including free-air

explosions and gunfire at all scales. Further, such a shock profile is the

primary component of pressure transients produced by transonic and

supersonic flow associated with aeronautical structures, turbines, and

rocket engines.

Typically, a DT consists of a cylinder closed at one end and open to

atmosphere at the other, with a means for rapidly injecting and igniting

an explosive mixture of a gaseous fuel and oxidizer. If the mixture is

ignited at the closed end of the tube, a high-pressure detonation wave

quickly forms in the mixture and propagates through the tube at a high

Mach number and out the open end (muzzle) (fig. 1, top). In free air, the

(initially) unipolar positive pressure pulse quickly converts to a bipolar

Figure 1. Schematic of

secondary shock-wave

formation by jet from

DT.

Combustion

products

Detonation

shock

Supersonic Jet shock

jet

Detonation Fuel/oxidizer

wave
2

shock-wave signature characterized by a rapid (within a few microseconds) rise to a positive peak, a pseudo-exponential fall to and through

ambient atmospheric pressure, and a lower amplitude but longer

duration negative pressure peak with a return to ambient pressure (see

fig. 1). As noted, this pressure-time signature is general and applies to a

wide range of shock-wave sources. For example, the signatures of a smallcaliber gunshot and a large high-explosive detonation at comparable

ranges differ primarily in the peak pressures reached in the (positive)

overpressure and (negative) underpressure phases and in the duration of

these pressure pulses. A useful approximation to this pressure-time

signature is given by p(t) = ∆ p (1 

–t/t

+) (1 

–t/τ) (1 – (t/τ)

2), 0 < 

t < τ , (1)

where ∆p is the magnitude of the peak positive pressure, t

+ is the duration of the positive pressure as measured at the ambient-pressure

baseline, and 

τ is the total duration of the pressure transient, including

both positive and negative phases [2]. For zero air particle velocity at the

end of the pulse (linear, or acoustic, regime), 

τ is constrained to about

3.58t

+. The waveform for this condition is shown in figure 2(a). The

acoustic spectrum for an impulse with t

+ = 1 ms from Fourier analysis is

shown in figure 2(b) and 2(c). The dominant acoustic frequency fp for this

pulse is ~238 Hz. Figure 2(d) shows the fractional integrated acoustic

power (normalized watts per square meter) for this 1-ms pulse as a

function of frequency. The median power frequency (the frequency at

which the integral from zero frequency of the fractional acoustic power

spectrum equals 0.5, also designated as the 50-percent cumulative power

frequency—50% CPF) is ~297 Hz. A small DT at ranges of a few meters

will typically produce an impulse with t

+ near 1 ms; therefore, much of

the radiated acoustic energy from such a source is in the range of a few

hundred hertz. The initial fast pressure rise contributes substantial energy

in the high audible frequency range above 1 kHz; as shown in figure 2(c),

these higher frequency components roll off at –20 dB per decade in

frequency (1/f spectrum).

For our applications, our goal was to shift the acoustic energy available

from a DT source to lower audible frequencies (below 100 Hz) and to

reduce the energy content in the range above 1 kHz, where the human ear

is particularly susceptible to damage. Since the frequency content of the

waveform produced by a single detonation tube is largely determined by

its size and therefore not easily altered for devices constrained to a certain

size range, we examined the possibility of synthesizing a waveform with

the desired frequency content by adding the outputs of multiple DTs. In

essence, we would fire several DTs in a predetermined rapid sequence

into a common air volume to cause their waveforms to interact and

generate a new waveform with altered characteristics.

If the tubes are close together (within a few tube diameters) and are fired

so that their detonation waveforms overlap in time, the shock fronts from

the tubes can interact at pressures in the medium shock regime (of the

order of 100 kPa). The second (and all later) shock fronts do not propagate

in the undisturbed ambient environment seen by the first shock, but in an
3

Figure 2. Idealized blast wave for t+ = 1 ms, τ = 3.58 ms: (a) time history of SPL, p(t); (b) low-frequency

acoustic spectrum; (c) logarithmic acoustic spectrum; and (d) integrated fractional acoustic power as a

function of frequency.

(a) (b)

1.2

1

0.8

0.6

0.4

0.2

0

–0.2

–0.4

t+ = 1 ms

τ = 3.58 ms

(zero particle velocity at end of pulse)

Time (ms)

–0.5 0 0.5 1 1.5 2 2.5 3 3.5

Relative pressure (arbitrary unit)

60

50

40

30

20

10

0

Relative pressure (arbitrary units)

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

Peak at f = 238 Hz

130

120

110

100

90

–20 dB/decade (1/f SPL)

Frequency (Hz)

10 100 1000 10000

dB (arbitrary reference)

1

0.8

0.6

0.4

0.2

0

50% at 297 Hz

Frequency (Hz)

10 100 1000 10000

Cumulative fractional power

(c) (d)

altered and highly dynamic environment. The shock waves in a series

will then propagate at different velocities and their amplitudes are expected to add nonlinearly; so simple superposition of the detonation tube

waveforms is not expected to hold. This shock-on-shock problem is not

well understood; further development of the theory of such interactions is

the subject of a companion paper [3]. Here we report the results of a

limited experimental and theoretical study of the interaction of shock

waves from multiple DTs.
4

2. Addition of Ideal Blast Waves

Even in the acoustic or linear regime (∆p << 100 kPa), we cannot accurately calculate the waveform generated by the addition of signals from

multiple impulsive acoustic sources by the simple addition of their

amplitudes (linear superposition). If the sources produce substantial

energy at frequencies f

p with corresponding wavelengths that are much

greater than the distance between the sources, and the time delays between the firing of these sources are much less than 1/f

p, the sources will

interact with each other and emit acoustic energy into a radiation resistance (acoustic impedance of the atmosphere) that is effectively altered

from its free-air value. The exact calculation of the resulting addition of

waveforms from multiple interacting sources is beyond the scope of this

report.

However, as a guide, we first examine the linear superposition of a series

of ideal blast waves (eq (1)), using (t+ = 1 ms, τ = 3.58 ms) waveforms as

an example (see fig. 1). An initial strategy for increasing the lowfrequency content of the waveform produced by adding blast waves

would be simply to stretch the effective duration of the summed pulses by

delaying successive individual DT pulses just enough to prevent the

instantaneous sound pressure level (SPL) p(t) from going negative after

the positive pressure phase from the previous pulse. We assume that the n

waves are identical in time history and amplitude and are successively

shifted in time with respect to each other by delay times ∆t

i

, where i = 1,

..., n – 1. Figure 3(a) shows the result for addition of five blast waves with

∆t

i

 = 1 ms. Note in the figure that p(t) repeatedly and increasingly goes

negative following the second pulse as the lower amplitude but longer

duration negative phases add in. Since the integral over τ of each of the

individual pulses is zero, the integral of a linear summation of these

pulses must also equate to zero. Consequently, a sufficiently long sequence of such pulses must reach a steady state, in which the positive and

negative pressure phases average to zero, and pulse stretching must

inevitably yield diminishing returns for large n. That is evident by pulse 4

for τ = 1 ms. Figures 3(b) to 3(d) show the low-frequency linear and fullrange logarithmic Fourier spectra and integrated power versus frequency

calculated for this case. Compared to the benchmark single pulse (fig. 2),

the longer duration of the five-pulse transient contributes some energy at

lower frequencies (peak at ~100 Hz in fig. 3(b); compare to fig. 2(b));

however, the 50-percent CPF has dropped only slightly, from 297 Hz to

281 Hz (compare fig. 2(d) and 3(d)). Almost 25 percent of the acoustic

energy in this five-pulse transient is now concentrated at the pulse repetition rate (1 kHz), and almost 40 percent of the total energy is at 1 kHz or

higher.

Figure 4 shows results for five pulses with ∆ti

 = 0.5 ms. The pulses are

close enough that p(t) stays positive for over 2.5 ms. Again, the peak

frequency (~159 Hz (fig. 4(b)) is associated with the total pulse duration.

More importantly, the 50-percent CPF is now 171 Hz, and relatively little

energy (<10%) is contributed by frequencies above 1 kHz.
5

Figure 5 shows results for five pulses with ∆t

i

 chosen for each pulse to

maximize the time that p(t) remains at or above the baseline (∆t

i

 = 1.0,

0.65, 0.55, and 0.47 ms). The spectrum shows a strong dominant frequency near 134 Hz, and the 50-percent CPF is now 164 Hz. Less than 15

percent of the energy is above 1 kHz.

Overall, our modeling suggests that multiple-pulse addition could shift

the acoustic energy of single DT pulses to substantially lower frequencies,

and that a promising strategy for the addition is to prolong the initial

positive p(t) transient as much as possible without allowing zero

crossings.

900

700

500

300

100

–100

–300

–500

02468

Relative pressure (artibrary units)

Time (ms)

80

70

60

50

40

30

20

10

0

Peak at f = 98 Hz

0 100 200 300 400 500 600 700 800 900 1000

Relative pressure

Frequency (Hz)

140

130

120

110

100

90

80

10 100 1000 10000

dB (arbitrary reference)

Frequency (Hz)

1

0.8

0.6

0.4

0.2

0

Frequency (Hz)

10 100 1000 10000

Cumulative fractional power

50% at 281 Hz

Figure 3. Linear superposition of five blast waves from figure 2 with delays of ∆ti = 1 ms: (a) time

history of SPL, p(t); (b) low-frequency acoustic spectrum; (c) logarithmic acoustic spectrum; and

(d) integrated fractional acoustic power as a function of frequency.

(a) (b)

(c) (d)
6

Figure 4. Linear superposition of five blast waves from figure 2 with delays of ∆ti = 0.5 ms: (a) time

history of SPL, p(t); (b) low-frequency acoustic spectrum; (c) logarithmic acoustic spectrum; and

(d) integrated fractional acoustic power as a function of frequency.

(a) (b)

(c) (d)

1.25

1.00

0.75

0.50

0.25

0

–0.25

–0.50

–0.75

–1.00

Time (ms)

0123 456

Relative pressure

140

120

100

80

60

40

20

0

Peak at f = 159 Hz

Frequency (Hz)

0 100 200 300 400 500 600 700 800 900 1000

Relative pressure

140

130

120

110

100

90

80

Frequency (Hz)

10 100 1000 10000

dB(arbitrary ref)

1

0.8

0.6

0.4

0.2

0

50% at 171 Hz

Frequency (Hz)

10 100 1000 10000

Cumulative fractional power
7

(a) (b)

Figure 5. Linear superposition of five blast waves from figure 2 with delays ∆ti = 1.0, 0.65, 0.55, and

0.47 ms: (a) time history of SPL, p(t); (b) low-frequency acoustic spectrum; (c) logarithmic acoustic

spectrum; and (d) integrated fractional acoustic power as a function of frequency.

(c) (d)

1.00

0.75

0.50

0.25

0

–0.25

–0.50

–0.75

–1.00

Time (ms)

012 34 56 7

Relative pressure

120

100

80

60

40

20

0

Peak at f = 134 Hz

Frequency (Hz)

0 100 200 300 400 500 600 700 800 900 1000

Relative pressure

140

130

120

110

100

90

80

dB (arbitrary reference)

Frequency (Hz)

10 100 1000 10000 10 100 1000 10000

Frequency (Hz)

50% at 164 Hz

1

0.8

0.6

0.4

0.2

0

Cumulative fractional power
8

Figure 6. Schematic

of experimental

arrangement.

Detonation tube array

Blast probe pressure

sensor

Range

3. Experimental Setup

A simplified schematic of the experimental configuration is shown in

figure 6. For most of the experiments described here, the DTs were cylinders with 6-in. interior diameters and length-to-diameter ratios greater

than 4:1. The fuel and oxidizer were near-stoichiometric mixtures of

ethylene and oxygen. These gases were rapidly injected into the tubes at

the closed (breech) end and then ignited within a fraction of a second by

an electric spark. The timing of the ignition for each tube could be

individually controlled within 0.1 ms; variations in the ignition and

detonation-wave formation processes resulted in tube-to-tube timing

uncertainties of ±0.3 ms. Up to six DTs were arranged in an array, with

their axes parallel and their open (muzzle) ends coplanar. The geometric

center of the array at the muzzles was defined as zero range.

The experiments took place outdoors over lightly vegetated ground in a

semi-arid environment. The axis of the DT array was approximately

horizontal, with a height above the gently sloping ground of 1.5 m at the

array and 2 m at a range of 15 m. We recorded the time history of the

pressure waveforms from the array with blast probe pressure sensors and

precision microphones located principally along the symmetry axis of the

array. The blast probes were 12-in.-long, 1-in.-diam torpedo probes with

side-mounted flush pressure transducers. PCB model 137A22 probes

were used at short ranges (<1 m), and model 137A23 probes were used at

longer ranges. Microphones were used at ranges of 5 m or greater: these

were Bruel & Kjaer models 4136 or 4135 1⁄4-in. condenser microphones,

with model 2669 preamplifiers and model 5935 power supplies. In addition, we recorded the pressure transients in the detonation tubes with

PCB model 113A24 pressure transducers mounted near the muzzle; these

signals served primarily as fiducial markers for each tube firing. We

recorded the signals with a Tektronix TDS684A four-channel digital

oscilloscope, an IOTech Wavebook/512 waveform digitizer operated by a

laptop computer, and a Sony PC204A digital audio tape (DAT) recorder.
9

4. Single Tube Experiments

4.1 Results

Figures 7 and 8(a) show p(t) waveforms recorded from the firing of a

single DT. Waveforms are shown in each figure from two separate DT

firings; the differences reflect minor pulse-to-pulse variations. For these

figures, the blast probes were located on the tube axis at ranges of 0, 0.17,

0.29, 0.53, 0.95, and 5.0 m (fig. 7) and 15.24 m (fig. 8(a)). (All ranges are the

distance from the muzzle of the DT to the pressure sensor on the side of

the blast probe.) The waveform recorded at the muzzle of the DT (fig.

7(a)) shows the expected detonation transient interior to the tube: a peak

positive pressure of over 2 MPa (about 300 psig) with a half-width of

about 0.1 ms, a lower positive-pressure phase associated with the static

combustion gas pressure behind the detonation wave, and a slow return

to ambient pressure (blow-down) as these gases exit the tube. There is no

negative-pressure phase in the tube on this time scale, since the tube is a

net source of gas.

Figures 7(b) and 7(c) show p(t) at 0.17 and 0.29 m from the tube (at about

one and two tube diameters from the muzzle). In this range, the detonation wave is expanding roughly spherically into free atmosphere; the

peak overpressure is dropping rapidly with range, a negative-pressure

phase is developing, and the resulting duration t+ of the positive pulse (as

now measured to the negative-pressure crossover at the baseline) roughly

doubles, from 0.22 to 0.48 ms. A delayed lower amplitude (near 100 kPa)

positive-pressure peak appears just after the start of the negative phase

(at about 0.7 ms in the figure). At 0.53 m (fig. 7(d)), the overpressure has

dropped to about 110 kPa (near 1 atm), and a compensating negative

phase is fully developed. The additional positive-pressure peak at about

1 ms with an amplitude near 70 kPa is now a major feature. This pattern

is repeated at 0.95 m, 5 m (fig. 7(e) and 7(f)), and 15.24 m (50 ft) (fig. 8(a)).

At 15.24 m, the overpressure has dropped to about 1 kPa (about 0.1 atm),

and t+ (not counting the secondary peak) has increased to about 1.2 ms.
10

(a) (b)

(c) (d)

(e) (f)

Figure 7. Time histories of SPL measured at various ranges r from a single DT: (a) r = 0.0 m,

(b) r = 0.17 m, (c) r = 0.29 m, (d) r = 0.53 m, (e) r = 0.95 m, and (f) r = 5.0 m.

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

2500

2000

1500

1000

500

0

–500

Instantaneous SPL (kPa)

85

86

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

1100

900

700

500

300

100

–100

Instantaneous SPL (kPa)

82

83

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

Instantaneous SPL (kPa)

87

88

400

300

200

100

0

–100

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

Instantaneous SPL (kPa)

89

90

110

90

70

50

30

10

–10

–30

–50

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

Instantaneous SPL (kPa)

70

60

50

40

30

20

10

0

–10

91

92

0 0.5 1.0 1.5 2.0 2.5 3.0

Time (ms)

Instantaneous SPL (kPa)

5

4

3

2

1

0

–1

–2
11

(a) (b)

(c) (d)

Figure 8. Pulse from a single DT measured at r = 15.24 m: (a) time history of SPL, p(t); (b) lowfrequency acoustic spectrum; (c) logarithmic acoustic spectrum; and (d) integrated fractional acoustic

power as a function of frequency.

0 5 10 15

Time (ms)

1500

1000

500

0

–500

–1000

Instantaneous SPL (kPa)

100

80

60

40

20

0

Relative SPL (kPa)

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

Peak at 92 Hz

140

130

120

110

100

90

80

Frequency (Hz)

10 100 1000 10000

dB (arbitrary reference)

1

0.8

0.6

0.4

0.2

0

50% at 122 Hz

Frequency (Hz)

10 100 1000 10000

Cumulative fractional power
12

4.2 Discussion

In figure 9, we plot the measured initial positive peak overpressure ∆p as

a function of range from the data in figures 7 and 8(a). We also plot the

magnitude of the second positive peak p2 estimated from the figures. In

figure 10, we plot the duration t+ of the initial positive peak as estimated

from the figures, excluding the contribution from p2. Assuming a spherically symmetric blast in the intermediate to weak shock regime, Raspet

[4] developed theoretical expressions for p(r) and the duration ts

 of the

shock front near maximum overpressure based on the thickness of the

compressed air shell at the shock front and its radial expansion rate.

These expressions are

p(r) = ∆p0 (R0/r) (1 + K (R0/ts0) ln(r/R0))–1/2 (2)

and

ts

 = ts0 (1 + K (R0/ts0) ln(r/R0))1/2 , (3)

where R0, t

s0, and ∆p0 are the equivalent spherical blast initial radius,

peak overpressure duration, and pressure, and K = (γ + 1)/(4γ c0) is a

constant (γ = 1.4 for air). Figure 9 is a plot of p(r) (dashed smooth curve);

the values of R0 and t

s0 were adjusted to 0.15 m and 0.01 ms, respectively,

to produce a reasonable fit to the experimental results. The value of R0 is

physically reasonable, since 0.15 m is the diameter of the DT muzzle. We

did not measure t

s

 directly; however, we expect t+ to be proportional to ts.

The quantity K1ts

 is plotted in figure 10 (dashed smooth curve); K1 was

adjusted to the value 13.2 to produce a close fit to t

+. The fits of p(r) and

K1ts to the data are quite reasonable in this intermediate to weak shock

regime.

The second positive-pressure peak, detectable at all ranges in figures 7

and 8(a) and plotted as p2 in figure 9, is not a ground reflection. In our

measurement geometry, ground reflections are delayed with respect to

the direct wave by at least 6 ms at ranges under 1 m. In contrast to p, p2

stays constant in magnitude with range from 0.17 to 0.53 m, where it

becomes comparable to p and then falls off in roughly the same manner as

p. An additional clue to the origin of p2 is that this signal is either absent

or strongly attenuated in measurements made at angles of more than 45°

off the DT axis. We interpret p2 as resulting from the blow-down of the

DT following the detonation. Following the hypersonic passage of the

detonation wave through and out of the DT, the high-pressure gas in the

wake of the detonation wave begins to flow out of the tube. Upon entering the ambient atmosphere, the exhaust gas flow goes supersonic and

forms a strong jet that is directed forward. The leading edge of this jet

interacts with the ambient atmosphere to form a shock front (see fig. 1(b)).

Since the jet is reasonably collimated, the jet shock front maintains intensity out to at least 0.5 m, in contrast to the leading detonation wave,

which is expanding spherically.

At ranges of 0.53 and 0.95 m (fig. 7(d) and 7(e)), the contribution of p2 to

the total energy in the DT acoustic wave is clear; at greater ranges, the

contribution is less obvious and must be distinguished from the primary
13

ground reflection. For our geometry, the ground reflection is expected to

be delayed with respect to the direct wave by 2 to 4 ms at 5 m and 1 to

2 ms at 15.24 m. (The uncertainty results from variably sloped terrain on

the measurement range.) The figures show the reflection arriving at 2.8

and 1.5 ms at 5 and 15.24 m, respectively. Particularly at these longer

ranges, the reflection contributes substantially to the total energy in the

acoustic signal.

Figures 8(c) and 8(d) show the logarithmic Fourier spectra and integrated

acoustic power for the single DT pulses at 15.24 m. While this real pulse

shows more frequency structure than the idealized pulse (compare to fig.

2), the overall characteristics are similar.

Figure 9. Range

dependence of single

DT overpressure

characteristics. Solid

line (squares) is

measured initial

positive overpressure

∆p; dashed line is fit

using equation (2).

Dashed line (circles)

is second overpressure

p2 attributed to jet.

10,000

1000

100

10

1

0.1

∆p

After Raspet

p2

Range r (m)

0.1 1 10 100

p(r) (Pa)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Mean power frequency (Hz)

10000

1000

100

10

Range r (m)

0.1 1 10 100

t+ (ms)

t+

50% CPF (Hz)

Scaled K1t

s

 (after

Raspet)

Figure 10. Range

dependence of single

DT overpressure

pulse duration. Solid

line (squares) is

measured positive

pulse duration t+, and

dashed line is scaled

fit K1ts from equation

(3). Solid curve

(circles) is measured

mean power frequency (50% CPF).
14

In addition to the measured pulse duration t

+, figure 10 also shows the

50% CPF values calculated from the measurements at the various ranges.

The increasing pulse duration with range is the major contributor to the

shift of the acoustic power spectrum to lower frequencies (atmospheric

absorption plays only a minor role at these ranges at frequencies below

20 kHz). At 15.25 m (50 ft), the 50-percent CPF is down to 122 Hz from

over 2 kHz at 0.165 m.
15

5. Multiple Tube Experiments

5.1 Simultaneous Firing of Six Tubes

Figure 11(a) shows the p(t) waveforms recorded at 15.24 m (50 ft) produced by the firing of a single DT (lower dotted curve) and of six identical DTs (solid curve) simultaneously (within ±0.3 ms). Although the

waveform recorded for the six-tube pulse at short range shows some sign

of the individual DT pulses resulting from the timing variations, these

individual pulses have coalesced into a single clean pulse plus the jet

shock at the 15.25-m range. In addition to the normal pulse lengthening

with range, this coalescence results from the tendency of the following

shock waves to catch up with the leading shock waves when the shocks

are close enough that the following shock is traveling in a high-pressure

low-density region immediately behind the leading shock [3]. The initial

overpressure ∆p for the single and multiple pulses is about 1.4 and

2.6 kPa, respectively. Because of the simultaneity or coalescence of the DT

pulses, the output of the six tubes is maximally coherent and, for a linear

(a) (b)

(c) (d)

Figure 11. Pulse measured at 15.24 m from six DTs fired with ∆ti = 0 ms: (a) Time history of SPL, p(t);

for six DTs (solid curve), single DT (dashes), and 61/2p(t)1 tube (dotted curve); (b) low-frequency

acoustic spectrum (six tubes); (c) logarithmic acoustic spectrum; and (d) integrated fractional acoustic

power as a function of frequency.

0 5 10 15

Time (ms)

4

2

0

–2

Instantaneous SPL (kPa)

6 tubes

1 tube

Square root

Peak at 92 Hz

100

80

60

40

20

0

Frequency (Hz)

0 100 200 300 400 500 600 700 800 900 1000

Relative SPL (Pa)

140

130

120

110

100

90

80

Frequency (Hz)

10 100 1000 1000

dB (arbitrary reference)

1

0.8

0.6

0.4

0.2

0

50% CPF = 88 Hz

Frequency (Hz)

10 100 1000 10000

Cumulative fractional power
16

interaction, we expect the resulting six-tube waveform to be that of the

single tube at six times the power level (not six times the amplitude, or 36

times the power), as would be predicted by simple superposition of

waveforms, or p(t)6 tubes ~ 61/2 p(t)1 tube. As shown in the figure, the

observed waveform differs from the 61/2 p(t)

1 tube curve with lower ∆p

amplitude, a longer pulse length, and a larger second peak (from the jet).

These differences, or nonlinear effects, are attributable to increased pulse

stretching because of the greater energy in the high-pressure shock interaction region close to the tubes and to an enhanced jet from the combined

outputs of the tubes. The stretched pulse length results in a 50-percent

CPF of about 90 Hz for the six-tube pulse (fig. 11(d)), which is substantially lower than the single-tube value of 122 Hz (fig. 8(d)).

5.2 Staggered Firing of Six Tubes

We then measured the acoustic output of the six tubes in the DT array

with delays between tube firings of from 20 to 0.5 ms. Figures 12 and 13

show results for programmed uniform delays between tube firings of 8

and 2 ms, respectively.

With delays of 8 ms, the individual pulses are sufficiently separated in

time that the sources do not interact substantially, and we expect that

simple linear superposition of the pulses should apply. The dotted curve

in figure 12(a) is the calculated p(t) waveform for the linear addition of six

identical pulses. The measured p(t) shows somewhat reduced peak

amplitudes, and the later pulses appear to be smeared out in time. We

attribute this effect to transmission of the later pulses through a disturbed

medium (turbulent hot gases) resulting from the exhaust jets from the

DTs. Note that with 8-ms delays, the positive pulse from successive

detonations arrives just as the negative phase from the preceding pulse is

ending. Thus, 8 ms is the minimum delay for which the pulses add

without overlap of negative and positive phases (without destructive

interference). The resulting p(t) in the figure may be described as a quasiperiodic waveform with superimposed high-frequency noise. At longer

delays, the pulses become almost completely separated in time and are

effectively independent.

Figure 12(b) shows the low-frequency fast Fourier transform (FFT) spectrum for this 8-ms delay. Note the large peak near 130 ± 8 Hz. (The frequency resolution for the FFT was 7.63 Hz.) This peak is attributable to

the 8-ms repetition rate of the detonations and is also the apparent fundamental frequency of the synthesized quasi-periodic pressure wave. Below

this peak are a series of peaks at ~22-Hz intervals; these frequencies may

be associated with the roughly 45-ms duration of the entire pulse train.

Above 1 kHz, the frequency spectrum shows the common overall –20 dB

per decade roll-off (fig. 12(c)). The large step in the cumulative power

curve for this pulse train (fig. 12(d)) results from the peak near 130 Hz;

this peak accounts for almost two-thirds of the measured total acoustic

power at 15.25 m from the source, and the 50-percent CPF is 126 Hz.

Therefore, the six-pulse sequence with 8-ms delays actually has a higher

mean frequency content than six tubes fired simultaneously (fig. 11(d)) or
17

(a) (b)

(c) (d)

Figure 12. Pulse measured at 15.24 m from six DTs fired with ∆ti = 8 ms: (a) measured time history of

SPL, p(t) (solid curve), and calculated p(t) based on single DT pulse (dotted curve); (b) measured lowfrequency acoustic spectrum; (c) logarithmic acoustic spectrum; and (d) integrated fractional acoustic

power as a function of frequency.

even a single tube (fig. 8(d)). However, the acoustic energy is strongly

concentrated near a single frequency. Matching the pulse delays ∆t to the

duration τ of a single pulse may be an effective strategy for maximizing

the acoustic power of a DT or other impulsive source at a single frequency. Since the pulse duration is a function of range for a shock-wave

source, this matching should be done for a particular range and its associated τ.

Figure 13(a) shows recorded p(t) (solid curve) for nominal delays of 2 ms

between the six DT pulses. Again, the dotted curve in the figure is the

calculated linear addition of six pulses. The actual measured delays

varied by ~ ±0.2 ms; the delays in the simulation have been altered to

match the measured values. In this sequence, successive pulses overlap in

time. The first three pulses add in such a way that p(t) is positive for the

initial 4 ms. The later pulses swing both positive and negative. The

overall behavior is reasonably well predicted for the earlier pulses by the

linear addition calculation. One exception is pulse 3, which falls well

below the calculated response. Other data suggest that this tube may

have misfired. At later times, the predicted and observed responses match

less well. As with the 8-ms pulse delays, the exhaust from the tubes may

perturb these later pulses. The low-frequency spectrum produced by the

2-ms pulse burst is shown in figure 13(b). The roughly 10-ms duration of

2

1.5

1

0.5

0

–0.5

–1

Instantaneous SPL (kPa)

0 5 10 15 20 25 30 35 40 45 50

Time (ms)

1

2 3 4 5 6

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

160

140

120

100

80

60

40

20

0

Relative SPL (Pa)

Peak at 130 Hz

140

130

120

110

100

90

80

Frequency (Hz)

10 100 1000 10000

dB (arbitrary reference)

1

0.8

0.6

0.4

0.2

0

50% at 126 Hz

Frequency (Hz)

10 100 1000 10000

Cululative fractional power
18

(a) (b)

(c) (d)

Figure 13. Pulse measured at 15.24 m from six DTs fired with ∆ti = 2 ms: (a) measured time history of

SPL, p(t) (solid curve), and calculated p(t) based on single DT pulse (dotted curve); (b) low-frequency

acoustic spectrum for measured p(t); (c) logarithmic acoustic spectrum; and (d) integrated fractional

acoustic power as a function of frequency.

the pulse train may account for the peak above 100 Hz; the (somewhat

variable) 2-ms repeat interval is probably responsible for the broad hump

in the spectrum around 500 Hz. The cause of the peak near 40 Hz, which

implies a period near 25 ms, is not clear. The 50-percent CPF for this

sequence is about 108 Hz; this value is lower than for the 8-ms sequence,

but not as low as for the six-tube simultaneous-fire sequence, and the

acoustic power is much more broadly distributed in frequency than for

the 8-ms delays.

Figure 14(a) shows p(t) data for a six-tube firing sequence in which the

delays have been adjusted to maximize the duration of the summed

positive-pressure pulse. As discussed in section 3, continuously shrinking

delays are required to achieve this. For this sequence, the intended delays

were 2.7, 2.2, 1.6, 1.5, and 0.5 ms between successive detonations; the

measured delays were 2.2, 1.5, 1.6, 1.2, and 0.5 ms. The delay times in the

linear-sum simulation (dotted curve) were again adjusted to fit the measured delays. With some minor crossovers, p(t) remains essentially positive

for 8 ms and then goes strongly negative following the last pulse. The

overall pulse train duration is about 15 ms. The spectrum of this pulse

train (fig. 14(b) and (c)) shows a large compound peak from 53 to 76 Hz, a

1

6

5

4

2 3

2.5

2.0

1.5

1.0

0.5

0

–0.5

–1.0

–1.5

Time (ms)

0 5 10 15

Instantaneous SPL (Pa)

Peak at 38 Hz

114 Hz

60

40

20

0

Relative SPL (Pa)

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

130

120

110

100

90

80

dB (arbitrary reference)

10 100 1000 10000

Frequency (Hz)

10 100 1000 10000

Frequency (Hz)

50% at 108 Hz

1

0.8

0.6

0.4

0.2

0

Cumulative fractional power
19

(a) (b)

(c) (d)

Figure 14. Pulse measured at 15.24 m from six DTs fired with measured ∆ti = 2.2, 1.5, 1.6, 1.2, and 0.5

ms: (a) measured time history of SPL, p(t) (solid curve), and calculated p(t) based on single DT pulse

(dotted curve); (b) measured low-frequency acoustic spectrum; (c) logarithmic acoustic spectrum; and

(d) integrated fractional acoustic power as a function of frequency.

lesser peak near 125 Hz, and a broad hump from about 400 to 800 Hz. We

may identify the 70-Hz peak with the overall pulse duration (f ~ 1/(15

ms)), the 125-Hz peak with the positive duration (f ~ 1/(8 ms)), and the

400- to 800-Hz hump as the contribution from the pulse delays (f ~

1/(2.5 ms) to f ~ 1/(1.25 ms)). For this sequence, the 50 percent CPF value

of 70 Hz falls within the frequency range of the 50- to 76-Hz compound

spectral peak.

Table 1 summarizes the acoustic spectrum results from the representative

measurements. For the 0-, 8-, 5-, and 2.2- to 0.5-ms delay cases, the 50-

percent CPF corresponds closely to the frequency of the spectral peak

with the greatest amplitude; i.e., the dominant and mean power frequencies tend to be the same. In these sequences, the acoustic power tends to

be concentrated near a single frequency. When the dominant and mean

power frequencies differ, the acoustic output power tends to be distributed across a broader band of frequencies. The 50-percent CPF is probably

the better overall measure of the effective frequency of the source, since it

properly weights the entire spectrum and is stable with respect to minor

spectral shifts. The 2.2- to 0.5-ms delay sequence is most effective at

shifting the source spectrum toward lower frequencies, even though the

1

1

4

5

3

2

2

6

2.0

1.5

1.0

0.5

0

–0.5

–1.0

–1.5

–2.0

Time (ms)

0 5 10 15

Instantaneous SPL (Pa)

1 Tube

6 stagger firings

6 firing simulation

Peak at 53 to 76 Hz

80

60

40

20

0

Relative SPL (Pa)

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

130

120

110

100

90

80

Frequency (Hz)

10 100 1000 10000

dB (arbitrary reference)

1

0.8

0.6

0.4

0.2

0

Cumulative fractional power

10 100 1000 10000

Frequency (Hz)

50% CPF at 70 Hz
20

DT pulse train used for this sequence has the shortest overall length of

those cases examined.

A consistent feature of the spectra of both the single and multiple DT

pulses is the –20 dB per decade overall roll-off at frequencies above

1 kHz. Altogether, these spectra may be viewed as a combination of a

series of low frequencies that are determined by the characteristics of the

individual DT and their firing delays, plus a 1/f high-frequency noise

component. Table 1 shows the percentage of the measured acoustic power

at frequencies above 1 kHz for the DT firings discussed above. The lowest

high-frequency content is associated with the six-tube simultaneous-fire

sequence. The 8-ms and variable delay sequence resulted in 5 to 6 percent

of the energy above 1 kHz; this is not surprising, since these timings

either translated the acoustic energy to lower frequencies or concentrated

it in a narrow frequency band. The 5- to 2-ms delay sequences show an

increased high-frequency content; as noted above, these delays are not

efficient at either translating or concentrating the acoustic spectrum.

Table 1. Dominant

and mean power

frequencies and

percent power above

1 kHz from single and

multiple detonation

tube firings measured

at 15.24-m range.

50%

Dominant cumulative Percentage of

spectrum power power above

Experiment peak (Hz) frequency (Hz) 1 kHz

Single tube 92 122 8.5

Six tubes, 0-ms delays 90 88 3.2

Six tubes, 8-ms delays 130 126 5.4

Six tubes, 5-ms delays 200 199 11.8

Six tubes, 4-ms delays 107 244 12.5

Six tubes, 2-ms delays 40 108 12.1

Six tubes, 2.2- to 0.5-ms 53–76 70 6.0

 delays
21

6. Conclusions

We have measured the pressure-time waveform and frequency content of

the acoustic output from single and multiple detonation tubes. We report

results for single tubes as a function of range in the moderate-shock to

acoustic range (from 1 MPa to below 1 kPa). The variation of the measured overpressures and pulse durations with range are well described by

existing models for spherical blast waves if an effective initial radius and

initial conditions are assumed near the source. The increase in pulse

duration with range accounts well for the measured decrease with range

of the mean frequency (50% CPF) of the acoustic power spectrum. At

ranges of 0.5 m or greater, the acoustic signal from the shock wave is

augmented by a second shock, attributable to supersonic jet formation

from the DT. Our experimental results confirmed our hypothesis that the

acoustic spectrum from a DT can be shifted to lower frequencies by the

timed addition of multiple DT pulses that, in effect, extend the positivepressure pulse duration. Judicious staggering of multiple pulses produced the largest down-conversion of acoustic energy and resulted in a

shift of the median energy from 122 Hz for a single tube to 70 Hz for six

tubes with optimal pulse staggering. The results also show that the

majority of the source acoustic power can be very effectively concentrated

in a narrow frequency band the pulse delays are tuned to match the

overall duration of an individual pulse.
22

References

1. L. I. Sedov, Similarity and Dimensional Methods in Mechanics, Academic

Press, New York (1959).

2. J. W. Reed, “Atmospheric Attenuation of Explosion Waves,” J. Acoust. Soc.

Am. 61 (1977), pp 39–47.

3. I. Kohlberg, H. E. Boesch, Jr., B. T. Benwell, and C. G. Reiff, Prediction of

Successive Radial Shock-on-Shock Interactions Using Thin Shell Model, submitted for presentation at the 38th AIAA Aerospace Sciences Meeting,

Reno, NV (10–13 January 2000).

4. R. Rasp, “Shock Waves, Blast Waves, and Sonic Booms,” Handbook of

Acoustics, M. J. Crocker, ed., J. Wiley & Sons, Inc. New York (1998), p 300.
23

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Modification of the Acoustic Spectrum of Detonation

Tube Shock Waves by Timed Multiple-Pulse Addition

May 2000 Final, 1/98–12/99

Detonation tubes are simple devices capable of producing substantial acoustic power that may be

useful for the simulation of high-level acoustic environments. We report results of an

investigation into the modification of the acoustic spectrum produced by detonation tubes by

timed addition of the shock-wave outputs of six detonation tubes fired in sequence. We first

examined the output of a single detonation tube as a function of range and found that it

conformed to existing models for spherical blast waves when appropriate initial conditions were

derived. We found timing schemes for the firing of the multiple tubes that (1) produce a

substantial shift of the acoustic energy to lower frequencies by maximizing the duration of the

positive pressure pulse, or (2) maximize the acoustic energy output in a narrow frequency range

by matching the pulse-to-pulse delay to the total duration (positive and negative pressure phases)

of a single detonation wave.

Shock wave, blast wave

Unclassified

ARL-TR-2203

0NEYYY

622120.140

A140

62120A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

31

Unclassified Unclassified SAR

2800 Powder Mill Road

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U.S. Army Research Laboratory

Attn: AMSRL-SE-DS eboesch@arl.mil

U.S. Army Research Laboratory

25

H. Edwin Boesch, Jr., Christian G. Reiff (ARL), Bruce T.

Benwell (Directed Energy Technologies)

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197