Proceedings of the Institute of Acoustics

 

 

Investigation of nonlinear propagation effects on sound from supersonic bullets

 

Erik Salomons¹, TNO Acoustics & Sonar Department, The Hague, The Netherlands

Frits van der Eerden, TNO Acoustics & Sonar Department, The Hague, The Netherlands

Frank van den Berg, TNO Acoustics & Sonar Department, The Hague, The Netherlands

 

ABSTRACT

 

ISO standard 17201-4 is a calculation method for sound levels generated by supersonic bullets. The standard takes into account nonlinear propagation effects, including broadening of a sound pulse with increasing propagation distance. Two elements of the ISO method are investigated in this paper: i) the accuracy for various Mach numbers, and ii) the effect of noise barriers. For the first element, calculation results are compared with measurement results. In general the agreement is good, but large deviations occur at Mach numbers near unity. Calculation parameters were varied in order to find an explanation for the deviations. For the second element, propagation paths along the horizontal top edge and the vertical side edges of a barrier of finite length are considered. For the path over the top edge, a possible approach is developed to account for the expected reduction of nonlinear propagation effects in the shadow region behind a barrier. For the paths along the vertical side edges, a simulation model with a dense distribution of point sources is employed. The three propagation paths are explained in terms of pairwise cancellation of Fresnel zones. The results of the simulation model are compared with results of an engineering model.

 

1. INTRODUCTION

 

International standard ISO 17201-4 [1] is part 4 of the series ISO 17201 dealing with noise from shooting ranges. Part 4 presents a calculation method for the sound exposure level of the sound pulse generated by a supersonic bullet. The sound pulse is represented as an N-wave [2]. Along the propagation path from the bullet to a receiver, various propagation effects occur, including nonlinear propagation effects. An important nonlinear propagation effect is a spectral shift to lower frequencies due to nonlinear broadening of the N-wave. The various propagation effects are taken into account in ISO 17201-4, which is based on nonlinear theory [2] and a Dutch method for predicting shooting sound levels [3].

 

In this article we present an analysis of the performance of the calculation method described in ISO 17201-4. First we compare calculation results with results of recent measurements on a rifle [4]. The comparison allows an assessment of the accuracy of the calculation method, for bullet speeds below 500 m/s. Next we consider the effect of a noise barrier near the bullet trajectory.

 

2. CALCULATION METHOD ISO 17201-4

 

This section presents an outline of the calculation method described in ISO 17201-4 [1]. The method yields the sound exposure level of the sound pulse generated by a supersonic bullet.

 

 

Figure 1: Illustration of the source point on the bullet trajectory, with the Mach ray normal to the wavefront. Areas I, II, and III are indicated.

 

Figure 1 shows the geometry, with the bullet trajectory along the horizontal axis. Three areas are indicated: areas I, II, and III. Bullet sound levels are highest in area II, which is also called the Mach area. The levels in areas I and III are much lower.

 

The sound pulse at a receiver in area II can be considered as originating from a source point on the bullet trajectory, as indicated in the figure. The line from the source point to the receiver is called the Mach ray, which is normal to the Mach wave front. Also indicated in the figure is the Mach angle μ, which is given by μ = arcsin(c/v), where c is the sound speed and v is the bullet speed at the source point. The Mach number M is defined as v/c, so we have μ = arcsinM ^-1. The bullet speed is assumed to decrease according to the relation v(x) = v₀ + kx, where v₀ is the speed at the muzzle of the weapon, k is the (negative) speed change per unit length, and x is the distance of the bullet from the muzzle.

 

The 1/3-octave band spectrum of the sound exposure level at the receiver is calculated with the following formula:

 

 

where 𝑓 is the 1/3-octave band center frequency, 𝐿_E,source is a source spectrum, 𝐴_div is a geometrical divergence term, 𝐴_nlin is a nonlinear correction to the divergence term, 𝐴_atm accounts for atmospheric absorption, and 𝐴_excess is the excess attenuation. All terms in Eq. (1) are functions of distance r along the Mach ray. Formulas for the terms can be found in Ref. [1]. For bullets with a constant speed ( k = 0), we have 𝐴_div = 10 lg 𝑟 (cylindrical divergence) and 𝐴_nlin = 2.5 lg 𝑟. The excess attenuation accounts for effects of ground reflections, atmospheric refraction, and screening by noise barriers.

 

The term 𝐿_E,source is a source spectrum calculated from the N-waveform, with characteristic frequency 𝑓_c as a parameter. This parameter is related to the frequency 𝑓_max where the spectrum has a maximum, through 𝑓_max = 0.65 𝑓_c. The spectrum 𝐿_E,source and characteristic frequency 𝑓_c are functions of the Mach number at the source point and two bullet dimensions: effective length l_p and diameter d_p (see Figure 2). The characteristic frequency 𝑓_c is also a function of propagation distance r, so the spectrum 𝐿_E,source is not a ‘true’ source spectrum. The variation of 𝑓_c with r is illustrated in Figure 3, for a situation with v₀ = 780 m/s, k = - 0.75 s^-1, l_p = 7.8 mm, and d_p = 31 mm. The source point was chosen at distance x = 160 m from the weapon, where the bullet speed is 660 m/s. The characteristic frequency is 4155 Hz at r = 10 m, and decreases proportional to r ^-1/4 to 1389 Hz at r = 800 m. The shift of the spectrum with distance r is a nonlinear propagation effect. It is related to the nonlinear broadening of a sound pulse [2].

 

Equation (1) shows that there is another nonlinear propagation effect, represented by the term 𝐴_nlin. This nonlinear effect represents the difference in geometrical divergence between an N-wave and a linear cylindrical wave. In Section 4, the two nonlinear effects will be analyzed for situations with a noise barrier.

 

 

Figure 2: Bullet dimensions l_p and d_p. The bullet ‘tail’ is ignored.

 

 

Figure 3: Example of the variation of characteristic frequency 𝑓_c with propagation distance r along the Mach ray.

 

3. COMPARISON WITH MEASUREMENT RESULTS

 

In this section calculation results of ISO 17201-4 are compared with results of recent measurements of bullet noise [4]. The measurements were performed with a 9 mm rifle, Beretta, using two types of ammunition, Action 4 and Geco. Bullet speed parameters are v₀ = 490 m/s and k = - 1.8 s^-1 for Action 4, and v₀ = 361 m/s and k = - 1.0 s^-1 for Geco. Bullet dimensions are l_p = 0.014 m and d_p = 0.009 m. Two series of ten shots were measured, one for Action 4 and one for Geco.

 

Figure 4 shows a top view of the measurement setup, with fifteen microphones (1-15). Mach rays are shown in the graphs. The lower speed of Geco bullets is reflected by smaller angles of the Mach rays to the bullet trajectory than for Action 4.

 

The height of the weapon and the (nearly) horizontal bullet trajectory was 6.2 m, so that ground reflections could be eliminated from the measured time signals. Further, the muzzle blast was suppressed by a damper around the weapon. In this way, the direct bullet pulse was determined (so we have 𝐴_excess = 0 in Eq. 1). The time signals showed the expected N-waveform.

 

From the time signals, 1/3-octave band spectra of the sound exposure level were determined. From these, broadband sound exposure levels were derived, and averaged over the series of ten shots. In addition, average values of the characteristic frequency 𝑓_c were determined, in two ways: i) from the maximum of the spectrum, using 𝑓_max = 0.65 𝑓_c, and ii) from the time signal, using 𝑓_c = 1/T, where T is the duration of the N-wave. The two values agreed within about 10%. The value obtained from the time signal was used for further analysis.

 

 

Figure 4: Top view of the measurement setup with fifteen microphones (1-15), for Action 4 (left) and Geco (right). The bullet trajectory is along the horizontal axis. Mach rays from the source points to the microphones are shown.

 

Figure 5 shows average values of the characteristic frequency 𝑓_c and the sound exposure level L_E, at microphones 1-15, for Action 4 and Geco. For Geco, no values are shown for microphones 6 and 15, as these are outside the Mach area (see Figure 4). The global trend is that 𝑓_c and L_E decrease with increasing propagation distance r, but not completely monotonic. The origin of this is that the two quantities depend not only on r but also on the Mach number at the source point (see Section 2), which is different for the different microphones. The lowest value of the (calculated) Mach number is 1.0003 at the source point of microphone 11, for Geco. The Mach number at the source point increases along the line of microphones 11-14, which is the origin of the increase of 𝑓_c along this line.

 

In general the agreement between ISO 17201-4 and the measurement results is good, in particular for L_E. The largest deviations for 𝑓_c occur at the microphones for which the Mach number at the source point is below 1.02: microphone 11 for Action 4 and microphones 11, 12, 3, 4, and 5 for Geco. The largest deviation is a factor of about 3 at microphone 11 for Geco: 𝑓_c = 358 Hz according to ISO 17201-4 versus 𝑓_c = 1184 Hz according to the measurement. Here the deviation of L_E is also largest: L_E = 98.4 dB versus L_E = 94.0 dB, respectively.

 

In order to find an explanation for the deviations, we varied some calculation parameters. First, we changed deceleration parameter k for Geco from -1.0 s^-1 to -0.3 s^-1. This caused a shift of the source point for microphone 11 from x = 20 m (see Figure 4) to x = 70 m. Consequently, propagation distance r is reduced, leading to higher values of both 𝑓_c and L_E: 𝑓_c = 700 Hz and L_E = 106.0 dB. The value of 𝑓_c is closer to the measured value of 1184 Hz, but the value of L_E is 12 dB higher than the measured value of 94 dB. So this variation does not lead to an improvement of the overall agreement. We performed similar variations for parameters v₀, l_p, and d_p, but we reached the same conclusion that these variations did not give an improvement of the overall agreement. A possible reason for the observed deviations for Mach numbers near unity is that the assumption of a needle-shaped body [2] is violated by the bullets. A practical solution is to simply use M = 1.02 for all values in the range 1.00 - 1.02, as this does improve the overall agreement.

 

 

Figure 5: Average values of the characteristic frequency (top) and the sound exposure level (bottom), at microphones 1-15, for Action 4 (left) and Geco (right).

 

4. SCREENING

 

For situations with a noise barrier near the bullet trajectory, we distinguish three propagation paths, as illustrated in Figure 6. The path over the horizontal top edge of the barrier (T) originates at a source point that coincides with the source point in the situation without barrier, in good approximation. The paths along the vertical side edges of the barrier (L and R) originate at different source points.

 

 

Figure 6: Illustration of a situation with a barrier and sound paths L, T, and R. The dashed line is the path in the situation without the barrier.

 

 

Figure 7: Geometry with six receivers along a Mach ray originating at a source point at 160 m from the weapon.

 

4.1. Path T

 

In this section we consider sound path T over the top of the barrier. In most practical situations, path T is the dominating sound path, and paths L and R can be neglected. We consider the geometry shown in Figure 7, with six receivers along a Mach ray, at distances of 10, 50, 100, 200, 400, and 800 m from the source point. Bullet parameters are v₀ = 780 m/s, k = - 0.75 s^-1, l_p = 7.8 mm, and d_p = 31 mm. A barrier is placed at the position of the first receiver, at 10 m from the source point (the barrier is not shown in the figure). Only path T over the top of the barrier is considered.

 

In the situation without the barrier, the characteristic frequency 𝑓_c decreases along the Mach ray, as shown in Figure 3 in Section 2. The value of 𝑓_c is 4155 Hz at 10 m and 1389 Hz at 800 m. The characteristic frequency is related to the frequency 𝑓_max where the spectrum of sound exposure level 𝐿_E has a maximum, by the relation 𝑓_max = 0.65𝑓_c. Figure 8 shows four sets of spectra calculated from Eq. (1) for the six receivers, without barrier (red lines) and with barrier (blue lines):

1. Without barrier. The spectra show that frequency 𝑓_max decreases with increasing distance r.

2. With barrier, including both nonlinear effects (shift of 𝑓_c and term 𝐴_nlin) over the full propagation path.

3. With barrier, ‘switching off’ the nonlinear frequency shift behind the barrier, so the only nonlinear effect behind the barrier is the term 𝐴_nlin.

4. With barrier, ‘switching off’ both nonlinear effects behind the barrier.

 

The idea here is that the barrier reduces the sound pressure in the shadow region behind the barrier, so nonlinear effects are expected to be reduced in the shadow region. As an extreme approach we ‘switched off’ one or both nonlinear effects in the shadow region. The dashed and dotted lines in the figure clearly show the effect of switching off the nonlinear frequency shift: with increasing distance r, frequency 𝑓_max remains around the value of 2.5 kHz at r = 10 m.

 

The reduction of the sound level caused by the barrier was calculated with ISO standard 9613-2 [5]. The formula for the barrier attenuation Abarrier in this standard is more or less similar to a well-known relation for A_barrier due to Maekawa (see Ref. [6]). Ground effects were ignored for simplicity, so in Eq. (1) we have 𝐴_excess = 𝐴_barrier for the situation with the barrier and 𝐴_excess = 0 for the situation without the barrier. It was assumed that the barrier extends 2 m above the line of sight, i.e. the line from the source point to the receiver.

 

Switching off the nonlinear effects behind the barrier is in most cases too extreme, except perhaps for very high barriers. A more realistic approach may be to convert the barrier attenuation to a shift in propagation range r, for the calculation of 𝑓_c and 𝐴_nlin as a function of r. Let us consider an example with a barrier at 10 m from the source point and a receiver at 400 m, and a barrier attenuation Abarrier of 10 dB. The barrier attenuation yields a shift in range by a factor of 10, assuming cylindrical spreading as an approximation. This means that the range interval from r = 10 m to 400 m is replaced with the interval from r = 100 m to 490 m, and consequently, the reduction of frequency 𝑓_c by a factor of 0.40 is replaced with a factor of 0.67 (see Figure 3). This leads to 𝑓_c = 2793 Hz instead of 𝑓_c = 1652 Hz at 400 m. To validate and possibly improve this approximate approach, we plan to perform measurements of reduction of bullet sound by a barrier.

 

 

Figure 8: Four sets of 1/3-octave band spectra 𝐿_E (𝑓) for the six receivers at r = 10, 50, 100, 200, 400, and 800 m (from top to bottom). Red lines are for the situation without barrier. Blue lines are for the situation with the barrier (as explained in the text).

 

4.2. Paths L,T,R

 

In this section we focus on the two additional paths L and R along the vertical side edges of a barrier (see Figure 6). This requires the consideration of Fresnel zones on the bullet trajectory [7,8,9]. Figure 9a illustrates the first three Fresnel zones around the source point S. The boundaries between successive Fresnel zones are defined such that the sound paths to the receiver (red lines) differ by half a harmonic period in arrival time at the receiver. Consequently, successive Fresnel zones largely cancel each other by destructive interference, and what ‘survives’ is the sound from a small region around the source point.

 

Figure 9b shows the same situation, but now with a barrier that screens the first two Fresnel zones, introducing two additional source points SL and SR. The additional source points emerge because the barrier disturbs the pairwise cancellation of Fresnel zones around these points.

 

Figure 9c illustrates a linear simulation model of bullet sound [7,8,9]. A dense distribution of point sources is placed on the bullet trajectory. Each point source emits a single period of a harmonic sound wave at the moment that the bullet arrives at the point source. The sound from all point sources is summed coherently at the receiver. This numerical approach simulates the pairwise cancellation of Fresnel zones and the emergence of a single source point corresponding to the path with the earliest arrival time.

 

 

Figure 9: Illustrations of Fresnel zones (a, b) and a simulation model (c). Figure (a) shows the first three Fresnel zones around source point S. Figure (b) shows the same situation with a barrier that screens the first two Fresnel zones, introducing two additional source points S_L and S_R. Figure (c) shows a simulation model with point sources separated by one tenth of a wavelength (λ / 10). The source point S corresponds to the path with the earliest arrival time.

 

Following Refs. [8,9], the source signal is modeled as a single period of a sine wave, with frequency 2 kHz, corresponding to a period T of 0.5 ms, and unit amplitude: 𝑓(𝜏) = sin(2𝜋 𝜏/ 𝑇) for 0 < 𝜏 <T. The signal at the receiver can be written as an integral over the bullet trajectory, from the muzzle at x = 0 to the target at  Here r(x) is the distance from point (x,0) to the receiver. The numerical model approximates the integral by a sum over point sources, separated by one tenth of a wavelength (λ / 10), which amounts to 0.017 m in this case.

 

 

Figure 10: Geometry (not to scale) with a 17 m long barrier that screens the first Fresnel zone.

 

We consider the geometry shown in Figure 10, with a 17 m long barrier that screens the first Fresnel zone. Figure 11 shows the calculated sound signal, for the situations without and with the barrier. For the situation without the barrier, we have a single sound pulse (blue line). The arrival time of 1.823 s (after the firing of the bullet) corresponds to the source point located at x = 121.3 m. The receiver is located in the Mach area. For receivers outside the Mach area, the simulation model yields much weaker sound signals, as expected.

 

For the situation with the barrier, we have two sound pulses (red line). The first one, labeled L+R, is actually the superposition of the two sound paths L and R, which arrive at the same time at the receiver. The sound pulse labeled T represents the sound path over the top of the barrier. This sound pulse is delayed by about 3 ms and is much weaker than the unscreened sound pulse. We assumed a barrier height of 5 m above the line of sight, corresponding to a barrier attenuation of about 25 dB, calculated with Maekawa’s formula (see [6]). The barrier attenuation and the delay was applied to the sound from all point sources that are screened, i.e. all point sources in the first Fresnel zone.

 

Figure 12 shows similar results for a 47 m long barrier, which screens eight Fresnel zones. The simulation model yields again two sound pulses: L+R and T. Pulse T is identical to pulse T in Figure 11. Pulse L+R, however, is weaker and more delayed than pulse L+R in Figure 11, as the barrier is longer than the 17 m long barrier for Figure 11. Also included in Figure 12 is the result of a practical engineering model for shooting sound [3]. This model yields separate contributions for the three sound paths L, T, and R. For each sound path, a barrier attenuation is applied with a formula that is similar to Maekawa’s formula. For the sound paths L and R, source points on the bullet trajectory are determined in such a way that the Mach rays arrive exactly at the vertical side edges of the barrier. At first sight, the results of the simulation model and the engineering model show a reasonable agreement. However, the result of the engineering model shows the two pulses L and R separately. When the two are summed, a sound pulse is obtained that has an amplitude that is about three times larger than the amplitude of the pulse L+R from the simulation model. Further research is required to investigate this discrepancy.

 

 

Figure 11: Sound signal at the receiver for the situation in Figure 10, without and with barrier.

 

 

Figure 12: Results of calculations for a situation with a 47 m long barrier that screens the first eight Fresnel zones (top).

 

5. CONCLUSIONS

 

The calculation method of ISO 17201-4 for bullet sound has been investigated for situations without and with a noise barrier. Possible approaches for improvement of the calculation method have been described. Further research is required to assess the accuracy of these approaches.

 

6. ACKNOWLEDGEMENTS

 

Interesting discussions with members of working group 51 for ISO 17201 are gratefully acknowledged. We are grateful to Mattias Trimpop (IFL) and the German Bundeswehr for providing the results of the measurements.

 

7. REFERENCES

 

1. ISO 17201-4, “Acoustics – Noise from shooting ranges – Part 4: Prediction of projectile sound”, International Standard, ISO 2006 (Geneva), URL https://www.iso.org/standard/30579.html

2. A.D. Pierce, “Acoustics. An introduction to its physical principles and applications” (AIP, New York, 1991), Chapter 11.

3. F.H.A. van den Berg, J. Vos, E.M. Salomons, H.E.A. Brackenhoff, “Handleiding ter bepaling van de geluidbelasting ten gevolge van schietactiviteiten”, November 2011.

4. The results of the measurements were provided by Mattias Trimpop (Institut für Lärmschutz, Germany) in the framework of ISO 17201-4 and ISO 17201-2.

5. ISO 9613-2, “Acoustics – Attenuation of sound during propagation outdoors – Part 2: General method of calculation”, International Standard, ISO 1996 (Geneva).

6. E.M. Salomons, “Noise barriers in a refracting atmosphere”, Appl. Acoust. 47 (1996) 217-238.

7. E.M. Salomons, “A coherent line source in a turbulent atmosphere”, J. Acoust. Soc. Am. 105(1999) 652-657.

8. J.M. Wunderli and K. Heutschi, “Simulation model for sonic boom of projectiles”, Acta Acustica 87, 86-90, 2001.

9. J.M. Wunderli and K. Heutschi, “Shielding effect for sonic boom of projectiles”, Acta Acustica 87, 91-100, 2001.

 

---

¹ erik.salomons@tno.nl