Explosion Source Models

Abstract

Explosive detonations produce shocked transients with highly nonlinear pressure signatures in the near field. This chapter presents the properties and defining characteristics of a suite of theoretical source pressure functions representative of detonations and deflagrations, and constructs criteria for defining reference blast pulses. Both the primary positive overpressure and the negative underpressure phases contribute to the temporal and spectral features of a blast pulse.

Appendices

Appendix 1. A Moment or Two

Practical explosion source pressure functions often take the form

$$ p\left( \tau \right) = \frac{{p\left( {\tau_{ref} } \right)}}{{f_{i} \left( {\tau_{ref} } \right)}}f_{i} \left( \tau \right) $$

(8.191)

where $\tau_{ref}$ is generally the scaled time of the peak overpressure or underpressure and $\tau$ is the scaled time referenced to the positive pulse duration $t_{p}$,

$$ \tau = \frac{t}{{t_{p} }}. $$

(8.192)

The magnitude of the derivative at the onset time and first zero crossing can be readily evaluated from

$$ \frac{{ dp_{i} \left( t \right)}}{dt} = \frac{1}{{t_{p} }} \frac{{ dp_{i} \left( \tau \right)}}{d\tau }. $$

(8.193)

Likewise, integration over time may be represented as

$$ \mathop \smallint \limits_{0}^{{t_{p} }} p\left( {\frac{t}{{t_{p} }}} \right)dt = \left( {t_{p} p_{p} } \right)\mathop \smallint \limits_{0}^{1} \frac{p\left( \tau \right)}{{p_{p} }}d\tau $$

(8.194)

$$ \mathop \smallint \limits_{0}^{{t_{p} }} \left[ {p\left( {\frac{t}{{t_{p} }}} \right)} \right]^{2} dt = \left( {t_{p} p_{p}^{2} } \right)\mathop \smallint \limits_{0}^{1} \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]^{2} d\tau . $$

(8.195)

The Fourier Transform pair used in this work is

$$ P\left( {j\omega } \right) = \mathop \smallint \limits_{ - \infty }^{\infty } p\left( t \right)e^{ - j\omega t} dt = t_{p} p_{p} \mathop \smallint \limits_{ - \infty }^{\infty } \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]e^{{ - j\left( {\omega t_{p} } \right)\tau }} d\tau $$

(8.196)

$$ p\left( t \right) = \frac{1}{2\pi }\mathop \smallint \limits_{ - \infty }^{\infty } P\left( {j\omega } \right)e^{j\omega t} d\omega . $$

(8.197)

Note that all the pulses in this chapter vanish for negative time, so only the positive time interval contributes to the Fourier transform. It is very easy to add an unnecessary factor of two if this is not taken into account.

The total impulse I of a balanced pulse must satisfy,

$$ I_{1} = \mathop \smallint \limits_{0}^{\infty } p_{1} \left( t \right)dt = P_{1} \left( {j\omega = 0} \right) = 0. $$

(8.198)

This is equivalent to a vanishing fist moment. Due to difficulties in capturing the peak pressure and rise time, the positive impulse—the integral over time of the positive phase of a recorded waveform—has been traditionally used as a fairly robust metric for explosions. The scaled first moment $M_{1p}$ of the positive phase can be expressed in terms of a reference positive impulse, the overpressure $p_{p}$, and the positive phase duration $t_{p}$ as

$$ I_{\text{p}} = \mathop \smallint \limits_{0}^{{t_{p} }} p\left( t \right)dt = \left( {t_{p} p_{p} } \right) \mathop \smallint \limits_{0}^{1} \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]d\tau $$

(8.199)

$$ M_{1} \equiv \frac{{2I_{\text{p}} }}{{t_{p} p_{p} }} = 2\mathop \smallint \limits_{0}^{1} \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]d\tau . $$

(8.200)

Parseval’s Theorem provides a measure of the signal exposure xE

$$ xE = \mathop \smallint \limits_{ - \infty }^{\infty } \left| {p\left( t \right)} \right|^{2} dt = \mathop \smallint \limits_{ - \infty }^{\infty } \left| {P\left( f \right)} \right|^{2} df = \frac{1}{2\pi }\mathop \smallint \limits_{0}^{\infty } S\left( \omega \right)d\omega $$

(8.201)

where S is the unilateral energy spectral density in Pa²/Hz². The expression for the total signal exposure has units of Pa² s and is a measure of the total energy in the signal. The exposure can be readily converted to decibels with the selection of an appropriate reference value (e.g., Garcés 2013). For finite pulses with a clear onset at zero time, the scaled second moment M₂ can be expressed as

$$ xE = \mathop \smallint \limits_{0}^{\infty } \left| {p\left( t \right)} \right|^{2} dt = \left( {t_{p} p_{p}^{2} } \right)\mathop \smallint \limits_{0}^{\infty } \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]^{2} d\tau $$

(8.202)

$$ M_{2} \equiv \frac{2xE}{{t_{p} p_{p}^{2} }} = 2\mathop \smallint \limits_{0}^{\infty } \left[ {\frac{p\left( \tau \right)}{{p_{p} }}} \right]^{2} d\tau $$

(8.203)

These definitions are a factor of two larger than the formal mathematical expressions for the first and second moments of normalized functions so that they can be scaled relative to the impulse and exposure of a reference triangular pulse approximation. Normalization by the second moment is generally required for the construction of wavelets. Scaled first moments for the positive phase and total second moments for the balanced canonicals are provided below.

The impulse-balanced triangular approximation for $\frac{{p_{p} }}{{\left| {p_{n} } \right|}} = 2$ is used as the reference impulse and exposure, where

$$ M_{{1\,\Delta }} = 2\frac{1}{2} = 1, $$

(8.204)

$$ M_{{2\,\Delta 2}} = 2\frac{1}{2} = \frac{2}{3} + \frac{1}{3} = 1, $$

(8.205)

with ~67% of the energy in the positive phase and a non-negligible 33% in the negative phase. The G17 pulse has a close match to the selected blast reference, with

$$ M_{1\,G17HE} = 2\frac{1}{2} = 1 $$

(8.206)

$$ M_{2\,G17HE} = 2\left( {\frac{1}{3} + \frac{2\sqrt 6 }{35}} \right)\,\approx\,\frac{2}{3} + 0.140\,\approx\,0.95. $$

(8.207)

and a similar distribution in power, as is expected since the G17 pulse uses the triangular approximation for the positive phase. The balanced Friedlander delivers about half of the power as the G17, with

$$ M_{1\,F46HE} = 2e^{ - 1} \,\approx\,0.74 $$

(8.208)

$$ M_{2\,F46HE} = 2\frac{1}{4} = 0.5. $$

(8.209)

The modified Friedlander positive moment is provided for completeness

$$ M_{1\,MF} = 2\left[ {\frac{{\alpha_{p} + 1 + e^{{ - \alpha_{p} }} }}{{\alpha_{p}^{2} }}} \right]. $$

(8.210)

The G95HE pulse slightly improves over the F46 in terms of power, and begins to show the contributions from the negative phase,

$$ M_{1\,G95HE} = 2\frac{1}{\sqrt 2 }e^{\sqrt 2 - 2} \,\approx\,0.79 $$

(8.211)

$$ M_{2\,G95HE} = 2\frac{{3\left( {2 + \sqrt 2 } \right)}}{32}\,\approx\,0.64. $$

(8.212)

The Reed (1977) pulse begins to resemble the positive triangular pulse and takes another step up in power delivery

$$ M_{1\,R77HE} = 2\frac{34,992}{78,125}\,\approx\,0.9 $$

(8.213)

$$ M_{2\,R77HE} = 2\frac{8,440}{21,609}\,\approx\,0.78 , $$

(8.214)

although it is not as efficient as the G17 pulse. The deflagration pulses are not expected to match the detonation criteria, and the differences can be immediately observed in the scaled first and second moments.

$$ M_{1\,B55LE} = 2\left( {1 + \sqrt 2 } \right)e^{ - \sqrt 2 } \,\approx\,1.17 $$

(8.215)

$$ M_{2\,B55LE} = 2\frac{{\left( {3 + 2\sqrt 2 } \right)}}{32}e^{4 - 2\sqrt 2 } \,\approx\,1.18. $$

(8.216)

The area under the curve is larger, and thus these pulses have a greater scaled positive impulse as well as a larger exposure. deflagration pulses are generally not as destructive as detonations because of their slower rise times. Finally, the large negative phase of the G95LE, coupled with the broad concave-downwards first phase, yields

$$ M_{1\,G95LE} = 2\frac{3 + 2\sqrt 3 }{3}e^{\sqrt 3 - 3} \,\approx\,1.21 $$

(8.217)

$$ M_{2\,G95LE} = 2\frac{123 + 71\sqrt 3 }{288}\,\approx\,1.71, $$

(8.218)

with almost double the exposure as the reference blast.

Appendix 2. Modified Friedlander Pulse Properties

The Hybrid Modified Friedlander equation implements different exponential decay parameters for the positive and negative phases of the blast pulse,

$$ p_{1p} = p_{p} f\left( {\tau , \tau_{0} ,\alpha_{p} } \right)/max\left( f \right) \quad for \quad 0 < \tau \le 1 $$

(8.219)

$$ p_{1n} = \left| {p_{n} } \right| f\left( {\tau , \tau_{0} ,\alpha_{n} } \right)/\left| {min\left( f \right)} \right| \quad for \quad \tau \ge 1 $$

(8.220)

$$ \tau = t/t_{p} $$

(8.221)

where p represents overpressure, p_p is the maximum blast pressure (peak overpressure), p_n is the minimum blast pressure (peak underpressure), t represents time, t_p is the positive pulse duration, with τ = 1 at t = t_p, and waveform parameters α_p and α_n are implemented for the positive and negative phases of the pulse, respectively.

The peak overpressure is as specified with p_p, but the peak underpressure must be corrected by

$$ \tau_{min} = 1 + \frac{1}{{\alpha_{n} }}, \quad f_{1} \left( {\tau_{min} } \right) = \frac{{e^{{ - \left( {1 + \alpha_{n} } \right)}} }}{{\alpha_{n} }}. $$

(8.222)

The negative Friedlander pulse e-folding time, or lifetime is defined as the time where the negative pressure reduces to 1/e of its (minimum) value. By defining the negative pulse duration as

$$ \frac{{t_{n} }}{{t_{p} }} = \frac{2\pi }{{\alpha_{n} }} $$

(8.223)

the pulse lifetime is

$$ t_{e} = t_{p} + t_{n} = t_{p} + \frac{{2\pi t_{p} }}{{\alpha_{n} }} $$

(8.224)

$$ \tau_{e} = \frac{{t_{e} }}{{t_{p} }} = 1 + \frac{2\pi }{{\alpha_{n} }} = \tau_{min} + \frac{2\pi - 1}{{\alpha_{n} }}, $$

(8.225)

where

$$ \frac{{\left| {p_{{1{\text{n}}}} \left( {\tau_{e} } \right)} \right|}}{{\left| {p_{n} } \right|}}\,\approx\,\frac{{2e^{ - 3} }}{\pi }\,\approx\,0.032 \sim e^{ - 1} . $$

(8.226)

As can be readily verified, a Friedlander pulse with $\alpha_{n} \ge 2\pi$ has a rapidly decaying negative pulse with a negative duration that is shorter than the positive pulse duration $t_{p}$.

For the special case of the balanced Friedlander, where $\alpha_{n} = 1$

$$ t_{n} \left( {\alpha = 1} \right) = 2\pi\,t_{p} . $$

(8.227)

Applying different waveform parameters and pressures in the positive and negative phases introduces an artificial and unsightly discontinuity in the derivative at the first zero crossing

$$ \left. {\frac{{dp_{1p} }}{dt}} \right|_{{t = t_{p} }} = - \frac{{p_{p} }}{{t_{p} }}e^{{ - \alpha_{p} }} $$

(8.228)

$$ \left. {\frac{{dp_{1n} }}{dt}} \right|_{{t = t_{p} }} = - \frac{{\left| {p_{n} } \right|\alpha_{n} }}{{t_{p} }}e^{1} . $$

(8.229)

Exact (unscaled) first and second moment expressions for the positive and negative components of the Friedlander pulse, where $f_{1p,n} = f_{F46HE} \left( {\alpha_{p} ,\alpha_{n} } \right)$, are

$$ \mathop \smallint \limits_{0}^{1} f_{1p} \left( \tau \right)d\tau = \frac{{e^{{ - \alpha_{p} }} + \alpha_{p} - 1}}{{\alpha_{p}^{2} }} $$

(8.230)

$$ \mathop \smallint \limits_{1}^{\infty } f_{1n} \left( \tau \right)d\tau = \frac{{ - e^{{ - \alpha_{n} }} }}{{\alpha_{n}^{2} }} $$

(8.231)

$$ \mathop \smallint \limits_{0}^{1} f_{1p}^{2} \left( \tau \right)d\tau = \frac{{2\alpha_{p}^{2} - 2\alpha_{p} + 1 - e^{{ - 2\alpha_{p} }} }}{{4\alpha_{p}^{3} }} $$

(8.232)

$$ \mathop \smallint \limits_{1}^{\infty } f_{1n}^{2} \left( \tau \right)d\tau = \frac{{e^{{ - 2\alpha_{n} }} }}{{4\alpha_{n}^{3} }} $$

(8.233)

and the Fourier transforms for the positive and negative phases are

$$ P_{1p} = \mathop \smallint \limits_{0}^{{t_{p} }} p_{1p} \left( t \right)e^{ - j\omega t} dt = \frac{{p_{p} t_{p} }}{{\alpha_{p}^{2} h_{p}^{2} }}\left( {\alpha_{p} h_{p} - 1 + e^{{ - \alpha_{p} h_{p} }} } \right) $$

(8.234)

$$ P_{1n} = \mathop \smallint \limits_{{t_{p} }}^{\infty } p_{1n} \left( t \right)e^{ - j\omega t} dt = - \frac{{\left| {p_{n} } \right|t_{p} }}{{\alpha_{n} h_{n}^{2} }}e^{{\left( {1 + \alpha_{n} } \right)}} e^{{ - \alpha_{n} h_{n} }} $$

(8.235)

where

$$ h_{p,n} = 1 + \frac{{j\omega t_{p} }}{{\alpha_{p,n} }}. $$

(8.236)

Note that the impulse is the Fourier transform evaluated at the zero (DC) frequency.

Appendix 3. Modified Brode Pulse

The canonical B55 pulse can be expressed as

$$ f_{2} = \tau \left( {1 - \tau } \right)e^{ - 2\alpha \tau } , $$

(8.237)

with slope at its zero crossing of

$$ \left. {\frac{{df_{2} }}{d\tau }} \right|_{\tau = 1} = - e^{ - 2\alpha } . $$

(8.238)

Its extrema are at

$$ \tau_{max} = \frac{{\left( {\alpha + 1} \right) - \sqrt {\alpha^{2} + 1} }}{2\alpha } , $$

(8.239)

$$ \hbox{max} \left( {f_{2} } \right) = f_{2} \left( {\tau_{max} } \right) = \frac{{\left( {\sqrt {\left( {\alpha^{2} + 1} \right)} - 1} \right){ \exp }\left[ { - \left( {\alpha + 1} \right) + \sqrt {\alpha^{2} + 1} } \right]}}{{2\alpha^{2} }} , $$

(8.240)

$$ \tau_{min} = \frac{{\left( {\alpha + 1} \right) + \sqrt {\alpha^{2} + 1} }}{2\alpha }, $$

(8.241)

$$ \hbox{min} \left( {f_{2} } \right) = f_{2} \left( {\tau_{min} } \right) = - \frac{{\left( {\sqrt {\left( {\alpha^{2} + 1} \right)} + 1} \right)\exp \left[ { - \left( {\alpha + 1} \right) - \sqrt {\alpha^{2} + 1} } \right]}}{{2\alpha^{2} }} . $$

(8.242)

The impulse is obtained from

$$ I_{2p} = \mathop \smallint \limits_{0}^{{t_{p} }} p_{2} \left( t \right)dt = \frac{{t_{p} p_{p} }}{{max\left( {f_{2} } \right)}}\mathop \smallint \limits_{0}^{1} f_{2} \left( \tau \right)d\tau , $$

(8.243)

$$ I_{2n} = \mathop \smallint \limits_{{t_{p} }}^{\infty } p_{2} \left( t \right)dt = \frac{{ - t_{p} \left| {p_{n} } \right| }}{{\left| {min\left( {f_{2} } \right)} \right|}}\mathop \smallint \limits_{1}^{\infty } f_{2} \left( \tau \right)d\tau . $$

(8.244)

The explicit solutions for the impulse are

$$ I_{2p} = t_{p} p_{p} \frac{{\left( {\alpha \,\cosh \alpha - \sinh \alpha } \right) exp\left[ {\left[ {1 - \sqrt {\alpha_{p}^{2} + 1} } \right]} \right]}}{{\alpha_{p} \left( {\sqrt {\left( {\alpha_{p}^{2} + 1} \right)} - 1} \right)}}, $$

(8.245)

$$ I_{2n} = - t_{p} \left| {p_{n} } \right|\frac{{\left( {\alpha_{n} + 1} \right) exp\left[ {1 - \alpha_{n} + \sqrt {\alpha_{n}^{2} + 1} } \right]}}{{2\alpha_{n} \left( {\sqrt {\left( {\alpha_{n}^{2} + 1} \right)} + 1} \right)}}. $$

(8.246)

One possible hybrid pulse can use the negative Brode pulse for the underpressure to balance the modified Friedlander overpressure, requiring the numerical solution of

$$ \frac{{\bar{p}_{p} }}{{\bar{p}_{n} }}M_{1} = \frac{{\left( {\alpha_{n} + 1} \right) exp\left[ {1 - \alpha_{n} + \sqrt {\alpha_{n}^{2} + 1} } \right]}}{{\alpha_{n} \left( {\sqrt {\left( {\alpha_{n}^{2} + 1} \right)} + 1} \right)}} $$

(8.247)

I follow Brode (1955) and Bonner et al. (2013) in using the B55 pulse as a fitting function for the negative phase of the pulse with duration $t_{n}$, where

$$ \tau_{B} = \frac{{t - t_{p} }}{{t_{n} }} $$

(8.248)

$$ p_{Bn} = - \frac{{\left| {p_{n} } \right|}}{{\hbox{max} \left( {f_{2} } \right)}}\tau_{B} \left( {1 - \tau_{B} } \right)e^{{ - 2\alpha_{n} \tau_{B} }} , \quad 0 \le \tau_{n} \le 1. $$

(8.249)

To facilitate evaluation relative to the same standardized time scale $t_{p}$, let

$$ \tau_{n} = \frac{{t_{n} }}{{t_{p} }} $$

(8.250)

$$ \tau_{B} = \frac{\tau - 1}{{\tau_{n} }} $$

(8.251)

$$ p_{Bn} = - \frac{{\left| {p_{n} } \right|}}{{\hbox{max} \left( {f_{2} } \right)}}\left( {\frac{\tau - 1}{{\tau_{n} }}} \right)\left( {\frac{{\tau_{n} + 1 - \tau }}{{\tau_{n} }}} \right)e^{{ - 2\alpha_{n} \frac{\tau - 1}{{\tau_{n} }}}} , \quad 1 \le \tau \le 1 + \tau_{n} $$

(8.252)

Brode (1955) and Bonner et al. (2013) use a value of $\alpha_{n} = 2$ to fit the negative phase of explosion blasts, with solution

$$ p_{Bn} = - \left| {p_{n} } \right|\frac{13.894}{{\tau_{n}^{2} }}\left( {\tau - 1} \right)\left( {1 + \tau_{n} - \tau } \right)e^{{ - 4\frac{\tau - 1}{{\tau_{n} }}}} , \quad 1 \le \tau \le 1 + \tau_{n} $$

(8.253)

$$ I_{Bn} \,\approx\,\frac{{ - t_{n} \left| {p_{\text{n}} } \right|}}{2.2} $$

(8.254)

Fig. 8.7

Comparison of Friedland (solid line) and Friedland–Brode (dashed line) hybrid, with similar slope discontinuities. The canonical balanced B55 pulse is shown for comparison

The hybrid curve fits shown in Fig. 8.7 introduce another discontinuity at the zero crossing, where the canonical Brode function would have otherwise smoothly continued its oscillation. The solutions for the Friedlander-Brode hybrids are rather similar to the Friedlander hybrids, and may not warrant the extra effort in most situations.

Appendix 4. Selected Positive and Negative Pulse Properties for Detonations

This section uses the Kinney and Graham (1985) tables and formulas as a starting point, as they are a time-tested and often quoted standardized set of metrics that are available in the open literature. The aim of this chapter is to provide a family of fitting function with methods that are transparent, reproducible, and transportable to blast data sets that can provide two key parameter pairs: the peak overpressure and underpressures, and the positive and negative phase durations (or their ratios). Since it can be challenging to accurately capture the peak pressures when the rise times are fast, the impulse—the pressure integrated through time—can be a more robust metric and could substitute the peak overpressure. It is generally possible to estimate the overpressure from the impulse of Friedlander-type positive phases. As suggested in the main chapter, it may also be useful to consider secondary pulses when estimating pulse lifetimes as they can elongate the negative phase. Secondary pulses may help recognize single-charge high explosive (HE) from nuclear (NE) surface blasts (e.g., Gitterman 2013). Another useful parameter that could be better documented is the observed rise time (referenced to the data collection system’s sample interval and passband) from ambient to peak pressure. The rise time could be a useful discriminant for detonations and deflagrations at short ranges, as well as assist in identifying propagation effects at longer ranges.

4.1 Kinney and Graham (1985) and ANSI S2.20 (1983) Relations for Detonations

Kinney and Graham (1985), hereafter referred to as KG85, build their equations on Hopkinson scaling from World War I and Sachs scaling from World War II, which encapsulate the hydrodynamics of explosions near the source. It is important to recognize that KG85 assumes a modified Friedlander blast pulse shape for the positive phase. They tabulate key blast parameters for both HE and NE using consistent metric units and provide useful expressions that behave almost linearly at large ranges with acoustic propagation velocities and finite amplitudes with inverse range pressure scaling. Table 8.8 summarizes the reference values used in this chapter.

Table 8.8 Reference values for blasts

Some tabulated parameters and functional relations for blast waves are also provided by ANSI S2.2 for 1 ktonne NE. The pertinent values extending beyond the KG85 tables are also rescaled to 1 kg NE and presented in Table 8.9.

Table 8.9 ANSI S2.2 standard for a yield of 1 kilotonne (10⁶ kg) NE as a function of actual range R scaled to a 1 kg yield at an equivalent scaled range Z

The peak overpressure at 10 km from 1 ktonne NE is predicted to be the same as would be measured at 1 km from 1 tonne NE, or at 100 m from 1 kg NE.

There are issues with most published tabulated blast values and equations, and KG85 is no exception. Amongst the issues in the KG85 tabulated values are formulas, which are given as follows:

• 1. Typo in the 30 m HE propagation time t_a. It has been replaced by 80.3 ms.

• 2. Inconsistent tabulated and equations units. Some times are in milliseconds, impulses are provided in bars, and some of the equations state incorrect units. Here they have all been converted to meters, kilogram, seconds (MKS), with pressures in Pa.

• 3. The tabulated reference pressures are incompatible with the equations unless the reference pressure P is 1 bar, or alternatively, that the pressures are given in bars. This can be confirmed by comparing the tabulated values and equations as well as by comparing Table X and Table XI, Part A and B (D. Russel, personal communication).

• 4. Tabulated values for the positive impulse are not consistent with positive pressure, duration, and waveform parameter values (Guzas and Earls 2010). Specifically, the condition

$$ \frac{{2I_{p} }}{{p_{p} t_{p} }} \equiv M_{1} = \frac{{2\left( {\alpha_{p} - 1 + e^{{ - \alpha_{p} }} } \right)}}{{\alpha_{p}^{2} }} = \le 1 $$

(8.255)

Fig. 8.8

Scaled positive impulse M₁ inferred from the KG85 tables for HE (dots) and NE (line). The positive impulse can only exceed the triangular area (unit scaled moment) if the positive phase has substantial downwards concavity, in which case it would no longer resemble the classic concave-upwards positive Friedlander pulse. This suggests the tabulated impulse and waveform parameters should be treated with caution, in particular for NE. Note the substantial increase in HE variability past 100 m/kg^1/3

is not satisfied with either HE or NE tabulated values for the impulse (Fig. 8.8).

• 5. The HE and NE tables do not overlap in scaled range.

• 6. The HE tables extend to the quasi-linear acoustics range (unit Mach number, small overpressure relative to ambient), whereas the NE tables stop short.

• 7. The NE tables extend deeper into the near-field hydrodynamic range, the HE tables do not get as close.

• 8. The reference yield for HE is 1 kg, and the reference for NE is 1 ktonne.

• 9. The positive period values for NE do not match the ANSIS2.20 standard.

• 10. There is no information on the negative phase of the blast pulse.

Despite these inconveniences, KG85 is one of the most consistently used standards and is a primary reference in this chapter. Although most of the material in this appendix is derivative, some corrections to published expressions are presented and new simplifying expressions are introduced.

In the detonation literature in general, and KG85 in particular, distance is scaled by atmospheric density and energy release, and may be expressed as

$$ Z = f_{d} R_{m} /W^{1/3} $$

(8.256)

where R is distance in units of meters, $f_{d}$ is a transmission factor for distance, and W is the equivalent explosive mass or energy. Although this is the traditional way of expressing the scaling relations, mathematical purist would rightfully recoil at the dimensional mayhem it creates. A more dimensionally correct formulation of the scaled distance would be,

$$ \bar{Z} = f_{d} \frac{R}{{R_{0} }}\left( {\frac{W}{{W_{0} }}} \right)^{{ - \frac{1}{3}}} $$

(8.257)

where R is distance in units of $R_{0}$, $f_{d}$ is a transmission factor for distance, and W is the equivalent explosive mass or energy in units of $W_{0}$. This dimensionless scaled distance introduces the problem of not tracking the reference yield, which can lead to ambiguity and confusion. For the sake of continuity and ease of use, I refer to the traditional KG85 scaled distance to track the units, while the formulas use the more correct nondimensional form.

The reference yield is traditionally mass in kg of TNT for high explosives (HE) and kilotonnes of TNT for nuclear blasts (NE). In this work, kt is used as the abbreviation for 10³ metric tons (10⁶ kg) where space is limited. Equivalent explosive yields are referenced to the nominal energy produced by the detonation of TNT, where 1 kg of TNT yields 4.184 × 10⁶ J = 4.2 MJ, and one kilotonne yields = 4.184 × 10¹² J ~ 4.2 TJ (Table 8.8).

$$ Z_{HE} = f_{d} R_{m} /W_{kg}^{1/3} $$

(8.258)

$$ Z_{NE} = f_{d} R_{m} /W_{ktonnes}^{1/3} $$

(8.259)

Scaled distances can be converted from HE to NE values (and vice versa) from

$$ Z_{NE} = f_{d} R_{m} /\left( {W_{{\rm kg}} \frac{1\,kt}{{10^{6}\,{\rm kg}}}} \right)^{1/3} = 10^{2}\,Z_{HE} . $$

(8.260)

The dimensionless distance transmission factor accounts for changes in atmospheric density $\rho$ at the source location, and can be expressed in terms of static ambient pressure P and temperature T in Kelvins as

$$ f_{d} = \left( {\frac{\rho }{{\rho_{0} }}} \right)^{{\frac{1}{3}}} = \left( {\frac{P}{{P_{0} }}} \right)^{{\frac{1}{3}}} \left( {\frac{{T_{0} }}{T}} \right)^{{\frac{1}{3}}} . $$

(8.261)

At the reference yield, pressure, and temperature, the scaled distance is the actual distance. Time may also be scaled relative to the reference conditions by accounting for the change in density and sound speed, c, and has a transmission factor

$$ f_{\tau } = \left( {\frac{\rho }{{\rho_{0} }}} \right)^{{\frac{1}{3}}} \frac{c}{{c_{0} }} = \left( {\frac{P}{{P_{0} }}} \right)^{{\frac{1}{3}}} \left( {\frac{{T_{0} }}{T}} \right)^{{\frac{1}{3}}} \left( {\frac{{T_{0} }}{T}} \right)^{{ - \frac{1}{2}}} = \left( {\frac{P}{{P_{0} }}} \right)^{{\frac{1}{3}}} \left( {\frac{T}{{T_{0} }}} \right)^{{\frac{1}{6}}} . $$

(8.262)

KG85 expresses the scaled time as

$$ t_{Z} = t f_{\tau } /W^{1/3} . $$

(8.263)

As with the scaled distance, a more correct dimensional representation would be

$$ \bar{t}_{Z} = f_{\tau } \frac{t}{{t_{0} }}\left( {\frac{W}{{W_{0} }}} \right)^{{ - \frac{1}{3}}} $$

(8.264)

with $t_{0}$ a reference time scale, such as MKS seconds. Since it is possible to track the scaled time reference through the KG85 scaled distance, this work uses the nondimensional scaling in the time formulas.

As an example, the duration of the positive phase of a blast pulse may be expressed as

$$ t_{p} = \frac{{t_{ Z p} }}{{f_{\tau } }}W^{{\frac{1}{3}}} = t_{ Z p} W^{{\frac{1}{3}}} \left( {\frac{P}{{P_{0} }}} \right)^{{ - \frac{1}{3}}} \left( {\frac{T}{{T_{0} }}} \right)^{{ - \frac{1}{6}}} . $$

(8.265)

Its equivalent dimensionally correct (but more cumbersome) form for HE would be,

$$ \begin{aligned} t_{p} & = \left( {1\,s} \right)\bar{t}_{ Z p} \left( {\frac{{W_{{\rm kg}} }}{1 {\rm kg}}} \right)^{{\frac{1}{3}}} \left( {\frac{P}{{P_{0} }}} \right)^{{ - \frac{1}{3}}} \left( {\frac{T}{{T_{0} }}} \right)^{{ - \frac{1}{6}}} \\ & = \left( s \right)\bar{t}_{ Z p} \frac{{W_{{{\text{kg}}}}^{{1/3}} }}{{{\text{kg}}^{{1/3}} }}\left( {\frac{P}{{P_{0} }}} \right)^{{ - \frac{1}{3}}} \left( {\frac{T}{{T_{0} }}} \right)^{{ - \frac{1}{6}}} , \\ \end{aligned} $$

(8.266)

where the reference units are explicitly expressed. The nondimensional forms are seldom used in the literature, and it is usually left up to the reader to interpret and reconcile the various units and standards. The site corrections for time and range may be small (on the order of a few percent) for near-surface shots, but can be significant for the case of high-altitude blasts.

As discussed in Sect. 8.2, a gauge pressure $p\left( t \right)$ can be readily obtained from a differential pressure sensor from

$$ p\left( t \right) = p_{a} \left( t \right) - \left\langle {p_{a} } \right\rangle $$

(8.267)

where $p_{a} \left( t \right)$ is the instantaneous blast pressure and $\left\langle {p_{a} } \right\rangle$ the mean atmospheric pressure within the instrument passband. This process would be equivalent to demeaning and detrending a pressure record. The peak gauge overpressure $p_{p}$ is one of the primary metrics of blast waves, and it is defined as the difference in the peak dynamic pressure relative to the averaged ambient pressure $\left\langle {p_{a} } \right\rangle$. The KG85 tables provide a scaled peak positive pressure from the gauge pressure over the reference barometric pressure $P$, not to be confused with the averaged long-period pressure over an instrument’s passband $\left\langle {p_{a} } \right\rangle$,

$$ \bar{p}\left( t \right) = \frac{{p_{a} \left( t \right) - \left\langle {p_{a} } \right\rangle }}{P} = \frac{p\left( t \right)}{P},\quad \bar{p}_{p} = \frac{{p_{p} }}{P}, $$

(8.268)

As noted earlier in this section, the reference pressure $P = P_{0}$ at sea level in Table XI of KG85 is 1 bar = 1.0132510⁵ Pa.

Since the peak gauge overpressure is scaled by the ambient pressure,

$$ p_{p} = \bar{p}_{p} P = \bar{p}_{p} P_{0} \left[ {\frac{P}{{P_{0} }}} \right] $$

(8.269)

where there may be a substantial overpressure correction for shots above sea level due to the exponential decay rate of pressure with elevation.

There are various corrections that should be considered. If a blast is detonated near the surface, the blast energy is distributed over a half-space and a hemispherical magnification factor to the yield should be applied. The nominal magnification correction is a maximum factor of two for a perfectly reflecting surface but would be unity for a perfectly absorbing surface (Baker 1973, p. 121). Thus, the equivalent yield of a surface shot is at most twice that of a free-air blast for a perfectly reflecting surface (KG85, Chap. 8; Kim and Rodgers 2016), and is bounded by,

$$ W_{spherical} \le W_{hemisphere} \le 2\,W_{spherical} . $$

(8.270)

Guzas and Earls (2010) recommend a magnification factor of 1.8 due to ground absorption, whereas Baker (1973, p.128) reports a factor of 1.7 along with a thorough historical perspective with abundant primary references.

A second factor of two arises in the rough conversion of NE to HE. Since only ~50% of a nuclear explosion is converted to mechanical energy, the general practice (ANSI S2.20-1983) is to divide the NE yield by a factor of two, or multiply the equivalent chemical yield by a factor of two,

$$ W_{NE kt} \,\approx\, 2\,W_{HE\,kt}. $$

(8.271)

Thus, the 2013 Chelyabinsk meteor, with an estimated yield of ~500 kt of TNT (HE), would have an equivalent yield of 1 megatonne of NE. Without some knowledge of the source conditions it is possible to be off by a factor of 4 in yield estimates, for a cubed root scaling factor of ~1.6 in blast parameter variability.

Surface reflections can also add uncertainty in pressure measurements, in particular for ground observations from sources aloft. Guzas and Earls (2010) provide an excellent summary of this topic. The classical linear acoustic correction for normal incidence onto a perfectly reflecting surface is a factor of two in the incident pressure. Other possible sources of uncertainty include the amount of detonated explosive and its relative effectiveness at the time of detonation. ANSI S2-20 specifies a minimum variation of 10–20% in airblast characteristics due to uncertainties in effective detonation mass, even for nuclear explosions. More telling is a footnote in the same standard, where it states that the type of “ton” used is well within this error margin, as the difference between short, long and metric tons is at most 12%, and cubed root scaling would reduce the variability of blast parameters to 4%. It is not unusual for researchers and organizations to develop scaling relations specific to their environments and data sets (e.g., Gitterman and Hoffstetter 2012) to reduce some of this variability. A review of the manifold scaling relations in the literature is beyond the scope of this work.

The next section of this appendix evaluates (and sometimes modifies) some of the tabulated KG85 blast parameters for the positive phase as well as includes some expressions for the negative phase parameters. Only scaled distances with tabulated values for the positive wave parameter are considered.

4.2 KG85 Positive Pulse Properties

For high explosives (HE) the KG85 reference yield is 1 kg TNT. For a scaled distance $Z_{HE}$ of

$$ 0.952\,{{\rm m}/{{\rm kg}^{1/3}}} \le Z_{HE} \le 500\,{{\rm m}/{{\rm kg}^{1/3}}} , $$

(8.272)

the dimensionless scaled peak overpressure is

$$ \bar{p}_{p\,HE} = \frac{{808\left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{4.5}} \right)^{2} } \right]}}{{\sqrt {1 + \left( {\frac{{\bar{Z}_{HE} }}{0.048}} \right)^{2} } \sqrt {1 + \left( {\frac{{\bar{Z}_{HE} }}{0.32}} \right)^{2} } \sqrt { 1 + \left( {\frac{{\bar{Z}_{HE} }}{1.35}} \right)^{2} } }}. $$

(8.273)

For nuclear explosives (NE), the reference yield is 1 ktonne = 1 Mkg TNT. For a scaled distance $Z_{NE}$ of

$$ 35\,{\rm m}/{{\rm ktonne}^{1/3}} \le Z_{NE} \le 5000\,{\rm m}/{{\rm ktonne}^{1/3}} , $$

the dimensionless scaled peak overpressure is

$$ \bar{p}_{p\,NE} = 3.2 \times 10^{6} \left( {\bar{Z}_{NE} } \right)^{ - 3} \sqrt {1 + \left( {\frac{{\bar{Z}_{NE} }}{87}} \right)^{2} } \left[ {1 + \frac{{\bar{Z}_{NE} }}{800}} \right]. $$

(8.274)

The duration of the blast wave is defined by KG85 as the time between the passing of the shock front and the end of the positive pressure phase. The scaled duration of the positive pressure phase, in seconds (s), can be expressed as

$$ t_{p\,HE} = \left( s \right)\frac{{W_{{{\rm kg}}^{1/3}} }}{{ {{\rm kg}^{1/3}} }}\frac{{0.98\left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{0.54}} \right)^{10} } \right]}}{{ \left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{0.02}} \right)^{3} } \right]\left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{0.74}} \right)^{6} } \right]\sqrt {1 + \left( {\frac{{\bar{Z}_{HE} }}{6.9}} \right)^{2} } }} $$

(8.275)

for 0.952 m/kg^1/3 $\le Z_{HE} \le$ 500 m/kg^1/3,

$$ t_{p\,NE} = \left( s \right)\frac{{W_{kt}^{1/3} }}{{ kt^{1/3} }}\frac{{0.18\sqrt {1 + \left( {\frac{{\bar{Z}_{NE} }}{100}} \right)^{3} } }}{{\sqrt { 1 + \left( {\frac{{\bar{Z}_{NE} }}{40}} \right)} \left[ {1 + \left( {\frac{{\bar{Z}_{NE} }}{285}} \right)^{5} } \right]^{{\frac{1}{6}}} \left[ {1 + \left( {\frac{{\bar{Z}_{NE} }}{50000}} \right)} \right]^{{\frac{1}{6}}} }} $$

(8.276)

for 35 m/ktonne^1/3 $\le Z_{NE} \le$ 5000 m/ktonne^1/3.

As mentioned earlier, nondimensional scaling with units of seconds are implemented as minor KG85 modifications. The positive pulse duration reaches a stable constant value at relatively short ranges, which makes the pulse duration (and its related quantity, the peak period) a relatively stable parameter for yield estimations.

An empirical equation for the positive impulse per area was obtained independent of the KG85 data for chemical explosives. As discussed earlier in this section, the KG85 impulse equation does not work well with the other parameters, and exceeds the maximum gamma value expected from a triangular pulse. Higher self-consistency can be obtained from the waveform parameter $\alpha$

$$ I_{p} = \mathop \smallint \limits_{0}^{{\tau_{p} }} p_{1} \left( t \right)dt = \frac{{p_{p} t_{p} }}{2}\frac{{2\left( {\alpha - 1 + e^{ - \alpha } } \right)}}{{\alpha^{2} }} = \frac{{p_{p} t_{p} }}{2}M_{1} $$

(8.277)

where the positive $\alpha$ for HE is estimated after Guzas and Earls (2010) using

$$ \alpha_{HE} = \mathop \sum \limits_{i = 1}^{N} c_{i} Z_{HE}^{i - 1} . $$

(8.278a)

The interpolation coefficients for HE are given in Table 8.10, where the range is extrapolated beyond the KG85 tables assuming the wavenumber parameter remains constant. This is not expected to be the case for much greater ranges, although the tabulated values trend this way.

Table 8.10 Polynomial coefficients for the HE decay coefficient, after Guzas and Earls (2010)

The waveform parameter coefficients for NE are can be represented by

$$ \alpha_{NE} = c_{1} e^{{ - c_{2} Z_{NE} }} + c_{3} \frac{1}{{Z_{NE} }} + c_{4} $$

(8.278b)

with coefficient provided in Table 8.11 for the same scaled range as the KG85 NE pressure and pulse duration. Both the HE and NE positive impulse are computed for interpolation and their resulting well-behaved $M_{1}$ is shown in Fig. 8.9. This work does not use the tabulated KG85 impulse values as they are inconsistent with the other parameters.

Table 8.11 Coefficients for the NE decay coefficient. Same scaled range as the KG85 NE table

Fig. 8.9

Scaled positive impulse $M_{1}$ for HE (solid) and NE (dashed) derived from the interpolated waveform parameters

The accuracy and utility of the tabulated waveform parameter are not clear, as it is not readily obtained from field measurements and seems to have a very narrow range of applicability. The corrected waveform parameter tables are presented here for the sake of completeness.

A curve fit was performed for the acoustic travel time accounting for the propagation speed to transition to the acoustic limit at a range where the Mach number is close to unity. The expression for HE is

$$ t_{a\,HE} = \left( s \right)\frac{{W_{kg}^{1/3} }}{{ kg^{1/3} }}\frac{0.2}{{\left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{10}} \right)^{ - 0.75795} } \right]^{3.14415} }} , Z_{HE} \le 10\,{\rm m}/{{\rm kg}^{1/3}} $$

(8.279)

$$ t_{a\,HE} = \left( s \right)\frac{{W_{kg}^{1/3} }}{{ kg^{1/3} }}\left\{ {\frac{0.2}{{\left[ 2 \right]^{3.14415} }} + \frac{{\left[ {\bar{Z}_{HE} - 10} \right]}}{340.294}} \right\} , Z_{HE} > 10\,{\rm m}/{{\rm kg}^{1/3}} . $$

(8.280)

For the NE case

$$ t_{a\,NE} = \left( s \right)\frac{{W_{kt}^{1/3} }}{{ kt^{1/3} }}\frac{50}{{\left[ {1 + \left( {\frac{{\bar{Z}_{NE} }}{100}} \right)^{ - 0.43385} } \right]^{9.76} }} , Z_{NE} \le 1\,{\rm km}/{{\rm ktonne}^{1/3}} $$

(8.281)

$$ t_{a\,NE} = \left( s \right)\frac{{W_{kt}^{1/3} }}{{ kt^{1/3} }}\left\{ {\frac{50}{{\left[ {1 + \left( {10} \right)^{ - 0.43385} } \right]^{9.76} }} + \frac{{\left[ {\bar{Z}_{NE} - 1000} \right]}}{340.294}} \right\} , Z_{NE} > 1\,{\rm km}/{{\rm ktonne}^{1/3}} $$

(8.282)

For 1 tonne at a range of 1 km the predicted time difference between the shock and acoustic propagation times is small and may require high sample rates, accurate origin information, and reliable environmental data to validate. A comparison of the tables and formulas are presented in Figs. 8.10, 8.11 and 8.12.

Fig. 8.10

KG85 tabulated data (dots) and equations (solid) for HE

Fig. 8.11

KG85 tabulated data (dots) and equations (solid) for NE, converted to a scaled distance of m/kg^1/3. The divergence in the positive impulse is due to the incompatibility of this parameter with all other tabulated data

Fig. 8.12

ANSI tabulated data (dots) and extrapolated equations (solid) for NE, converted to a scaled distance of m/kg^1/3

4.3 Overpressure to Underpressure Ratios

Although addressed in the early literature (e.g., Sect. 5.9 of Bethe et al. 1958), the properties of negative phase of a blast have been largely ignored not only because it is considered secondary to the main positive blast phase, but also because it can be difficult to measure (Baker 1973). Recent interest in the effects of the negative phase (e.g., Rigby et al. 2014) have helped in evaluating and consolidating the literature on the topic. The negative pressure at sea level was estimated by Larcher (2008) relative to 1 kg HE by two linear approximations, and is consistent with the curves presented in Smith and Hetherington (1994). A lowpass filter functional form is implemented here to match the constant value at short scaled distances and the asymptotic linear slope at large scaled distances,

$$ \left| {\bar{p}_{n\,HE} } \right| = \frac{{10^{4}\,{\rm Pa}}}{P }\left[ {1 + \left( {\frac{{\bar{Z}_{HE} }}{3.5}} \right)^{2} } \right]^{{ - \frac{1}{2}}} \to \frac{1}{{\bar{Z}_{HE} }}\frac{{3.5 \times 10^{4}\,{\rm Pa}}}{P}. $$

(8.283)

The negative pressure for NE can also be estimated from the 1 kton DNA reference (Needham and Crepeau 1981) using a similarly coarse approximation at large scaled distances,

$$ \left| {\bar{p}_{n\,NE} } \right| = \frac{{3.35425 \times 10^{4}\,{\rm Pa}}}{{P_{0} }}\left[ {1 + \left( {\frac{{\bar{Z}_{NE} }}{80}} \right)^{2} } \right]^{{ - \frac{1}{2}}} . $$

(8.284)

The ratios of peak overpressure to underpressure for 1 kg of HE and NE at a reference range of $100\,{\rm m}/{{\rm kg}^{1/3}}$ are

$$ \frac{{p_{p\,HE} }}{{\left| {p_{n\,HE} } \right|}} \,\approx\, \frac{840}{350} \,\approx\,2.4, Z_{HE} = 100\,{\rm m}/{{\rm kg}^{1/3}} , $$

(8.285)

$$ \frac{{p_{p\,NE} }}{{\left| {p_{n\,NE} } \right|}} \,\approx\, \frac{503}{268} \,\approx\,1.9, Z_{NE} = 100\,{\rm m}/{{\rm kg}^{1/3}} . $$

(8.286)

The negative pulse durations present some challenges because of their vulnerability to secondary shocks and ground effects. I concentrate on estimating the ratio of the negative and positive pulse duration. For impulse-balanced blast pulses, this ratio is set by the waveform shape and the overpressure to underpressure ratio. The simplest example is for the triangular pulse. As discussed in the main text, an impulse-balanced triangular pulse must satisfy

$$ \frac{{t_{\Delta n} }}{{t_{\Delta p} }} = \frac{{p_{p} }}{{\left| {p_{n} } \right|}} $$

(8.287)

where the pressure ratio and the positive phase duration sets the negative phase duration. In other words, for simple impulse-balanced pulses of a prescribed shape, at most three blast parameters can be specified.

Similarly, the theoretical pulse duration ratio for the impulse-balanced hybrid modified Friedlander (HMF) pulse can be estimated from

$$ \frac{{t_{n} }}{{t_{p} }} = M_{1} \frac{\pi }{{e^{1} }}\frac{{p_{p} }}{{\left| {p_{n} } \right|}} \le \frac{\pi }{{e^{1} }}\frac{{p_{p} }}{{\left| {p_{n} } \right|}} \,\approx\,1.16\frac{{p_{p} }}{{\left| {p_{n} } \right|}} , $$

(8.288)

which is consistent with the triangular pulse approximation. Computed pressure and time duration ratios are shown in Fig. 8.13 as a function of scaled range using the KG84 values for the waveform parameter, which determine the first moment $M_{1}$ (Fig. 8.9) in Eq. 8.288. It should be noted that the KG85 values lead to overdampened negative phases (Figs. 8.5 and 8.7) for both HE and NE, and the negative phase duration shown in Fig. 8.13 is almost surely underestimated. The HE ratios reach a near-constant value after 20 m/kg^1/3, whereas the NE ratios appear to stabilize closer to 100 m/kg^1/3. This suggests that a reference range of 100 m/kg^1/3 would be reasonable for both HE and NE blasts.

Fig. 8.13

overpressure to underpressure ratio (upper panel) and negative to positive phase duration (lower panel) for HE (solid) and NE (dashed) using the lowpass filter approximation for the underpresure and the impulse balance for the negative phase duration

In the literature, Fig. 8.5 of ANSI 2.20 shows the positive and negative pulse durations. It is possible to estimate from the upper limit of the ANSI curve as $t_{p\,NE}$ ~ 0.52 s, $t_{n\,NE}$ ~ 0.96 s at a yield of 1 ktonne and a range of 10 km, which is equivalent to 1 tonne at 1 km, or 1 kg at 100 m,

$$ \frac{{t_{n\,NE} }}{{t_{p\,NE} }} = \frac{0.96}{0.52} \,\approx\,1.85 , Z_{NE} = 100\,{\rm m}/{{\rm kg}^{1/3}} , $$

(8.289)

corresponding to an overpressure to underpressure ratio of approximately two. The HE negative pulse duration at sea level is reported by Larcher (2008) as

$$ t_{n\,HE} = \left( s \right)\frac{{W_{kg}^{1/3} }}{{ kg^{1/3} }}1.39 \times 10^{ - 2} , 1.9\,{\rm m}/{{\rm kg}^{1/3}} < Z_{HE} . $$

(8.290)

Using the positive pulse duration at the reference range,

$$ \tau_{n\,HE} = \frac{{t_{n\,HE} }}{{t_{p\,HE} }}\,\approx\,\frac{0.0139}{0.0042} \,\approx\,3.31, Z_{HE} = 100\,{\rm m}/{{\rm kg}^{1/3}} , $$

(8.291)

which would correspond to a pressure ratio of approximately three, and is likely to be affected by ground effects and secondary pulse oscillations. On the other hand, $\tau_{n} = 3$ corresponds to a pulse lifetime $\tau_{e} = 4$, which is the effective spectral pseudo period of the balanced G77 (GT) pulse.

4.4 Synopsis

Substantial effort was placed in revisiting the KG85 and ANSI standards as starting points for the next iteration of blast parameter estimation. This work proposes a reference range and yield of 1 km for 1 tonne (100 m for 1 kg, 10 km for 1 ktonne) for HE and NE blasts. At this scaled range, shocked blast pulses propagate near acoustic speeds, and the overpressure is a fraction of atmospheric pressure at sea level, and decays roughly as the inverse distance.

The aim of this chapter is to illustrate different approaches to blast pulse characterization. The duration ratios appear to be less stable than the pressure ratios, and in the interest of simplicity and clarity a single-digit-precision reference peak overpressure to underpressure ratio of 2 is used consistently as a representation of HE and NE blasts, and is expressed as

$$ \left. {\frac{{p_{p} }}{{p_{n} }}} \right|_{ref} \equiv 2. $$

(8.292)

It is also possible to make some useful simplifying reductions of the KG85 relations near the scaled distance $Z_{HE} \sim100\,{\rm m}/{{\rm kg}^{1/3}}$, where geometrical spreading is near spherical. Near the yield-scaled reference range, but before the far field, we can express the overpressure and positive phase duration (or pseudo period) as

$$ \frac{{p_{p} }}{{p_{ref} }} = \frac{{R_{ref} }}{R}\left( {\frac{W}{{W_{ref} }}} \right)^{{\frac{1}{3}}} , $$

(8.293)

$$ \frac{{t_{p} }}{{t_{ref} }} = \left( {\frac{W}{{W_{ref} }}} \right)^{{\frac{1}{3}}} , $$

(8.294)

where the reference values could be provided by Table 8.1 or Table 8.6. These simplifications should provide reasonable values near the reference range ± a factor of two, and may be practical for rapid first-order estimates when losses due to spherical spreading are dominant.

Appendix 5. Minor Uncle and Distant Image 2 kt Case Studies

The Distant Image and Minor Uncle tests were performed at White Sands Missile Range (WSMR), New Mexico, approximately 6.5 km south from the ground zero of Trinity. Trinity had an equivalent yield of ~20 kt TNT, and the Distant Image (DI) and Minor Uncle (MU) events were approximately one order of magnitude smaller. The MU had the same ground zero as DI after the previous crater was filled in. The apparent MU crater radius was 42 m with an apparent depth of 22 m below surface (R. Reinke, personal communication). The center of the crater can be estimated from Google Maps at 33.619953°N and 106.477619°W at an elevation ~1500 m above sea level. The origin times and location for the DI for MU tests are presented in Table 8.12.

Table 8.12 Origin times and locations for Distant Image and Minor Uncle, courtesy of R. Reinke

There may be some uncertainty in the actual weight of detonated explosives. For the purposes of this discussion, it is assumed the ammonium nitrate, fuel oil (ANFO) is completely and simultaneously detonated. The next step is to convert the yield of the surface hemispherical charges in long (imperial) tons to the equivalent weight of a spherical charge in free air for comparison with the KG85 tables. The conversion from imperial tons to metric tonnes is straightforward, but there is some uncertainty on the relative effectiveness (RE) factor of ANFO with respect to TNT. At the time of the tests, an ANFO RE of 0.8 was recommended, whereas the current recommended ANFO RE is closer to 0.83 (R. Reinke, personal communication). An additional source of variability is the magnification factor from the reflected energy off the ground. In a perfectly reflecting boundary, the magnification factor doubles the yield. However, the resulting crater is evidence of energy losses to the ground, and in practice, a magnification factor of 1.8 is recommended (e.g., Guzas and Earls 2010). Table 8.13 below presents some upper and lower estimates for yield conversion of a surface hemispherical charge to an equivalent spherical charge in free air. The exact values are not as important as the range of values and their effect on the scaling of the blast pulse parameters.

Table 8.13 Estimated upper and lower bounds for the equivalent spherical DI and MU yields in ktonnes

In addition to the yield scaling, the temperature and pressure at ground zero will affect the range scaling, peak pressure, positive pulse duration, and pulse propagation time (KG85). The KG85 tables are provided relative to a reference pressure of 1013.25 mbar (1013.25 × 102 Pa) and a temperature of 288.15 K. The standard atmosphere pressure and temperature at a height 1500 m above sea level are 845.6 mbars and 278.4 K. However, the measured pressure and temperature for Minor Uncle were 853 mbar and ~293 K (20 C), respectively, which was unseasonably cool due to a frontal passage (R. Reinke, personal communication). The average temperature at White Sands in June at ~10 AM local time is expected to be closer to 303 K (~86 F) as the ground warms up in early summer. For the purposes of illustration, the measured ambient pressure and temperature are used for Minor Uncle, and the standard atmosphere pressure and average temperature are used for Distant Image to compare the effects local weather variability on the blast parameter estimates. Table 8.14 summarizes estimated site and yield corrections from MU and DI, as well as percent errors that may be introduced by rounding up in yield or not implementing the corrections. The last column of Table 8.14 only keeps two significant figures. The peak overpressure correction seems to be most vulnerable due to its linear dependence on the ambient pressure, in contrast to the cubed root dependence of the temporal and spatial correction factors. The distance correction is the next most vulnerable, with the time transmission factor being the least sensitive. The differences between the MU and DI site-dependent correction factors are small, which is reassuring as it is not always possible to have access to the local meteorology during unscheduled detonations.

Conservative predicted pulse parameters from KG85 could be evaluated by using a magnification factor of 1.8 and an ANFO RE factor of 0.8. However, it would be reasonable to represent both the MU and DI tests as detonations with an equivalent TNT yield of 1.8 ktonnes of TNT (with a ~4% error) at a 1500 m asl elevation with a standard atmospheric pressure of 845.6 mbar and an average temperature of 303 K at White Sands between the hours of 9–10 AM local time in June. As shown in Table 8.14, this would correspond to a peak overpressure correction of 0.84, a distance transfer function fd = 0.93, and a time transfer function ft = 0.95. Thus, at the end of this analysis, the 2 kt surface shot scales to ~2 ktonnes of TNT in free air (11% error). This variability is well within the ANSI expectations. In the main text, I refer to the combination of these two shots as the 2 kt case study.

Table 8.14 Uncertainty estimates for yield scaling and site corrections

Representative waveforms for 2 kt case study are shown in Figs. 8.1 and 8.2 in the main text. They are scaled by the peak pressure and the positive pulse duration. To construct this scaling, it is necessary to estimate the blast arrival time ta, the peak overpressure pp, and the positive pulse duration tp. The observed parameters can be yield-corrected and compared directly with standard blast parameter tables and formulas to assess their predictive accuracy and design refinements. The rise time to reach the overpressure, coupled with the data collection sample rate, would also be of interest, and could use the pulse onset threshold needed to estimate the arrival time. The negative pulse parameters can also be computed and expressed in terms of the peak overpressure and the positive pulse duration. Pertinent negative pulse parameters include the scaled pressure and time of the peak underpressure, tmin, the underpressure duration tn, the total pulse lifetime te, and the scaled pressure threshold used to evaluate the pulse lifetime. With these parameters in place, it is possible to numerically compute positive and negative impulses, evaluate pulse shape parameters, as well as refine equivalent energy estimates for individual blast signatures.

The 2 kt case study also brings up another important issue of cubed root scaling and the differences between the 1 kg and 1 ktonne tables. Table 8.15 shows a comparison between the observed and predicted site-corrected KG85 overpressure and peak period for 2 ktonnes HE, and Table 8.16 shows the same comparison for 4 ktonnes NE, which should be equivalent to 2 ktonnes NE. For the large DI and MU yields, the NE tables provide closer agreement to the measured 2 ktonne yield, in particular for the positive phase duration. Variability in the positive pulse duration is to be expected due to the different burn times of NE and ANFO HE (e.g., Petes et al. 1983), as well as between TNT and ANFO (Reinke, personal communication). Table 8.17 shows that the predicted ratios between the peak overpressure and underpressure are consistent for the HE and NE predictions in Appendix 4 for the scale ranges pertinent to this case study, as was the case for ANFO (Petes et al. 1983).

Table 8.15 KG85, 2 ktonne HE

Table 8.16 KG85, 4 ktonne NE (2 ktonne HE equivalent)

Table 8.17 Negative phase properties