THE SHAPE OF THE BLAST WAVE: STUDIES OF THE
FRIEDLANDER EQUATION
John M. Dewey
Dewey McMillin & Associates, 1741 Feltham Road, Victoria, BC V8N 2A4, Canada
Professor Emeritus, Dept. Physics & Astronomy, University of Victoria, BC Canada
ABSTRACT
The time-histories of pressure and other physical properties of centred blast waves, have a characteristic shape.
Friedlander (1946) suggested a simple relationship, now known as the Friedlander equation, which accurately
describes this shape over a wide range of shock strengths. Introduction of an additional coefficient allows the
modified Friedlander equation to describe blast wave time histories over an even wider range.
This paper explores the ranges over which the modified and un-modified Friedlander equations can be used to
describe the various physical properties of blast waves. It also shows the application of these equations to blast
wave radius-profiles at various times after detonation.
A number of the unique characteristics of the Friedlander equation are explored. It has long been suggested that
the simplicity of the equation indicates a fundamental relationship with the explosive process. This possibility is
investigated, and an explanation is presented, relating the Friedlander equation to the spherical piston path that
drives the blast wave from a centered explosion.
THE FRIEDLANDER EQUATION
The time histories of hydrostatic overpressure and several other physical properties of
centered blast waves, as measured by gauges at fixed locations, have a characteristic shape.
At the arrival of the primary shock there is a virtually instantaneous increase in pressure,
followed by a modified exponential-type decay such that after a time, t+, the pressure returns
to the ambient value and moves into the negative phase. Friedlander (1946) suggested that
this characteristic shape could be described by an equation of the form
P  PS e

t
t
t 

1    ,
 t 
(1)
where P is the overpressure at a fixed location, PS is the peak overpressure immediately
behind the primary shock, t is the time after the arrival of the primary shock at that location
and t+ is the positive duration. This simple two-coefficient relationship has long been known
to be an excellent descriptor of overpressure time-histories produced by a variety of
explosives over a wide range of peak overpressures.
This is illustrated in Fig. 1, which shows, in red, the overpressure time history
measured using a piezo-electric transducer at a distance of 523.5 m from the centre of a
surface-burst, hemispherical, 2.205 ktonne ammonium nitrate – fuel oil explosion (MINOR
UNCLE) (Slater et al, 1995). This gauge record has been chosen for illustrative purposes
because independent measurements were also made of the density-time and the total-pressure
time-histories at the same location. It is rare, if not unique, to have such independent gauge
measurements of several physical properties at the same location in the same blast wave.
80
Overpressure (kPa)
60
Least squares fit to Friedlander equation
Pressure gauge signal
40
20
0
-20
-200
0
200
400
600
800
1000
Time (ms)
Figure 1. Measured pressure time history, in red, at a distance of 523.5 m from the centre of a surface-burst,
hemispherical, 2.205 ktonne ammonium nitrate – fuel oil explosion (Slater et al, 1995). The black curve is the
least squares fit of these data to the Friedlander equation (1).
Fig. 1 clearly illustrates the excellent way in which the Friedlander equation can be used to
describe the pressure time-history, at that pressure level, and is the preferred way of
determining the peak pressure and the positive duration from a noisy gauge signal.
The density and total (pitot) pressure time-histories were also measured at the same location,
and from the hydrostatic and total pressures the dynamic pressure (½ρu2) time-history was
calculated. Each of these time histories are also fitted by least squares to the Friedlander
equation, and the results shown in Figs 2, 3 and 4. The manner in which the Friedlander
equation is able to describe these time histories, using only two coefficients, PS and t+, is
typical for this strength of blast wave produced by a variety of high explosives, both air- and
surface-burst.
0.8
Density (kg m-3)
0.6
Measured Density
Least Squares fit to Friedlander Equation
0.4
0.2
0.0
-0.2
0
100
200
300
400
500
Time (ms)
Figure 2. Density time-history, in red, measured by Slater et al (1995) using a β absorption gauge. The black
curve is the least squares fit of these data to the Friedlander equation (1).
80
Total Overpressure (kPa)
60
Friedlander Fit
Measured Total Pressure
40
20
0
-20
0
100
200
300
400
500
Time (ms)
Figure 3. Total (Pitot) pressure, in red, measured by Slater et al (1995). The black curve is the least squares
fit of these data to the Friedlander equation (1).
16
14
Friedlander Fit
Dynamic pressure
12
Dynamic Pressure (kPa)
10
8
6
4
2
0
-2
-4
0
100
200
300
400
500
Time (ms)
Figure 4. Dynamic pressure (½ρu2), in red, calculated from the hydrostatic and total pressures measured by
Slater et al (1995). The black curve is the least squares fit of these data to the Friedlander equation (1).
THE MODIFIED FRIEDLANDER EQUATION
At peak overpressures above about one atmosphere, the Friedlander equation is no longer
able to describe accurately the pressure time-histories, and it is necessary to introduce another
coefficient, α, in a modified version of the equation, viz.
P  PS e t (1 
t
).
t
(2)
(It may be noted that some authors use β/t+ in place of α in the modified Friedlander equation.
If the same data are fitted to the two equations, it will be found that    t  , and that the
fitted coefficients PS and t+ are not identical in the two cases. It is recommended that the form
of (2) be used as the modified Friedlander equation.)
Fig. 5 shows an attempt to fit the Friedlander equation to the pressure time-history, derived
from the AirBlast1 data base for the free-field explosion of 1 kg TNT at a distance of 1.3m
(3.7 atm). The green curve shows the same data fitted to the modified Friedlander
equation (2). The modified Friedlander equation appears to provide a good description of
overpressure time-histories up to a peak hydrostatic overpressure level of about 7 atm, and is
probably the best method of obtaining the peak pressure and positive duration from a noisy
gauge signal, as illustrated in Fig. 5.
1
AirBlast is an interactive data base of blast wave properties provided by Dewey McMillin & Associates.
Hydrostatic Overpressure (atm)
4
3
Friedlander Fit
AirBlast Overpressure (FF1.3m)
Modified Friedlander Fit
2
1
0
-1
0.0
0.5
1.0
1.5
2.0
Time (ms)
Figure 5. The data points, in red, show the hydrostatic overpressure time history derived from AirBlast at
1.3 m from a free-field 1 kg TNT explosion. The black curve is the Friedlander (1) fit to those data. The green
curve is the modified Friedlander (2) fit to the same data.
SOME PROPERTIES OF THE FRIEDLANDER EQUATION
In addition to its value as a descriptor of the pressure time-history of the blast wave from a
free-field or surface-burst explosion in the region below one atmosphere peak hydrostatic
overpressure, the Friedlander equation has some other interesting and valuable properties.
The impulse during the positive phase can be found by integrating the Friedlander equation
over that period, viz.
t
t
t 

I    PS e  t  1     0.368PS t  .
 t 
0
(3)
The result of this simple relationship shows excellent agreement with the positive impulse
calculated by the numerical integration of gauge signals.
It may be of interest to note that the total impulse of a Friedlander blast wave is zero, viz.

I Tot   PS e
0

t
t
t 

1     0 ,
 t 
(4)
so that the impulse of the positive phase equals that of the negative phase. In practice, this
does not occur because of the arrival, in the negative phase, of the secondary and subsequent
shocks.
Thornhill (1959) points out another interesting property of the Friedlander equation. The
relaxation time, t*, of an exponential decay is defined as the time for the property to decay to
1/e of its peak value. Appling this in the Friedlander equation gives
t*
PS
t*
  

 PS t 1  
e
 t

 .

(5)

 ,

(6)
Therefore,
1 e
 t*
 1
 t





 t*
1  
 t
and the solution of (6) is
t   2.31t * .
(7)
This can be a useful relationship when interpreting a noisy gauge signal from which it is
difficult to make an accurate measurement of the positive duration. The relaxation time, t*,
may be easier to measure, and can be used in (7) to calculate t+. Similarly, by combining (3)
and (7), the positive impulse can be calculated from
I   0.85 PS t * .
(8)
Thornhill (1956) suggests that the valid region for application of the Friedlander equation can
be extended above the peak hydrostatic overpressure level of one atmosphere by replacing t+
in the Friedlander equation (1) by 2.31t* from (7), but this is not correct because the two
equations are algebraically identical.
The time of the minimum pressure in the negative phase of a blast wave that conforms to the
Friedlander equation can be found by differentiating the equation and equating it to zero.
This gives
tmin  2t   4.6t * ,
(9)
which gives the minimum overpressure in the negative phase as
Pmin   PS e 2  0.135 PS .
(10)
BLAST WAVE DISTANCE PROFILES
The pressure time-history, measured at a fixed location in a blast wave, is a somewhat
inadequate descriptor of the wave in that it is an Eulerian measurement of a Lagrangian
process. In a uniform, non-decaying wave the time-history of a process can be used to
describe the wave profile, but this is not the case for a decaying wave such as a blast wave.
The wave profile of a blast wave, namely the variation of a physical property with distance at
a fixed time, can be measured by photogrammetric techniques such as the particle trajectory
analysis (PTA) method described by Dewey (1971) and Dewey & McMillin (1981, 1987).
This technique was use to construct the AirBlast data base, from which blast wave distance
profiles can be obtained.
Fig. 6 shows, in red, the hydrostatic overpressure wave profile, obtained from AirBlast, at
2 ms after the free-field detonation of 1 kg TNT. The black curve is the Friedlander fit to
these data, where the Friedlander equation as been rewritten in terms of distance rather than
time, viz.
r
  S e r (1 

r
),
r
(11)
where r = RS – R, RS is the radius of the primary shock, R the radius of any other point in the
wave profile, and r+ is the distance behind the primary shock where the overpressure, P, first
returns to its ambient value.
2.0
1.8
Overpressure (atm)
1.6
1 kg TNT FF at 2 ms
Friedlander Fit
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
Radius (m)
Figure 6. Hydrostatic overpressure wave profile at 2 ms after a free-field detonation of 1kg TNT. The red
points are derived from AirBlast, and the black curve is the least squares fit to the Friedlander equation (11).
It will be noted that the peak overpressure is above 1 atm, the upper limit of the Friedlander
equation for time histories. In the case of wave profiles, the Friedlander equation is valid for
peak overpressures above 1 atm, but not below. This is illustrated in Fig. 7.
Hydrostatic Overpressure (atm)
0.4
0.3
1 kg TNT FF at 10 ms
Modified Friedlander Fit
Friedlander Fit
0.2
0.1
0.0
-0.1
4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
Radius (m)
Figure 7. Hydrostatic overpressure wave profile at 10 ms after a free-field detonation of 1kg TNT. The red
points are derived from AirBlast. The black and green curves are, respectively, the least squares fits to the
Friedlander and modified Friedlander equations
Wave profiles with peak overpressures below 1 atm can no longer be described by the
Friedlander equation, but are well described by the modified equation, written as
  S e r (1 
r
),
r
(12)
where r and r+ are defined above, and PS, α and r+ are the fitted coefficients.
DISCUSSION
Several authors, including Thornhill (1956), noting that the simple Friedlander equation, with
only two fitted coefficients, is such an excellent descriptor of the blast waves produced by a
variety of free-field and surface-burst explosives, have asked if there may not be a more
fundamental relationship between the physical processes of an explosion and this equation,
i.e. that it is not merely an empirical relationship.
Taylor (1946) suggested that any centered blast wave can be produced by a spherical piston
with an appropriate radius time trajectory. That trajectory can be determined experimentally
by high speed photogrammetry of a flow tracer, i.e. a smoke puff or trail established close to
an explosive source shortly before initiation. This technique has been used extensively during
the past fifty years to reconstruct the blast wave properties produced by high explosives
ranging from a few kilograms to several kilotons. Fig. 8 shows the particle trajectory used to
reconstruct the blast wave from a hemispherical surface-burst explosion of 2,445 tons of
ANFO (Operation MISERS GOLD).
Figure 8. The upper curve is the trajectory of a smoke tracer originally formed at about 150 m from the center
of a 2,445 ton ANFO explosion. The points on the curve were measured and used as the spherical piston path to
reconstruct the physical properties of the blast wave.
The piston-path points in Fig. 8 have been rotated and fitted by least squares to the
Friedlander equation, as shown in Fig. 9. It is now clear that the spherical piston which drives
a centered blast wave has the form of the Friedlander equation, and thus, it should not be
unexpected that the resulting wave profiles and time histories also have that form. Future
work will investigate the form to be expected from an expanding pressurized sphere.
100
Inverse Radius (m)
80
Friedlander Fit
Misers Gold Piston Path
60
40
20
0
0
100
200
300
400
Time (ms)
Figure 9. Points from the spherical piston path of MISERS GOLD fitted by least squares to the Friedlander
equation.
In order to obtain a suitable orientation for the points shown in Fig. 9, the origin of the piston
path was selected as the radius and time when the primary shock first reached the smoke
tracer, and the resulting radii were subtracted, arbitrarily, from 80 m. The “ambient” line was
determined as an additional coefficient in the Friedlander fit, which then appears as
R  R0  Rmax e

t
t
t

1  
 t

,

(13)
where R is the inverse radius defined above, and R0, Rmax and t+ are the fitted coefficients.
ACKNOWLEDGEMENTS
The author gratefully acknowledges Doug McMillin, who first observed the Friedlander form
of the spherical-piston path.
REFERENCES
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Proc. Roy. Soc., A 324, 275-299.
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Waves reflected from Real and Ideal Surfaces, Can. J. Phys., 59, 10, pp 1380-1390.
Dewey, J. M. & McMillin, D. J., 1987, Numerical Reconstruction of the Flow Field in Experimentally
Observed Mach Stem Blast Waves, Proc. 10th Int. Symp. Military Appl. Blast Simulation, MABS
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Symp. Rept., Defense Nuclear Agency, POR 7453-4, Vol. 4, 2, 1-98.
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273-292.
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