THE SHAPE OF THE BLAST
WAVE: STUDIES OF THE
FRIEDLANDER EQUATION
by
John M. Dewey
Dewey McMillin & Associates
1741 Feltham Road, Victoria BC Canada
(www.blastanalysis.com)
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Characteristic Shape
Hydrostatic Overpressure (kPa)
80
60
Ps
40
20
tt++
0
-20
-200
0
200
400
600
Time (ms)
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800
1000
Friedlander 1946
Friedlander suggested that the classic
pressure time-history could be described by
P  PS e
t
 
t
t 

1   
 t 
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Press. vs Time 523.5m SB ANFO 2.205 kt (MINOR UNCLE)
Hydrostatic Overpressure (kPa)
80
60
40
20
0
-20
-200
0
200
400
Time (ms)
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600
800
1000
Friedlander Fit
80
Overpressure (kPa)
60
Least squares fit to Friedlander equation
Pressure gauge signal
40
20
0
-20
-200
0
200
400
Time (ms)
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600
800
1000
Density vs Time
0.8
-3
Density (kg m )
0.6
Measured Density
Least Squares fit to Friedlander Equation
0.4
0.2
0.0
-0.2
0
100
200
Time (ms)
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300
400
500
Total (Pitot) Pressure vs Time
80
Total Overpressure (kPa)
60
Friedlander Fit
Measured Total Pressure
40
20
0
-20
0
100
200
Time (ms)
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300
400
500
Dynamic Pressure (½ρu2) vs Time
16
14
12
Dynamic Pressure (kPa)
10
Friedlander Fit
Dynamic pressure
8
6
4
2
0
-2
-4
0
100
200
300
Time (ms)
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400
500
Friedlander Fails at Higher
Overpressures
Hydrostatic Overpressure (atm)
4
3
Friedlander Fit
AirBlast Overpressure (FF1.3m)
Modified Friedlander Fit
2
1
0
-1
0.0
0.5
1.0
Time (ms)
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1.5
2.0
P  PS e t (1 
t
)
t 
Modified Friedlander Equation
Additional coefficient α
t
t
P  PS e (1   )
t
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t
t
t 

I    PS e  t  1     0.368 PS t 
 t 
0
Properties of Friedlander Equation
Impulse in positive phase
t
I    PS e
0

t
t

t

1  
 t


  0.368 PS t

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Properties of Friedlander Equation
Total Impulse

I Tot   PS e
0
t

t

t

1  
 t
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
0

Properties of Friedlander Equation
Minimum Pressure
t min  2t
Pmin  PS e
2

 0.135PS
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Blast Wave Profile
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
Radius (m)
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1.8
1.9
2.0
2.1
Blast Wave Profile
Low Overpressures
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
Radius (m)
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5.0
5.2
5.4
5.6
Particle Tracer Photogrammetry
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Spherical Piston Path
MISERS GOLD
2.445 KT ANFO
SURFACE BURST
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Friedlander Fit to Piston Path
100
Inverse Radius (m)
80
Friedlander Fit
Misers Gold Piston Path
60
40
20
0
0
100
200
Time (ms)
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300
400
Conclusions
1. The time histories of the physical
properties of centered blast waves are
well described by the Friedlander
equation at peak overpressures less than
1 atm.
2. The wave profiles of the physical
properties are well described by the
Friedlander equation at peak
overpressures greater than 1 atm.
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Conclusions
3. The trajectory of the spherical piston that
drives a centered blast wave has the
form of the Friedlander equation
4. Are there physical reasons why it should
be expected that a point source release
of energy would generate a spherical
piston path of this shape?
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Properties of Friedlander Equation
Relaxation Time t*
PS
t
  
 PS t 1  
e
 t
t
*
*

t  2.31t
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*


