THE RANKINE-HUGONIOT EQUATIONS: THEIR EXTENSIONS AND
INVERSIONS RELATED TO BLAST WAVES
J. M. Dewey
Dewey McMillin & Associates Ltd
1741 Feltham Road, Victoria, BC V8N 2A4, Canada
(dewey@blastanalysis.com)
ABSTRACT
The Rankine-Hugoniot equations were developed in their original forms independently by
Rankine (1870a, b) and Hugoniot (1887, 1889). The equations describe the relationships
between the physical properties in the two possible states of a moving compressible gas for
which mass, momentum and energy are conserved. The contact surface between these two
states is a shock front. In their original forms, the equations described the relationships for a
supersonic gas passing through a stationary shock into a subsonic state. For blast wave
applications, the equations are transformed to describe the physical properties of the gas
behind a shock moving into a stationary ambient atmosphere, in terms of the Mach number of
the shock.
Assuming the ratio of specific heats of air to be 1.4, the Rankine-Hugoniot equations provide
exact values of the physical properties behind the primary shock of a blast wave, with
accuracy such that this is the preferred method to calibrate electronic transducers. For
overpressures above about 10 atm, additional degrees of freedom of the air molecules are
excited, and the ratio of specific heats must be modified to allow for these real-gas effects.
The most commonly used forms of the equations relate the hydrostatic overpressure, density
and particle velocity to the shock Mach number. In this paper, these equations are extended
to give eleven different physical properties in terms of the ratio of specific heats, γ, and for
γ = 1.4. In the latter case, the equations are also inverted to provide the shock Mach number,
and thus the other ten properties, in terms of a specified physical property.
To simplify the use of these relationships, they have been incorporated into a series of
spreadsheets that provide all of the physical properties behind a shock in terms of any
specified property. The properties may be expressed in terms of the ambient conditions ahead
of the shock, or in Imperial or SI units. The spreadsheets, which will be demonstrated, can be
used on any desktop, laptop or palmtop computer, and are available for licensing.
INTRODUCTION
The Rankine-Hugoniot equations were developed in their original forms independently by
Rankine (1870a, b) and Hugoniot (1887, 1889). The equations describe the relationships
between the physical properties in the two possible states of a moving compressible gas for
which mass, momentum and energy are conserved. The contact surface between these two
states is a shock front. In their original forms, the equations described the relationships for a
supersonic gas passing through a stationary shock into a subsonic state. For blast wave
applications, the equations are transformed to describe the physical properties of the gas
behind a shock moving into a stationary ambient atmosphere, in terms of the Mach number of
the shock.
The Rankine-Hugoniot relationships have been in use for more than a century, and subject,
therefore, to intense evaluation. All such evaluations have demonstrated the absolute validity
of the relationships, assuming the correct value of the ratio of specific heats has been used.
The relationships have the same order of reliability as the classical equations of
thermodynamics. As a result, the most reliable method of calibrating gauges and transducers
used to measure the physical properties of blast waves, is to measure the shock speed using
two simple time-of-arrival detectors, calculate the shock Mach number, and then use this in
the appropriate Rankine-Hugoniot relationship to determine the change of the physical
property across the shock. This value can then be related to the output of the transducer.
For the study of blast waves, the primary shock usually is moving into the air of the ambient
atmosphere, which can be considered as a perfect gas with a ratio of specific heats, γ = 1.4.
When the overpressure ratio across the primary shock is greater than about 10 atmospheres,
corresponding to a shock Mach number of about three, additional degrees of freedom are
excited in the air molecules, and the ratio of specific heats can no longer be considered
constant. For shocks with a Mach number greater than 3.5, i.e. hydrostatic overpressure of
13 atm (193 psi, 1330 kPa), this assumption leads to an error of more than 1%. For shocks
with a Mach number greater than 5.5, i.e. hydrostatic overpressure of about 34 atm (500 psi,
3458 kPa), the error is more than 5%.
Many of the Rankine-Hugoniot relationships relating the ratio of the physical properties
across a shock to the shock Mach number, can be inverted to provide the shock Mach number
as functions of the properties ratios. These inverse relationships are useful when one of the
physical properties behind a shock is known, e.g. the hydrostatic overpressure, and it is
desired to find other properties at that point, such as density, temperature, or local Mach
number. The inverse relationships are presented below.
THE ORIGINAL RANKINE-HUGONIOT RELATIONSHIPS
(1)
(2)
P1
P2
ρ1
ρ2
w1 (M1>1)
w2 (M2<1)
Figure 1. A perfect gas moving through a stationary discontinuity from state 1 to state 2. The
relevant physical properties of the gas are: hydrostatic pressure P; density ρ, and particle
speed w.
Consider a perfect gas moving in a cylinder of constant cross-section and passing from state 1
to state 2 through a stationary discontinuity, as shown in figure 1. In state 1 the hydrostatic
pressure, density and particle speed are P1, ρ1 and w1, respectively, and w1 is supersonic. In
state 2 the corresponding properties are P2, ρ2 and w2, and w2 is subsonic.
The equations representing the conservation of mass, momentum and energy across
the discontinuity may be expressed, respectively, as
P2
2
2 w2  1w1 ,
(1)
P2   2 w22  P1  1w12 , and
(2)
 e2 
1 2 P1
1
w2 
 e1  w12 ,
2
1
2
(3)
where e is the internal energy of the gas, per unit volume.
The enthalpy of the gas, h, is defined as
h
P

 e.
(4)
For a thermally and calorically perfect gas
P  RT , and
h

 1
RT ,
(5)
(6)
where R is the universal gas constant and T the absolute temperature.
Using (1) to (6), the following relationships can be derived,
1    1 


 2    1 
, and
1   1

2   1
(7)
  1 P2

2
  1 P1
w

 1.
1   1 P2  1 w2
  1 P1
(8)
P2

P1
1
These relationships may also be written in terms of the Mach number of the flow in region 1,
M1, where
M1 
w1
, and
a1
(9)
a1, the speed of sound in region 1 is given by
a12  
P1
1
.
(10)
THE RANKINE-HUGONIOT EQUATIONS FOR A BLAST WAVE SHOCK
At this point it is convenient to change the nomenclature for the situation in which a shock,
with speed VS , is moving into an ambient atmosphere at rest. Suffix 0 will be used to identify
the physical properties of the ambient gas, and suffix S to identify those immediately behind
the shock, as shown in Figure 2. For most blast waves, these "peak" values decay rapidly in
an almost exponential fashion, as functions of both time and distance.
MS=VS/a0
(S)
(0)
PS
P0
ρS
ρ0
uS
u0=0
Figure 2. Configuration of a shock moving with speed VS into a stationary gas in
state 0. The gas behind the shock is in state S. P, ρ, and u are the hydrostatic pressure,
density and particle speed, respectively. The shock Mach number MS = VS/a0, where a0 is the
ambient sound speed
Applying these changes to (7) to (10) gives,
PS 2M S2    1

, and
P0
 1
(11)
  1M S2 .
S

 0   1M S2  2
(12)
Hydrostatic Pressure
The hydrostatic pressure is the pressure measured by a transducer moving with the flow, or
the pressure measured by a transducer that is side-on to the flow such that it senses no
component of the flow normal to the transducer. It is the pressure caused by the vibrational
motion of the gas molecules, only. Some texts refer to this as the static pressure, but here the
term hydrostatic pressure is used to emphasize the difference from the total or stagnation
pressure. The hydrostatic pressure in excess of the ambient pressure is the hydrostatic
overpressure.
When describing blast waves, hydrostatic pressure is usually measured in units of kiloPascals
(kPa), pounds-weight per square inch (psi) or non-dimensionally in atmospheres (atm). An
informal tripartite (US, UK and Canada) agreement recommended that the standard
atmosphere to be used when describing blast waves is that at NTP (Normal Temperature and
Pressure), viz. 15 C (288.16 K) and 101.325 kPa (14.696 psi). This was considered to be
more appropriate than STP (Standard Temperature and Pressure) for which the standard
temperature is 0 C.
Equation (11) gives the hydrostatic pressure across the primary shock as
PS 2M S2    1
,

P0
 1
(13)
and the hydrostatic overpressure, OPS/P0, the pressure in excess of the ambient pressure P0 ,
as
OPS PS  P0 PS
2M S2    1
2



1 
1 
M S2  1 .
P0
P0
P0
 1
 1
(14)
For γ = 1.4, (13) becomes
PS 7 M S2  1

,
P0
6
(15)
OPS 7
 M S2  1 .
P0
6
(16)
and (14) becomes
The inverses of (15) and (16), which give the shock Mach number in terms of the pressures,
are, respectively,
MS 

1  6PS

 1 , and
7  P0

(17)
6 OPS
1.
7 P0
(18)
MS 
Density
Density (ρ) is the mass per unit volume of a gas, measured in units of kilograms per cubic
metre (kg m-3), pounds-mass per cubic foot (lb ft-3) or pounds-mass per cubic inch (lb in-3).
Non-dimensionally, it may be measured relative to the density of air at NTP, which is
1.225 kg m-3 (0.076475 lb ft-3, 4.4256 x 10-5 lb in-3). When a gas is traversed by a shock,
there is a rapid increase of the density. For an ideal gas with a ratio of specific heats of
γ = 1.4, there is an upper limit of 6.0 for the ratio of the densities, ρS/ρ0, across an infinitely
strong shock. In practice, very strong shocks produce other changes to the gas so that the
ratio of specific heats, γ, does not remain constant, and larger density changes can occur.
The density ratio across the shock is given by (12) in terms of the shock Mach number, as
  1M S2 .
S

 0   1M S2  2
(19)
S
6M 2
 2 S , and
0 M S  5
(20)
For γ = 1.4
the inverse is
MS 
5 1  0
.
6  1  0
(21)
Particle velocity
Particle velocity (u) is the translational velocity of the gas within a blast wave, measured in
metres per second (m s-1) or feet per second (ft s-1). Non-dimensionally, the particle velocity
may also be quoted relative to either the sound speed in the ambient gas (a0), or the sound
speed at the location of the gas particle (a). u/a0 is not a Mach number, but a dimensionless
quantity that is useful in scaling blast waves for different charge sizes and atmospheric
conditions. The particle velocity in terms of the sound speed at the same position in the blast
wave, u/a, is a true Mach number, and is known as the local Mach number of the flow. When
considering the blast interaction with a structure, it is important to know if the local Mach
number is less than or greater than one. If the local Mach number is greater than one, i.e.
supersonic, a bow shock forms around the structure and this further change the properties of
the gas before it can interact with the structure, as described in the section on total pressure,
below.
The co-ordinate transformation for a shock moving into a stationary gas and the
corresponding change of nomenclature, using (8), gives
 S w1
VS
MS



,
 0 w2 VS  u S M S  u S a0
(22)
where a0 is the sound speed in the ambient gas. Thus, using (22) and (19),
MS 
uS

 M S 0 , and
a0
S


   1M S2  2 
uS
  M S  0  1   M S 
 1
2
a0
 S

   1M S

2  M S2  1 



  1  M S 
.
(23)
For γ = 1.4, (23) becomes
uS 5  M S2  1 
 , and
 
a0 6  M S 
(24)
2


 uS 
1  uS

M S   3  9   25  .
5  a0

 a0 


(25)
the inverse of (24) is
Temperature
Absolute temperature (T) is the temperature of a gas measured from the absolute zero,
-273.16 C, in Kelvin (K). The absolute temperature at NTP is therefore 288.16 K. The
absolute temperature of the gas behind the shock, TS, is obtained in terms of the ambient
temperature, T0, using (5), (15) and (19) as
TS
PS  0 PS  0 2M S2    1   1M S2  2



.
  1M S2
T0
 S P0 P0  S
 1
2M    1.  1M

2
S
  12 M S2
2
S
2

.
(26)
For γ=1.4,
TS 7 M S2  1M S2  5
,

T0
36 M S2
(27)
and the inverse is


1  TS
 36  34  
MS 
14  T0


2


TS 
 34  36   140  .
T0 



(28)
Sound Speed
The sound speed of a gas (a) is the speed at which a weak compression or rarefaction wave is
transmitted through the gas. For one mole of an ideal gas P/ρ = RT, where R is the universal
gas constant, and the speed of sound is given by a = √(γP/ρ) = √(γRT). Therefore, the speed
of sound is proportional to the square root of the absolute temperature. The sound speed
immediately behind the shock, aS, in terms of the ambient sound speed, a0, is obtained from
(26) as,
aS
TS
1


a0
T0   1M S
2M    1.  1M
2
S
2
S

2 .
(29)
For γ = 1.4, (29) becomes
aS
1

a0 6 M S
7 M
2
S
 1M S2  5 ,
(30)
and the inverse of (30) is

1   a S
MS 
 36
14    a0

2


  34 



a
34  36 S

 a0



2
2



  140  .



(31)
Local Mach number
The Mach number of the flow immediately behind the shock, uS/aS, using (23) and
(29), is
uS
u S a0
2  M S2  1 

.


aS
a0 a S   1  M S 

2M  1

2
S
2M    1.  1M
2
S
 1M S
2M    1.  1M
2
S
2
S
2

2
S
2
.
(32)
For γ = 1.4,
5M S2  1
uS

aS
7M
2
S
 1M S2  5
,
(33)
and the inverse of (33) is
MS 
u
17 S
 aS
2
4

u 
u
  25  6 9 S   25 S

 aS 
 aS
u
25  7 S
 aS



2



2
.
(34)
Dynamic Pressure
Dynamic pressure (PD) is defined as one half the gas density times the square of the particle
velocity, i.e. ½ ρ u2. Dynamic pressure is a scalar property of the gas, and since gas is a
compressible fluid, is not equal to the stagnation or total pressure exerted on a surface at
which the gas is brought to rest, for which see below. Although the dynamic pressure does
not represent the pressure exerted on any surface in the blast wave, it is a useful measure of
the relative importance of the drag forces produced by the wave as compared to the
hydrostatic forces. The dynamic pressure immediately behind a shock, using equations (19)
and (24), is given in terms of the ambient pressure, as
2
2
PDS
1  S u S2 1  S  uS   0 a02 1  S  uS   0 P0 1  S  u S 
 
 
 



 
P0
2 P0
2  0  a0  P0
2  0  a0  P0  0 2  0  a0 
  1M S2 . 2  M S2  1 
1
 


2   1M S2  2    1  M S 
2
2
.
(35)
M S2  1
2
  1   1M S2  2
2

For γ = 1.4,
PDS 35 M S2  1
,

P0
12 M S2  5
2
(36)
and the inverse is

1  PD
MS 
 12  70 
70  P0

2
 PD

 PD
 
12  70   70012  7   .
 P0

 P0
 
(37)
Total Overpressure
Total overpressure (OPT), or stagnation overpressure, is defined as the increase of pressure
sensed by a transducer or a surface which is face-on to the flow, as a result of the flow being
brought to rest isentropically. Work is done both to bring the gas to rest and to compress it
adiabatically. The total overpressure is not the same as the reflected overpressure, for which
see below, or the dynamic pressure, see above.
Figure 3. Numerical simulation showing the shock reflected from a rigid structure. The
Mach number of the incident shock, MS = 2.5. The total pressure, PT, is the stagnation
pressure on the front face of the structure. (Courtesy A. A. van Netten)
When the primary shock of a blast wave strikes a rigid surface face on, it will be reflected,
and the reflected shock will run back into the flow, as shown in Figure 3. If the flow
produced by the primary shock is subsonic, i.e. uS/aS < 1, the reflected shock will continue to
move back through the flow, and will disperse. In this case, the face-on rigid surface will
continue to be exposed to the decaying flow in region 1 behind the primary shock. If the flow
produced by the primary shock is supersonic, i.e. uS/aS > 1, the reflected shock will move
back into the flow until a point is reached when its speed is equal to that of the flow, i.e. MR =
VR/aS = uS/aS. It will then form a bow shock through which the flow must pass before being
stagnated at the rigid surface. The physical properties of the gas are changed as it passes
through the bow shock and these changes must be taken into consideration when calculating
the total overpressure.
(a) Subsonic case
The overpressure at a stagnation point in a gas may be written as (Prandtl and Tietjens, 1934,
p227)


w 2   1   1
  1,
q  p1 
2 p  

(38)
where p is the pressure in the free stream, ρ the density, w the flow speed, and γ the ratio of
specific heats.
Writing (38) in terms of the flow conditions immediately behind a shock gives the total
overpressure, OPTS, as


 u 2   1   1
  1.
OPTS  PS 1  S S
2 PS  

(39)
Writing the physical properties as ratios of their values in the ambient atmosphere, (39)
becomes

OPTS PS

P0
P0

1  S u S2 P0  0 a02   1   1
1 
 1
2  0 a02 PS P0
 


PS 
1  S  u S  P0

1
  1



P0 
2  0  a0  P1


2

since
 0 a02


 1
,
(40)
1
 0 P0
  . Writing the ratios across the shock in terms of the shock Mach
P0  0
P0
number, MS, using (16), (19) and (23), gives
  1M S2  2  M S2  1    1  1 
OPTS 2M S2    1 
1 


2
2


P0
 1
   1M S  2    1  M S   2M S    1


2



 1
1
(41)
2M S2    1  M S2 2 2  1M S2    3  1  1
   1M 2  22M 2    1   1.
 1
S
S


For γ = 1.4,
OPTS 7 M S2  1  12 M S2 M S2  2  

 M 2  57 M 2  1
P0
6
S
 S

3.5
 1.
(42)
It has not been possible to find a simple analytical inversion of (42), but the following
empirical relationship provides a good description of the shock Mach number in terms of the
total overpressure in the subsonic case,
 OP
OP
M S  1.0816  0.06119 TS  1.0903 TS
P0
 P0



1
2
 OP
 0.81528 TS
 P0
1
 3
 .

(43)
(b) Supersonic case
Equation (32) gives the local Mach number of the flow behind a shock in terms of the shock
Mach number, MS , and the ratio of specific heats, γ. Putting the flow Mach number equal to
1, and γ = 1.4 gives a value for the shock Mach number of 2.0681. For shocks stronger than
this value, the flow immediately behind the shock is locally supersonic. In this case, a bow
shock forms ahead of any stationary object or structure enveloped by the primary shock of the
blast wave. The supersonic flow will pass through the bow shock and be made locally
subsonic before being brought to rest by the object or structure.
The total pressure behind a reflected shock in terms of the hydrostatic pressure,
sometimes called the Rayleigh supersonic pitot formula, is given as equation (6.3), in
Liepmann and Roshko, 1957. Using this equation gives the total overpressure, OPTS, as
 1
OPTS PT 2
P P

1  T2 S 1 
P0
P0
PS P0

M 12 

 2


 1
 2
  1
M 12 


  1 
 1
1
 1
PS
 1,
P0
(44)
where PT2 is the total pressure behind the reflected shock, P0 is the pressure of the ambient
atmosphere ahead of the incident shock, and PS and M1 are the hydrostatic pressure and flow
Mach number, respectively, behind the incident shock.
Using (13) and (32) in (44) gives

 1


2  1M  1


2
2
OPTS  2M S    1  1M S  2 

P0
2

8 M S2  1

2
2
   12M S    1  1M S
2
S
For γ = 1.4,
2
 2 M S2  1 
   1  1


 2

  1

  1
1
 1
OPTS
67920.1M S2  1

 1.
P0
M S2  542 M S4  96M S2  452.5
1.
(45)
7
(46)
No analytical solution of (46) could be found to give the shock Mach number in terms of the
total pressure in the supersonic case, but the following empirical relationship, (47), gives a
good description for shock Mach numbers from 2.06 to 3.5. For shock Mach numbers greater
than 3.5, real-gas effects begin to become important, and these equations become increasingly
less accurate.
 OP
M S  1.1421  0.3091 TS
 P0



0.5305
.
(47)
Reflected Overpressure
Reflected overpressure (OPRS) is the overpressure exerted on a plane surface face-on to the
shock front, immediately after the shock reflection. If the plane, reflecting surface is finite in
size, the reflected overpressure will be relieved by a rarefaction wave generated as the
reflected shock diffracts around the boundary of the reflecting surface (Figure 4). The
reflected pressure, PRS, is given by
PRS  2M S2    1  3  1M S2  2  1 
,


  1M S2  2 
P0 
 1

and the reflected overpressure, OPRS, is
(48)
 2M S2    1  3  1M S2  2  1 
OPRS PRS

1  
1.

  1M S2  2 
P0
P0
 1


For γ = 1.4,
OPRS 7 M S2  14 M S2  1

 1,
P0
3M S2  5
(49)
(50)
and the inverse of (50) is
2
 OP 
 OP   OP 
1
MS 
196  42 RS   42 196  196 RS    RS  .
128
 P0 
 P0   P0 
(51)
Figure 4. Numerical simulation of a Mach 2.5 shock incident on a rigid structure. The gas
behind the reflected shock has been brought to rest non-isentropically, and the pressure on the
structure is the reflected pressure, PR. As the reflected shock diffracts around the edge of the
structure, a rarefaction wave is produced which moves at the speed of sound behind the
reflected shock, aR. (Courtesy A. A. van Netten)
Reflected Temperature and Sound Speed
It is necessary to know the sound speed behind a reflected shock in order to calculate the time
for the rarefaction wave, produced when the reflected shock diffracts around the edges of the
reflecting surface, to move across the reflecting surface. After the arrival of the rarefaction
wave, the excess pressure on the surface will reduce to the total overpressure, OPTS, described
above.
The absolute temperature, TR, and sound speed, aR, in the region behind a reflected shock are
derived in terms of the incident shock Mach number, MS, as follows, using (8) and (48)
TRS
P 
P 
 RS 0  RS 0
T0
 RS P0
P0  R
   1 PRS     1 PRS 
1 
 

, and

   1 P0     1 P0 
2  1M S2  3    3  1M S2  2  1

  12 M S2

PRS
P0

a RS
TRS


a0
T0


2  1M  3   3  1M
2
S
(52)

 1M S
2
S
.
 2  1
(53)
For γ = 1.4,
TRS M S2  2 4 M S2  1

, and
T0
9 M S2
aR

a0
M
2
S
(54)
 2 4 M S2  1
.
3M S
(55)
The inverse of (54) is
2
T 
T
T
1
MS 
18 RS  14  6 9 RS   14 RS  9 ,
4
T0
T0
 T0 
(A 56)
and of (55) is
2
4
2
a 
a 
a 
1
MS 
18 RS   14  6 9 RS   14 RS   9 .
4
 a0 
 a0 
 a0 
(57)
Entropy
The entropy (S) of a gas is defined in terms of its change, dS, when a quantity of heat, dQ, is
added at a temperature T, viz.
(58)
dS  dQ T .
Since dQ = cv dT + PdV, for a perfect gas, the total change of entropy across a shock is given
by
T1
V1
dQ
dT
dV
S  
 cv 
 R
,
(59)
T
T
V
T0
V0
where PV = RT, and the universal gas constant R = 8.3143 J mole-1 K-1.
R = cp-cv, therefore
 cp

 cp

 

 1 
cv  c p  R  R  1  R
 1  R
 1  R
 .
c c



1


1




R

p
v


(60)
(59) and (60) give

 TS 
 VS 
T
S
1
dT
dV
1



ln    ln    ln  S


R
  1 T0 T V0 V
  1  T0 
T
 V0 
 0
T1

T
 ln  S
 T0




V1
1
 1



1
 1
 VS

 V0

  0 
 
  S 

(61), (27) and (19) give
1


2
2
2


S
 2M S    1  1M S  2  1    1M S  2  
 ln 
 

2
R
  12 M S2
    1M S  




1


2
2



1
  2M S    1    1M S  2  1 
 ln  
    1M 2  
 1
S
 
 








.
.
(61)
(62)
For γ = 1.4 and R = 8.3143 J mole-1 K-1, the change of entropy across a shock is
2.5
2

 7 M S  1
S  8.3143 ln 
46656


 M S2  5 


2
 MS 
3.5


-1 -1
 J mole K .


(63)
RANKINE-HUGONIOT SPREADSHEETS
In order to make the above equations easy to use, they have been incorporated into a set of
Excel© spreadsheets. The first spreadsheet requires the input of the shock Mach number, MS,
and calculates the changes of all the physical properties across the shock in terms of the
ambient properties, and in SI and Imperial units. Illustrated below is the output for an input
shock Mach number of 1.5.
Hydrostatic Overpressure (OP)
Dynamic Pressure (DP)
Ms
Atm
psi
kPa
Atm
psi
kPa
1.5 1.458333 21.43167 147.7656 0.628592 9.237787 63.69208
Total Overpressure (TOP)
Reflected Overpressure (ROP)
Atm
psi
kPa
Atm
psi
kPa
2.146445 31.54415 217.4885 4.425287 65.03402 448.3922
Density (D)
Flow Speed (U)
D/Do
lb/ft^3
kg/m^3 U/Ao
U/A
ft/s
m/s
1.862069 0.142448 2.281034 0.694444 0.604387 775.6097 236.4089
Sound Speed (A)
Temperature (T) Inv. Press Reflect Sound Speed
A/Ao
ft/s
m/s
T/To
T (C)
Po/P
ft/s
m/s
1.149007
1283.3 391.1552 1.320216 107.2735 0.40678 1447.214 441.1164
The other spreadsheets allow the input of the change of any of the physical properties across
the shock, in any unit system. For example, the spreadsheet shown below is that for an input
value for the dynamic pressure of 15 psi.
Hydrostatic Overpressure (OP)
Dynamic Pressure (DP)
Ms
Atm
psi
kPa
Atm
psi
kPa
1.623125 1.906956 28.02463 193.2223 1.020686
15 103.421
Total Overpressure (TOP)
Reflected Overpressure (ROP)
Atm
psi
kPa
Atm
psi
kPa
3.062135 45.00114 310.2709 6.263559 92.04926 634.6551
Density (D)
Flow Speed (U)
D/Do
lb/ft^3
kg/m^3 U/Ao
U/A
ft/s
m/s
2.070487 0.158392 2.536347 0.839191 0.708236 937.2739 285.6848
Sound Speed (A)
Temperature (T) Inv. Press Reflect Sound Speed
A/Ao
ft/s
m/s
T/To
T (C)
Po/P
ft/s
m/s
1.184904 1323.393 403.3755 1.403996 131.4156 0.344002 1524.991 464.8233
The spreadsheets can be run on any PC, including laptops and palm computers. In calculating
the total overpressure, allowance is made for both supersonic and subsonic flow behind the
shock. Warnings are presented if the input values exceed the range of validity. The
spreadsheets are demonstrated on www.blastanalysis.com, and information is provided on
procedures for obtaining a licence for their use.
ACKNOWLEDGEMENTS
The assistance of A. A. van Netten in making the numerical simulations for Figures 3 and 4,
using the AWAF code, is gratefully acknowledged.
REFERENCES
Hugoniot, P. H., 1887, Mémoire sur la propagation du mouvement dans les corps et ples spécialement
dans les gaz parfaits, 1e Partie, J. Ecole Polytech. (Paris), 57, 3-97.
Hugoniot, P. H., 1889, Mémoire sur la propagation du mouvement dans les corps et plus spécialement
dans les gaz parfaits, 2e Partie, J. Ecole Polytech. (Paris), 58,1-125.
Liepmann, H. W. and Roshko, A., 1957, Elements of Gas Dynamics, Galcit Aeronautical Series, John
Wiley & Sons, New York.
Prandtl, L., Tietjens, O. G., 1934, Fundamentals of Hydro- and Aerodynamics, Eng. Soc. Monographs,
reprinted by Dover, New York, NY, 1957.
Rankine, W. J. M., 1870a, On the thermodynamic theory of waves of finite longitudinal disturbance,
(read 16 Dec., 1869), Phil. Trans. Roy. Soc. London, 160, 277-286.
Rankine, W. J. M., 1870b, supplement to "On the thermodynamic theory of waves of finite longitudinal
disturbance", Phil. Trans. Roy. Soc. London, 160, 287-288.