Compressible flow, Convected wave equation, Incompressible flow, Lorentz factor, Prandtl-Glauert factor, Special relativity


International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 
DOI: 10.5923/j.ijtmp.20170705.02 

 

Deriving Special Relativity from the Theory of   
Subsonic Compressible Aerodynamics 

Michael James Ungs 

Tetra Tech, Lafayette, USA 

 

Abstract  Mathematical transformations that convert the convected wave equation in subsonic compressible flow to one 
based on incompressible flow have profound implications in understanding the physical basis of the transformations used in 
the theory of special relativity. The evolution of using incompressible flow solutions for airfoil design in lieu of conducting 
high speed wind tunnel tests is briefly reviewed. This in turn evokes the forgotten history of aerodynamicists using the 
Prandtl-Glauert method of spatial contraction as a substitute for compressibility effects before WWII. Matrix expressions 
identical in form to those representing relativistic velocity, acceleration, and mass are developed from linear 
transformations relating compressible versus incompressible flow systems and fixed-to-vehicle versus fixed-in-space 
coordinate reference frames. The mathematical intersection of special relativity and compressible flow theory is generally 
not understood nor appreciated outside the field of subsonic aerodynamics, making it a compelling subject for us to 
explore. 

Keywords  Compressible flow, Convected wave equation, Incompressible flow, Lorentz factor, Prandtl-Glauert factor, 
Special relativity 

 

1. Introduction 
The development of a comprehensive theory of flight has 

been a slow, frustrating study in progress. An interesting 
account of its complex history up to the 1930s in Europe is 
given by [1]. Briefly stated, the question of interest revolves 
around using the approximate, ideal fluid theories of 
Bernoulli and Euler versus using the more exact, non-linear 
Navier-Stokes equations. By the beginning of the 1940’s, a 
lot of low speed data had been collected from wind tunnel 
testing of standardized airfoil sections for subsonic 
aerodynamic research by the National Advisory Committee 
for Aeronautics [2, 3]. Using incompressible flow theory, 
the wind tunnel data were processed and formulated into 
tables and graphs suitable for aircraft development and 
design. The near universal assumption of incompressible 
flow for manned flight was quite reasonable at that time 
considering air speeds less than 200 miles per hour were 
typically involved, with only the tips of high speed 
propellers approaching the speed of sound. With the 
development of faster planes and jets, such as the 500 mph 
Messerschmitt Me262, it became imperative that methods 
be devised that would allow the older low speed, 
incompressible  based tabulations to  be used with simple  

 
* Corresponding author: 
michael.ungs@tetratech.com (Michael James Ungs) 
Published online at http://journal.sapub.org/ijtmp 
Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved 

correction factors so that the effect of compressibility could 
be accounted for when designing aircraft for higher speeds. 
The alternative was to build faster wind tunnels and to 
painstakingly redo the airfoil tests and tabulations. 

Analytical methods to compensate incompressible based 
lift calculations are called compressibility corrections. The 
Prandtl-Glauert method is one such approach that assumes 
two-dimensional, incompressible flow parallel to an airfoil 
cross section and then reduces the length of the airfoil chord 
in the lift and moment equations to account for 
compressibility [4]. The effect of mathematically 
compressing the parallel axis coordinate can be easily 
accounted for in two dimensions. However, in three 
dimensions, stretching and compressing the parallel axis 
coordinate results in lateral coordinate effects that produce 
non-linear changes in wing forces and moments [5, 6]. To 
the aeronautical engineer, contracting chord length and 
other aircraft dimensions as a function of speed in 
expressions formulated on the basis of incompressible flow 
is not controversial. It will be shown that these spatial 
contractions are mathematical artefacts induced while 
performing transformations from one coordinate system to 
another. 

Interest expressed by the aeronautical community in 
using coordinate transformation methods to convert 
incompressible to compressible flow has substantially 
decreased since the 1950s. It has been replaced with 
methods that directly solve the fluid dynamic equations of 
compressible flow using fast computers and software based 



114 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

on computational fluid dynamics. None the less, there is 
still interest in special applications such as for steady-state 
flow problems. Examples of this are seen by the use of a 
power series expansions in terms of the Mach number [7] 
and using a variable density and flow angle for mapping in 
two dimensions [8]. 

A link between special relativity and compressible fluid 
dynamics is not a new concept [9, 10]. But it is rarely 
pursued outside the shadow of unconventional physics. The 
purpose of this article is to systematic examine the 
mappings between compressible and incompressible flow 
systems in different coordinate frames and to show the 
profound similarity between special relativity and the 
convective wave equation expressions. 

We shall assume air is a continuous fluid with fluid 
properties that are irrotational, inviscid, barotropic, and 
isentropic. There are also two flow systems considered that 
affect the form of the wave equation when solving for the 
perturbation velocity potential. In both cases, the X-axis of 
the coordinate system is aligned with the direction of the 
free-stream velocity vector. The x=0 coordinate is located at 
the airfoil’s leading edge and coordinate values increase 
towards the trailing edge. Compressible flow refers to the 
representation of the wave equation for the perturbation 
velocity potential in which there are cross-derivative terms 
between the X, Y, Z spatial coordinates and the time 
coordinate. Incompressible flow is defined as flow within 
which the fluid density remains constant with pressure. It is 
represented with a wave equation for the perturbation 
velocity potential in which there are no cross-derivative 
terms. These two flow systems are linked by coordinate 
transformations. 

Two reference frames are also considered. There is the 
non-inertial, fixed-to-vehicle (FTV) reference frame with 
coordinates attached to the leading edge of the vehicle. The 
vehicle remains at rest as the fluid medium flows against 
the leading edge with free-stream velocity 

∞
V , where 

1 1 2 2 3 3ˆ ˆ ˆe V e V e V∞ ∞ ∞∞ = + +V . The other is the inertial, 
fixed-in-space (FIS) reference frame with the fluid medium 
initially at rest and the vehicle moving with speed 

∞
V . 

2. Convected Wave Equation 
2.1. Transient Case 

Consider the fluid dynamics of a moving fluid medium 
that is compressible. The adiabatic compression of a fluid 
ˆ sρκ  can be defined as the relative change in the local fluid 

density ρ̂  and adiabatic compressibility coefficient sκ  in 
response to a change in the local fluid pressure p̂ , such that 
ˆ ˆ ˆ( )s pρκ ρ= ∂ ∂ s . In addition, the free-stream 

squared-speed of sound c∞  is inversely proportional to the 

adiabatic compression ˆ1 sc ρκ∞ =  evaluated at the 
free-stream fluid density ρ∞ , such that: 

( )2 ˆˆˆc p ρ ρρ ∞∞ == ∂ ∂     (1) 
The Cauchy momentum equation can be written in the 

absence of viscous and external forces (e.g., gravity) as 
follows: 

ˆ ˆ ˆD Dt pρ = ∇u     (2) 
Expression (2) basically states that a pressure gradient at 

any point in the medium produces an acceleration of the fluid. 
The convective derivative term ˆD D tu  is written out as 
the sum of two contributions, where ˆ∇u  is a tensor 
derivative of the fluid velocity û : 

ˆ ˆ ˆD Dt t t= ∂ ∂ + ∂ ∂ ⋅∇u u x u     (3) 

The vector term t∂ ∂x  is defined as the fluid velocity 
û  for a stationary medium. However, when the medium is 
also moving with a free-stream velocity ∞V , then the 
free-stream contribution can be separated from the 
perturbation velocity component u , such that 

t ∞∂ ∂ = +x V u  ˆ= u . 
The convective form of the continuity equation for the 

conservation of mass can be expressed in terms of the 
divergence of the fluid velocity, such that: 

ˆ ˆ ˆD Dt Divρ ρ= − u     (4) 
The gradient of a velocity potential cΦ  can be expressed 

in terms of the fluid velocity for irrotational flow, such that: 

cΦˆ ∇=u      (5) 

The velocity vector of the fluid becomes tangent to the 
streamline for streamline flow conditions [3, 11], such that in 
Cartesian coordinates: 

2 1ˆ ˆ ;d y d x u u= 3 2ˆ ˆ ;d z d y u u= & 31 ˆˆ uuzdxd =  
(6) 

Consider the case for unsteady compressible flow 
expressed in terms of a velocity potential, with a FTV (or 
laboratory) reference frame, irrotational (i.e., 0urlc ≡u ), 
inviscid (i.e., 0cµ ≡ ), barotropic (i.e., cρ ≡ ( cρ )), 
isentropic (i.e., constant entropy) flow conditions, and in the 
absence of external forces (e.g., gravity), such that [12, 13]: 

( )

( )

2 2 2 2Φ Φ Φ Φ
1

Φ Φ Φ
2

a c c c c

c c c

c t t∇ = ∂ ∂ + ∂ ∇ ⋅∇ ∂

+ ∇ ⋅∇ ∇ ⋅∇
   (7) 

The subscript “c” indicates compressible flow conditions 
and ac  is the local speed of sound. The wave described by 
(7) continues onwards to infinity since no viscosity terms are 
included in the formulation to dissipate the wave. Terms in 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 115 
 

 

(7) can be rearranged such that the second-order derivatives 
of the velocity potential are combined to form the following 
unsteady wave equation in Cartesian coordinates for 
compressible fluid flow [14, 15]: 

( )( ) ( )( )
( )( )

( )

( )

2 2

2

2 2

2

Φ 1 Φ Φ 1 Φ

Φ 1 Φ

1 1
Φ 2 Φ Φ Φ Φ Φ Φ

1
2 Φ Φ Φ Φ Φ Φ Φ Φ Φ

c xx c x a c yy c y a

c zz c z a

c t t c x c xt c y c y t c z c z t
a a

c x c y c xy c x c z c xz c y c z c yz
a

c c

c

c c

c

− + −

+ − =

+ + +

+ + +

 

(8) 
Poisson [16] was apparently the first to derive the 

one-dimensional version of (8). Some of the earliest 
discussions concerning the steady-state, two-dimensional 
version can be found in [17, 18]. The cross-derivative terms 
such as Φc xt  and Φc xy  in the wave equation (8) represent 
nonlinearities generated after differentiating 
squared-velocity quantities in (7). The cross-derivative terms 
vanish as the flow speed goes to zero but otherwise remain 
non-zero valued as the speed increases. Hence, any attempt 
to eliminate the cross derivatives by means of coordinate 
transformations will also make the transformed fluid a 
fictitious fluid. The non-dimensional variable jM  is 
defined as the Mach number in the jth direction of flow. It is 
evaluated as the quotient of the fluid velocity in the jth 
direction and the local speed of sound ac , such that: 

Φ1 c
j a

a j
M

c x
∂

=
∂

    (9) 

Substitute the Mach number expressions from (9) back 
into the unsteady velocity potential equation (8) for a 
compressible fluid medium with a FTV reference frame, 
such that in Cartesian coordinates: 

( ) ( ) ( )
( )

( )

2 2 2
1 2 3

1 2 3 2

1 2 1 3 2 3

Φ

1
2 Φ Φ Φ Φ

2 Φ Φ

Φ 1 Φ 1 1

1

Φ

a

c xx a c yy a c zz a

a c xt a c y t a c z t c t t
a

a a c xy a a c xz a a c yz

M M
c

M M M

M M M

M
c

M M M

− + − + − =

+ + +

+ + +

 

(10) 
Of course the solution to (10) is not unique until both 

initial conditions and boundary conditions are prescribed 
for the velocity potential. Discussion of these are outside 
the scope of the paper but flow around aircraft vehicles is 
typically assumed to be tangent to all solid surfaces. 
Expression (10) holds for subsonic and supersonic flow 
conditions but not transonic flow since additional nonlinear 
terms representing compression shock and temperature 
loses have not been included. It should also be clear that a 

singularity, called a Prandtl-Glauert singularity, occurs 
when one of the jth Mach numbers approach the value of 
one. It is a mathematical singularity, rather than a physical 
singularity, arising from the absence of additional nonlinear 
terms in (10). 

An additional assumption can be introduced to further 
simplify the compressible flow expression (10) for 
barotropic and isotropic flow conditions. Consider the case 
where a solid body disturbing the flow field is slender and 
is traveling at either subsonic or supersonic velocities. 
Using a Legendre transformation, replace the velocity 
potential term Φc  with one representing a free-stream or 
undisturbed flow speed 1| | cV ∞=∞V  (i.e., speed of the 
solid body) in the X-direction and that of a small perturbed 
velocity potential component cϕ . Under these conditions, 
the following linearized expansion using a perturbation 
velocity potential cϕ  for compressible flow conditions can 
be used, such that: 

1 1

Φ cc
c c c

c c
V V u

x x
ϕ

∞ ∞

∂∂
= + = +

∂ ∂
; 

cc
c

c c
v

y y
ϕ∂∂Φ

= =
∂ ∂

; 

Φ cc
c

c c
w

z z
ϕ∂∂

= =
∂ ∂

; c c
c ct t

ϕ∂Φ ∂
=

∂ ∂
  (11) 

Expression (10) further simplifies if the velocity potential 
derivatives are much smaller than the free-stream speed 
| |∞V  and if flow is aligned along the X-axis. The resultant 
unsteady perturbation velocity potential equation reduces as 
follows: 

2
21 1

1 2
1c xx c yy c zz c xta c t t a

aa

M M
cc

ϕ ϕ ϕ ϕ ϕ − + + = + 
 

 (12) 

2.2. Steady-State Case 

The steady Prandtl-Glauert equation is obtained from the 
general velocity potential equation (10), such that: 

( ) ( )

( )
22 2

1 32

1 2 1 3 2 3

Φ 1 Φ 1 Φ 1

2 Φ Φ Φ

c xx a c yy c zz aa

c xy c xz c yza a a a a a

M M M

M M M M M M

 − + − + − 
 

= + +
 

(13) 
The steady-state expression (13) will dramatically 

simplify if the free-stream velocity ∞V  is also aligned 
along the X-axis, such that: 

( )21Φ 1 Φ Φ 0c xx c yy c zzM ∞− + + =    (14) 
The simplified form of (14) does not hold near the 

leading edge of an airfoil where the velocity perturbation is 
of the same order-of-magnitude as the free-stream velocity, 
unless the free-stream Mach number is itself small. It forms 



116 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

the basis for many aerodynamic analysis methods. 
The presence of cross-derivative terms in the convected 

wave equation (10) still makes its numerical computation a 
challenging process despite the recent advantage in having 
fast computers and more efficient algorithms. 

3. Incompressible Flow 
The more general problem of describing the convective 

wave equation in a compressible fluid medium will now be 
simplified for subsonic speeds. A brute force selection 
procedure will be presented to transform the coordinate 
system to various fictitious, incompressible flow systems. 
This approach will demonstrate that the resultant 
transformations are mathematical constructs that only 
depend on satisfying the desired form of the partial 
differential equations being transformed. 

Convert coordinate time intervals 0c ct t−  and 

0i c i ct t−  with initial times 0ct  and 0i ct  to distance 

intervals cτ  and i cτ  by introducing the following 
change in variables: 

( )0 1c c c ct t Vτ ∞= −  and ( )0 1i c i c i c i ct t Vτ ∞= −  
(15) 

The linearized, unsteady, elliptic convected wave 
equation of velocity potential for a compressible fluid 
medium previously given in expression (10) reduces as 
follows for Cartesian coordinates upon application of the 
linearization expansion procedure given in (11) [13, 14, 19]: 

( )
2 2 2

2
12 2 2

2 2

2 2
1 12

1

2

c c c
c

c c c

c c
c c

c cc

M
x y z

M M
x

ϕ ϕ ϕ

ϕ ϕ
ττ

∞

∞ ∞

∂ ∂ ∂
− + + =

∂ ∂ ∂

∂ ∂
+

∂ ∂∂

   (16) 

Note that the Mach number 1cM ∞  in the convected 
wave equation (16) is defined in (9) and (11) as a function 
of the local perturbation velocity cu  for a compressible 
fluid medium. 

The wave equation (16) for a compressible fluid with 
coordinate system ( ), , ,c c c cx y z τ  will now be transformed 
to an equivalent incompressible flow coordinate system 
( ), , ,ic ic ic icx y z τ  in Cartesian coordinates. Incompressible 
terms are indicated by using the subscript “ic”. A subsonic 
free-stream velocity 1cV ∞  and free-stream speed of sound 

cc ∞  in the compressible flow system will be assumed, 
such that: 

i c c cϕ β ϕ=         (17) 

2
11c cMβ ∞= −  1 1ciff M ∞ <      (18) 

i c c c c c

i c c

i c c

i c c c c c

x a x b

y y

z z

c x d

τ

τ τ

= + 


= 


= 
= + 

       (19) 

ciff x− ∞ < < + ∞  & ct− ∞ < < + ∞ . Note that 
the X  and T  line coordinates must be defined over an 
infinite domain in order for these transformation methods to 
be valid. The coefficient cβ  in (17) is called the 
Prandtl-Glauert factor in aerodynamics and the Lorentz 
contraction factor in the theory of relativity [20]. It should 
be obvious from (19) that the new variable i cτ  is no 
longer a real time variable since its value is translated by 
the X space coordinate term .c cc x  This means the newly 
introduced i cx , cτ  and i cτ  coordinates are part of a 
fictitious mathematical construct that will be used to simplify 
the partial differential equations. Term Det  is set equal to 
the Jacobian determinant of the ( ),c cx τ  transformation 
matrix given in (19): 

c c c cDet a d b c= −       (20) 
Take the second-order partial derivatives of the 

perturbation velocity potential cϕ  with respect to the 

compressible Cartesian coordinate cx  and the 

compressible time variable cτ . Replace the coordinate 
derivatives in (16) using the chain rule on (19), such that: 

2 2 2

2

2

2

2c c i c c i cc c c
c c i c i c i c i c i c i c

c i c
c

i c i c i c

a a c
x x x x x

c

ϕ ϕ ϕ ϕ ϕ
ϕ ϕ τ

ϕ ϕ
ϕ τ τ

∂ ∂ ∂ ∂ ∂
= +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂
+
∂ ∂ ∂

(21) 
2 2 2

2 2

c c i c c i c
c c c c

c c i c i c i c i c i c i c

c i c c i c
c c c c

i c i c i c i c i c i c

a b a d
x x x x

c b c d
x

ϕ ϕ ϕ ϕ ϕ
τ ϕ ϕ τ

ϕ ϕ ϕ ϕ
ϕ τ ϕ τ τ

∂ ∂ ∂ ∂ ∂
= +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂
+ +
∂ ∂ ∂ ∂ ∂ ∂

  (22) 
2 2 2

2

2

2

2c c i c c i cc c c
c c i c i c i c i c i c i c

c i c
c

i c i c i c

b b d
x x x

d

ϕ ϕ ϕ ϕ ϕ
τ τ ϕ ϕ τ

ϕ ϕ
ϕ τ τ

∂ ∂ ∂ ∂ ∂
= +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂
+
∂ ∂ ∂

  

(23) 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 117 
 

 

Group all of the XX partial derivative terms upon 
substitution of (21), (22), and (23) into the compressible 
convected wave equation (16). Then set the resultant 
expression equal to the corresponding XX partial derivative 
term given below in (28) that represents the incompressible 
wave equation with a FTV reference frame. Repeat the 
process of grouping the XT and TT partial derivatives. 
Solve the resultant grouped coefficient expressions and the 
Jacobian determinant (20), such that: 

2 2 2 2 2
1 12c c c c c c c XXa M a b M b Sgnβ ∞ ∞− − =   (24) 

( )2 2 21 12 2 2 0c c c c c c c c c c ca c M a d c b M b dβ ∞ ∞− + − =
 (25) 

2 2 2 2 2 2
1 1 12c c c c c c c TT i cc M c d M d Sgn Mβ ∞ ∞ ∞− − = −   (26) 

c c c c Deta d b c Sgn− =    (27) 

The Y and Z axis components to the transformation 
coefficients will always equal to one since the free-stream 
velocity c∞V  vector is assumed to be aligned parallel with 
the X axis. Hence, the transformation problem reduces to 
finding the remaining coefficients for just the X and T 
components. There remain four unknown coefficients ca , 

cb , cc , cd  and four constraint equations in (24) to (27). 
The value of the three sign terms used in (24) to (27) are 
unknown but they are restricted to plus or minus one. The 
end product of the transformation process is a partial 
differential equation that represents the wave equation of the 
perturbation velocity potential for an incompressible fluid 
medium with a FTV reference frame, such that: 

2
1i c i c i c i c i c i c i c i cXX i c x x i c y y i c z z TT i c i c

Sgn Sgn M τ τϕ ϕ ϕ ϕ∞+ + =
 (28) 

The expression (28) is also known as the acoustic wave 
equation in fluid dynamics and the d'Alembert equation for 
electromagnetic waves in physics when the two Sgn  terms 
equal plus one [21]. 

3.1. Brute Force Algorithm 

The more general problem to be addressed here is to 
present a brute force algorithm that is used to perform a 
systematic search of coefficients for the linear homogeneous 
transformation with just X and T components. These 
coefficients will convert the convected wave equation for a 
compressible fluid medium with a FTV reference frame into 
a wave equation for an incompressible fluid medium with 
either a FTV or FIS reference frame. The algebraic matrix 
used in this search is as follows: 

a b

i c c

i c cc d

C C
x xE E

C C
E E

τ τ

 
    
 = ⋅            
 

   (29) 

The transformation coefficients first presented in (19) are 
related to the generic case of (29) as follows: 

c aa C E= ; c bb C E= ; c cc C E= ; and 

c dd C E= . A search was conducted using eleven trial 
formulas that were substituted into the numerator 
coefficients aC , bC , cC , and dC . The eleven trial 
formulas are as follows: 0 , 1 , 1− , M ∞ , M ∞− , β , 
β− , 2β , 2β− , 2M ∞ , & 

2M ∞− . In addition, seven trial 
formulas were systematically substituted into the 
denominator coefficient E . The seven trial formulas are as 
follows: 1 , β , M

∞
, 3 2M ∞ , 

3 2β , 2M ∞ , & 
2β . A 

systematic search using the above combination of numerator 
and denominator trial formulas resulted in a total of 102,487 
combinations that were tested. The three sign terms used in 
(24) to (27) are only allowed to have values equal to plus or 
minus one: 1XXSgn ±= ; 1TTSgn = ± ; & 1DetSgn = ± . 
The search using these three sign coefficients requires an 
additional eight times of effort, resulting in a total of 819,896 
searches. The actual search is easily performed using a 
numerical algorithm by setting the Mach number M ∞  to an 
arbitrary subsonic value, such as 0.3, looping through all of 
the different formulas and signs, and saving only those trial 
formulas that exactly satisfy the four constraining 
expressions (24) to (27). Only six unique transformation sets 
were found during these searches. These are listed in Table 1. 
Term Det  is the Jacobian determinant of the 
transformation matrix shown in the third column. Three of 
the transformations have a Jacobian determinant equal to 
plus one and three have a Jacobian determinant equal to 
minus one. 

It should be obvious that making changes in the sign of the 
Jacobian determinant and in the partial differential equation 
terms of (28) will result in profound differences in the 
physical system being represented by the incompressible 
equation and in any boundary conditions associated with the 
original compressible flow system given by (16). 

All six of the transformations listed in Table 1 start with 
the partial differential equation for the linearized, unsteady, 
convected wave equation for a compressible fluid medium 
with a FTV reference frame given in (16) for subsonic flow 
conditions. During the forward process, the transformed 
system represents various partial differential equations for 
the linearized, unsteady, wave equation for an 
incompressible fluid medium with a FTV reference frame. 
Both the original convected wave equation (16) and the six 
transformed wave equations of Table 1 are classified as 
elliptic partial differential equations. The remainder of the 
paper will only focus on the first two solutions given in Table 
1. 

The matrix shown on line a in Table 1 is the inverse matrix 
for a Galilean transformation. The matrix shown on line b of 
Table 1 will be labeled the Miles transformation matrix and 
is described in [13]. The Lorentz transformation can be 



118 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

obtained by pre-multiplying the Galilean matrix with the 
Miles matrix: 

2 2 2

1 0 1 11 11 1
0 1 1M Mββ β∞ ∞

    
=           

    (30) 

The complete family of transformations will be described 
shortly. 

There are at least four different ways to write the unsteady 
wave equation for perturbation velocity potential [13] in 
Cartesian coordinates when the free-stream velocity vector is 
made parallel to the X-axis. These come about from 

transforming between compressible versus incompressible 
flow systems and from transforming between FTV and FIS 
reference frames. The four partial differential expressions 
are listed in Table 2. Of these four, only line a in Table 2 
represents air as a real fluid by using a compressible FTV 
wave equation with cross-derivative terms. The other three 
represent the equations for different fictitious fluids. 
Obviously the boundary conditions of the wave equations, 
which are not discussed here, would also be transformed 
accordingly. The fourth column of Table 2 gives a two letter 
abbreviation to distinguish each of the flow and coordinate 
frame assumptions. 

 

Table 1.  Summary of the six coordinate transformations that are unique and that satisfy the four constraints given in (24) to (27) 

 Transformation Det  Transformed PDE 

a 
1 1
0 1

− 
 
 

 1 
2

Xi c i c i c i c i c i c i c i ci c X i cY Y i c Z Z i cMϕ ϕ ϕ ϕ∞+ + = T T  

b 
2 2

1 01
Mβ β∞

 
 
 
 

 1 
2

i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞+ + =  

c 
2 2

0 11
M M M∞ ∞ ∞

 
 
 − 

 1 
2

i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞− + + = −  

d 
1 1

0 1
− 
 
 

 -1 
2

i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞+ + =  

e 
2 2

1 01
Mβ β∞

− 
 
 
 

 -1 
2

i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞+ + =  

f 
2 2

1

0 11
M M M∞ ∞ ∞

 
 
 − 

 -1 
2

i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞− + + = −  

Note that the Mach number M ∞  is defined as 1ˆ( )M c∞ ∞ ∞= ⋅e V . The free-stream speed of sound c ∞  and vehicle speed ∞V  are assumed 
to be the same in all reference frames. The compressible and incompressible subscripts of these and associated speed variables are henceforth dropped. 

Table 2.  Partial differential equations of the unsteady, perturbation velocity potential in Cartesian coordinates 

 Flow Type Frame Label Coordinates Unsteady Wave Equation 

a compressible FTV CV ( ),c cx τ  
2 2 2(1 ) 2

c c c c c c c c c cc x x c y y c z z c c xM M Mτ τ τϕ ϕ ϕ ϕ ϕ∞ ∞ ∞− + + = +  

b incompressible FTV IV ( ),ic icx τ  2i c i c i c i c i c i c i c i ci c x x i c y y i c z z i cM τ τϕ ϕ ϕ ϕ∞+ + =  

c compressible FIS CS ( ),c cX T  2c c c c c c c cc X X cY Y c Z Z cMϕ ϕ ϕ ϕ∞+ + = T T  

d incompressible FIS IS ( ),ic icX T  2i c i c i c i c i c i c i c i ci c X X i cY Y i c Z Z i cMϕ ϕ ϕ ϕ∞+ + = T T  
 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 119 
 

 

Given four partial differential equations, with abbreviates 
CV, CS, IV, & IS listed in Table 2, there are a total of twelve 
ways to inter-transform them. These transformation 
combinations are as follows: { }, ,IV IS CS CV→ ; 

{ }, ,CV CS IS IV→ ; { }, ,IS IV CS CV→ ; and 
{ }, ,CS CV IS IV→ . The coefficients for these are derived 

in the same brute force manner previously described or by 
using simple substitution or inversions between matrices. 
These twelve transformation matrices are listed in Table 3. 

In a manner similar to that developed for (15), the 
coordinate time variables cT  and ciT  for a FIS reference 
frame are related to units of distances cT  and ciT  through 
the following change in variables: 

( )0 1c c cT T V ∞= −T  and ( )0 1i c i c i cT T V ∞= −T
(31) 

What is most striking about the brute force approach 
described above is that it can be used to sculpture virtually 
any transformation matrix between two different partial 
differential equations of the same order. This should be seen 
as a cautious warning that coordinate transformations are 
convenient mathematical constructs that create a fictitious 
set of equations with fictitious fluid properties that are 
simpler to solve. It does not imply the resultant fluid or flow 
system exists or is even plausible. 

3.2. Summary of Subsonic Coordinate Transformations 

A list of the twelve transformation matrices used to 
convert the convected wave equation for subsonic speeds 
from compressible to incompressible flow conditions or the 
reverse process are listed in Table 3. Two letter 
abbreviations of the reference frame are listed in the second 
and fourth columns and the classical name associated with 
the transformation matrix is listed in the last column. 

It is worth repeating that only the compressible FTV (i.e., 
CV) wave equation with cross derivatives in Table 2 
represent air as a real fluid. The other three represent the 
wave equations for three different fictitious fluids. The brute 
force algorithm used to produce Table 3 demonstrates, from 
a heuristic point of view, that the transformation matrices 
enforce a particular linkage between time and space 
coordinates. This linkage eliminates the generation of 
cross-derivative terms involving time when transforming 
between the compressible wave equation CV and the three 
fictitious wave equation CS, IV, and IS. As stated at the 
bottom of Table 3, the brute force algorithm assumed the 
speeds cc ∞  and i cc ∞  to be the same in all reference frames, 
i.e., c i cc c∞ ∞= . However, the constraints of (24) to (27) do 
not require this speed constant to be the maximum allowed 
speed. 

Consider the IV CV→  transformation formulas 

c i cd x d xβ=  for the X-coordinate and c i cd dτ τ β=  
for the T-coordinate given on line f of Table 3. These 
formulas convert spatial and temporal changes measured in 
an incompressible flow system to changes in a compressible 
flow system. The Prandtl-Glauert factor β  is always less 
than one for subsonic speeds. It then follows that an observer 
working in a compressible flow system might be tempted to 
conclude that spatial measurement i cd x  decreases in length 
and that temporal measurement i cdτ  increases in duration 

with an increase in the free-stream speed ∞V . Of course 
we know this is only a mathematical artefact arising from the 
decision that someone previously had used a theoretical 
model based on the assumption of incompressible flow. This 
is exactly the case faced by aeronautical engineers before the 
availability of high speed wind tunnels and fast computers to 
run computational fluid dynamic software before WWII. No 
discrepancy in measurement occurs with speed if the 
observer had been consistent in comparing measurements 
against a theoretical model based on compressible flow. 
However, the situation is much more difficult and 
paradoxical to resolve if the observer were to deny the very 
existence of a compressible flow system. Space and time 
would then be concluded to be warped or bent with speed. 

3.3. Velocity Transforms between Reference Frames 

This section will describe how the perturbed velocity 
components developed in one reference frame can be 
related to those in another reference frame using the various 
coordinate transformations introduced in the previous 
section. The basic approach of [20] will be followed. 

Let there be two different reference frames, called the 
 and  systems. Assume the  system moves 

relative to the  system with velocity  in a 
direction that is parallel to the X-axis of both systems. 
Define the X-axis velocity component u d x d t=  as the 
particle velocity for the  system and the X-axis velocity 
component u d x d t′ ′ ′=  of the same particle in the  
system, such that in general: 

u d x d t

v d y d t

w d z d t

= 


= 


= 

       (32) 

u d x d t

v d y d t

w d z d t

′ ′ ′= 
′ ′ ′= 


′ ′ ′= 

   (33) 

 
  

K K′ K′
K ∞V

K
K′



120 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

Table 3.  Summary of coordinate transformations for subsonic flow conditions 

Reference frame on left side of 
the equal sign 

Transformation 
Reference frame on right side of the 

equal sign 

a IV 
2

1 11
1

i c i c

i c i c

x X

Mτ β ∞

    
= ⋅        

     T
 IS Lorentz 

b IV 
2

1 11
1

i c c

i c c

x X

Mτ β ∞

    
= ⋅        

     T
 CS Lorentz 

c IV 
2 2

1 01i c c
i c c

x x
Mτ ββ τ∞

    
= ⋅        

    
 CV Miles 

d CV 
1 1
0 1

c c

cc

x X
τ
    

= ⋅           T
 CS Galilean 

e CV 
1 1
0 1

i cc

i cc

Xx

τ

    
= ⋅           T

 IS Galilean 

f CV 

2

2

01
1

i cc

i cc

xx

M

β
τβτ ∞

    
 = ⋅      −    

 IV Miles inverse 

g IS 
2

1 11
1

i c i c

i c i c

X x

M τβ ∞

−    
= ⋅        −    T

 IV Lorentz inverse 

h IS 
1 0
0 1

i c c

i c c

X X    
= ⋅           T T

 CS Unit 

i IS 
1 1
0 1

i c c

i c c

X x

τ

   − 
= ⋅            T

 CV Galilean inverse 

j CS 
1 1
0 1

cc

c c

xX
τ
 −   

= ⋅          T
 CV Galilean inverse 

k CS 
1 0
0 1

c i c

c i c

X X    
= ⋅           T T

 IS Unit 

l CS 
2

1 11
1

i cc

i cc

xX
M τβ ∞

−    
= ⋅        −    T

 IV Lorentz inverse 

Note that the expressions in Table 3 make the assumption that free-stream velocity components 1 1 1c i cV V V∞ ∞ ∞= =  and speed of sound values 

c i cc c c∞ ∞ ∞= =  are equal. It then follows that the Mach numbers 1 1c i cM M M∞ ∞ ∞= =  and Prandtl-Glauert factors 
2 2 2 21c i c Mβ β β ∞= = = −  are equal. 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 121 
 

 

3.4. Transforming Velocities from FIS to FTV Reference 
Frames 

The transformation between FIS and FTV reference 
frames is represented by the matrix on line a of Table 3 for 
incompressible flow systems, such that upon taking 
incremental differences in the coordinates: 

22 1 Mβ ∞= −  1iff M ∞ <     (34) 

( )

( )

1

2
1 1

1

1

i c i c i c

i c i c

i c i c

i c i c i c

Xd x d d T V

d y d Y

d z d Z

d t V M d X d T V

β

β

∞

∞ ∞ ∞


= + 


= 


= 


= +


  (35) 

The incremental changes in time variable i cτ  associated 

with the FTV reference frame and i cT  associated with the 
FIS reference frame are related to incremental changes in 
coordinate time variables i ct  and i cT  by the relationships 
(15) and (31), such that: 

1i c i cd d t Vτ ∞=  and 1i c i cd d T V ∞=T    (36) 

Divide the incremental spatial variable expressions in the 
first three lines of (35) by the expression for i cd t  from the 
fourth line in (35), such that: 

1

1

2
1

i c i c

i c i ci c

i c

d x V d X
V

d t d Td X
M V

d T

∞
∞

∞ ∞

 
= +    

+  
 

  (37) 

1

2
1

i c i c

i c i ci c

i c

d y V d Y
d t d Td X

M V
d T

β ∞

∞ ∞

=
 

+  
 

   (38) 

1

2
1

i c i c

i c i ci c

i c

d z V d Z
d t d Td X

M V
d T

β ∞

∞ ∞

=
 

+  
 

   (39) 

Define the following perturbation flow velocity 
{ , , }i c i c i cu v w  of the FTV reference frame for an 
incompressible flow system using expressions from (5) and 
(37): 

i c i c
i c

i c i c

d x
u

x d t
ϕ∂

= ≡
∂

      (40) 

i c i c
i c

i c i c

d y
v

y d t
ϕ∂

= ≡
∂

      (41) 

i c i c
i c

i c i c

d z
w

z d t
ϕ∂

= ≡
∂

    (42) 

Define the following perturbation flow velocity 
{ , , }i c i c i cU V W  of the FIS reference frame for an 
incompressible flow system: 

i c
i c

i c

d X
U

d T
≡      (43) 

i c
i c

i c

d Y
V

d T
≡      (44) 

i c
i c

i c

d Z
W

d T
≡      (45) 

Substitute the perturbation velocities from (40) through 
(45) back into (37), (38), & (39), replace the Mach number 

1M ∞  with 1M V c∞ ∞ ∞= , and rearrange the resultant 
terms, such that for incompressible flow systems: 

( )1
1

2
1

i c
i c

i c

U V
u

V
U

c

∞

∞

∞

+
=

 
 +
 
 

        (46) 

1

2
1

i c
i c

i c

V
v

V
U

c

β

∞

∞

=
 
 +
 
 

        (47) 

1

2
1

i c
i c

i c

W
w

V
U

c

β

∞

∞

=
 
 +
 
 

        (48) 

2 21for Mβ ∞= −  & 1M V c∞ ∞ ∞= . The 
expressions in (46) through (48) are collectively called the 
velocity-addition formulas [20] or the composition law for 
velocities. They are mathematical manifestations of the 
Lorentz transformation used in (30) to transform the FIS 
reference frame to a FTV reference frame in incompressible 
flow systems. It is easy to show with numerical simulation 
that the magnitude of the perturbed velocity set 
{ , , }i c i c i cu v w  in formulas (46) to (48) will always vary 

between 0 and the fluid’s characteristic speed c∞  when 
the magnitude of the FIS perturbation velocity set 
{ , , }i c i c i cU V W  varies between 0 and c∞ ; and when the 
magnitude of the free-stream velocity 1V ∞  is restricted to 

vary between 0 and c∞ . 



122 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

Consider for a moment the simple coordinate 
modification for the FTV and FIS velocity representations 
in the X-Y plane of the incompressible flow system: 

i cu u Cosθ= , i cv u Sinθ= , i cU U Cosθ′ ′= , and 

i cV U Sinθ′ ′= . Substitute these trigonometric relationships 
into the Y-axis component of (47): 

( )2 2 2111 1
i cv u Sin

U V c Sin U Cos V c

θ

θ θ

=

′ ′ ′ ′= − + ∞∞ ∞ ∞
(49) 

Perform a binomial series expansion of the square root 
and denominator terms in (49) when the ratio 1 /V c∞ ∞  is 

much less than one such that: u Sin U Sinθ θ′ ′− =  
2 2

1U Sin Cos V cθ θ ∞′ ′ ′− ∞ . Further assume the special 

case where the velocity components u  and U ′  are equal 
to speed c∞ : 1 /Sin Sin Sin Cos V cθ θ θ θ ∞ ∞′ ′ ′− = − ,

iff u c∞≡  & U c∞′ ≡ . Substitute in the sine subtracting 

function Sin Sinθ θ′ − =
( ) ( )2 ( ) / 2 ( ) / 2Cos Sinθ θ θ θ′ ′+ − , such that: 
( ) ( )2 ( ) / 2 ( ) / 2Cos Sinθ θ θ θ′ ′+ −

1 /Sin Cos V cθ θ ∞ ∞′ ′= . Define Δ i cθ θ θ′= −  as the 
difference between the velocity coordinate angles θ′  and 
θ  in the X-Y plane of the incompressible flow system. The 

(( ) / 2)Cos θ θ′ +  and Cosθ′  terms cancel each other 
when Δ i cθ  goes to zero, such that: 

1Δ i c Sin V cθ θ ∞ ∞′= , where 1 / 1V c∞ ∞ << , u c∞≡ , 

U c∞′ ≡ , and Δ 0i cθ → . This final expression for Δ i cθ  
is called the aberration of light formula in relativistic 
physics [20] when c∞  is interpreted as the speed of light in 
vacuum. 

Now consider the relationship between the speed of the 
FTV perturbation velocity set { , , }i c i c i cu v w and the speed 
of the FIS perturbation set { , , }i c i c i cU V W . Define the FTV 

speed i cu  and FIS speed i c || U  for incompressible 

flow as follows: 

2 2 2
i c i c i cu v wi c = + +u     (50) 

2 2 2
i c i c i c i cU V W= + +U        (51) 

Substitute the velocity relationships derived in (46) to (48) 

into the FTV expression for the squared-speed 
2

i cu  in 
(50), such that: 

( )

2

2
12 2 2 2 2 2

1 1 2

1

2

2

2

1

i c

i c i c i c i c i c i c

i c

V
U U V V V W V W

c

V
U

c

∞
∞ ∞

∞

∞

∞

=

 
+ + + + − + 

 
 

 
+ 

 
 

u

 

(52) 

Substitute squared-speed 
2

i cu  from (52) into the 

formula 
2

21 i c c∞− u  and take the square root of the 
resultant expression, such that for incompressible flow: 

2 2

1

2 2 2

1

1 1 1
i c i c

i c i c
V

U
c c c

β

−

∞

∞ ∞ ∞

 
 − = − +
 
 

u U
 

(53) 
The derivation of the velocity addition formulas will not 

be given here for the reverse transformation of going from a 
FTV to a FIS reference frame in an incompressible flow 
system. The final solution is given by the matrix on line g 

of Table 4. In addition, the squared-speed 
2

i cU  can be 
derived with the help of (51) as follows: 

( )

2

2
12 2 2 2 2 2

1 1 2

1

2

2

2

1

i c

i c i c i c i c i c i c

i c

V
u u V V v w v w

c

V
u

c

∞
∞ ∞

∞

∞

∞

=

 
− + + + − + 

 
 

 
− 

 
 

U

  (54) 

The square root of the formula for 
2

21 i c c∞− U  is 
given by: 

2 2

1

2 2 2

1

1 1 1
i c i c

i c
V

u
c c c

β

−

∞

∞ ∞ ∞

 
 − = − −
 
 

U u
 

(55) 
Rearrange (53) and (55) for the Prandtl-Glauert factor β  

and multiply the resultant expressions together, such that: 

1 12

2 2
1 1i c i c

V V
U u

c c
β ∞ ∞

∞ ∞

   
   = + −
   
   

  (56) 

3.5. Summary of Subsonic Velocity Transforms 

This section summarizes in Table 4 all twelve of the 
velocity relationships developed from the twelve coordinate 
transformations listed in Table 3 for subsonic speeds. 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 123 
 

 

Table 4.  Summary of subsonic velocity transformations when using FTV versus FIS reference frames and compressible versus incompressible flow 
systems 

Left side of The equal sign Formula Right side of the equal sign 

a IV 

1

1

2

1 1 0 0
1 0 0 0

0 0 0

i c i c

i c i c i c

i c i c

u V U
V

v U V
c

w W

β
β

− ∞
∞

∞

                 = + + ⋅                          

 IS 

b IV 

1

1

2

1 1 0 0
1 0 0 0

0 0 0

i c c

i c c c

i c c

u V U
V

v U V
c

w W

β
β

− ∞
∞

∞

                 = + + ⋅                          

 CS 

c IV 
12

2

1 1 0 0
0 0
0 0

i c c

i c c c

ci c

u u
V

v u v
c ww

β β
β

−

∞

∞

          = + ⋅                 

 CV 

d CV 

1

0
0

c c

c c

c c

Vu U
v V
w W

∞    
    = +    

    
    

 CS 

e CV 

1

0
0

i cc

c i c

c i c

V Uu
v V
w W

∞    
    = +    
           

 IS 

f CV 
1

2

1 0 0
1 0 1 0

0 0 1

i cc

c i c i c

c i c

uu
V

v u v
cw w

β
β

−

∞

∞

           = − ⋅                 

 

 

IV 

g IS 

1

1

2

1 1 0 0
1 0 0 0

0 0 0

i c i c

i c i c i c

i c i c

U V u
V

V u v
c

W w

β
β

− ∞
∞

∞

    −             = − + ⋅                          

 IV 

h IS 

i c c

i c c

i c c

U U

V V

W W

   
   

=   
      
   

 CS 

i IS 

1

0
0

i c c

i c c

i c c

U V u

V v

W w

∞   − 
    

= +    
           

 CV 

j CS 

1

0
0

c c

c c

c c

VU u
V v
W w

∞−    
    = +    

    
    

 CV 



124 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

Left side of The equal sign Formula Right side of the equal sign 

k CS 

1

0
0

c c

c c

c c

VU u
V v
W w

∞−    
    = +    

    
    

 IS 

l CS 

1

1

2

1 1 0 0
1 0 0 0

0 0 0

c i c

c i c i c

c i c

U V u
V

V u v
c

W w

β
β

− ∞
∞

∞

    −             = − + ⋅                          

 IV 

 
 
The subsonic velocity-addition formulas are shown to be 

entirely based on the characteristics of the partial differential 
equations listed in Table 2 using linearized expressions 
derived from classical fluid dynamic principals, the 
Navier-Stokes equation for compressible flow, and Galilean, 
Miles, and Lorentz transformations. The formulas (46) to (48) 
and the matrix on line a in Table 4 are also identical to those 
developed for special relativity using clocks, rods, the 
Lorentz transformation, and first-order expansions [22]. 
However, it was also pointed out that these velocity-addition 
formulas are only valid when used with fictitious fluid 
systems of incompressible flow where one embeds the 
compressibility effects directly into the definition of the 
coordinates when using the Lorentz transformation. 

Even though the Lorentz transformation is almost 
unanimously associated today with special relativity, it can 
be derived, as shown above, entirely with classical 
hydrodynamic principals for compressible fluids. 

3.6. Transforming Velocities from FIS to FTV Reference 
Frames 

This section describes how the perturbed acceleration 
components developed in one reference frame can be 
related to those in another using the various transformations 
introduced in the previous sections. Let there be two 
different reference frames, called the K  and K′  
systems. Assume the K′  system moves relative to the K  
system with constant velocity ∞V  in a direction that is 
parallel to the X-axis of both systems. Define the X-axis 
acceleration component ua d u d t=  as the particle 
acceleration for the  system and the X-axis acceleration 
component ua d u d t′ ′ ′=  of the same particle in the K′  
system, such that in general: 

u

v

w

a d u d t

a d v d t

a d w d t

= 



= 


= 

    (57) 

u

v

w

a d u d t

a d v d t

a d w d t

′ ′= 

′ ′= 


′ ′= 

    (58) 

3.6.1. Transforming Accelerations from FIS to FTV 
Reference Frames 

Make an incremental change in the velocity terms ciu  
and ciU  in the matrix of line a in Table 4, keeping the 
free-stream velocity ∞V  and speed ∞c  constant, such 
that: 

12

2

2

1i c i c i c
V

d u U dU
c

β

−

∞

∞

 
 = +
 
 

    (59) 

1

2

1 1

2 2

1

2

1

1

i c i c i c

i c i c i c

V
d v U dV

c

V V
U V dU

c c

β

β

−

∞

∞

−

∞ ∞

∞ ∞

 
 = +
 
 

 
 − +
 
 

 (60) 

1

2

1 1

2 2

1

2

1

1

i c i c i c

i c i c i c

V
d w U dW

c

V V
U W dU

c c

β

β

−

∞

∞

−

∞ ∞

∞ ∞

 
 = +
 
 

 
 − +
 
 

(61) 

Divide the three incremental velocity expressions in (59) 
on the left side of the equal sign by 1i cd t V ∞  from the 
fourth line in (35) and on the right side of the equal sign by 

2
1( )i c i c i cd T M d X d T V β∞ ∞+ , such that for 

incompressible flow: 

K



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 125 
 

 

13

2

3

1i c i ci c
i c i c

d u V dU
U

d t d Tc
β

−

∞

∞

 
 = +
 
 

     (62) 

12

2

1 12

2 2

2

3

1

1

i c i c
i c

i c i c

i c
i c i c

i c

d v V dV
U

d t d Tc

V V dU
U V

d Tc c

β

β

−

∞

∞

−

∞ ∞

∞ ∞

 
 = +
 
 

 
 − +
 
 

(63) 

12

2

1 12

2 2

2

3

1

1

i c i c
i c

i c i c

i c
i c i c

i c

d w V dW
U

d t d Tc

V V dU
U W

d Tc c

β

β

−

∞

∞

−

∞ ∞

∞ ∞

 
 = +
 
 

 
 − +
 
 

(64) 

Define the following perturbation flow acceleration 
components { , , }u i c v i c wi ca a a  of the FTV reference 
frame for an incompressible flow system, such that: 

i c
u i c

i c

d u
a

d t
≡       (65) 

i c
v i c

i c

d v
a

d t
≡      (66) 

i c
wi c

i c

d w
a

d t
≡      (67) 

Define the following perturbation flow acceleration 
components { , , }u i c v i c wi cA A A  of the FIS reference 
frame for an incompressible flow system, such that: 

i c
u i c

i c

dU
A

d T
≡      (68) 

i c
v i c

i c

dV
A

d T
≡      (69) 

i c
wi c

i c

dW
A

d T
≡      (70) 

Substitute the accelerations terms of (65) to (70) back into 
the expressions (62) through (64). The final expressions are 
listed on line a of Table 5 [23]. 

 

Table 5.  Summary of subsonic acceleration transformations as a function of using fixed-to-vehicle versus fixed-in-space reference frames and 
compressible versus incompressible flow systems 

Left side of the 
equal sign 

Formula 
Right side of the 

equal sign 

a IV 

13

2

3

1u i c u i c i c
V

a A U
c

β

−

∞

∞

 
 = +
 
 

 

IS 1 1 12 2
2 2 2

2 3

1 1v i c v i c i c i c u i c i c
V V V

a A U V A U
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 

1 1 12 2

2 2 2

2 3

1 1wi c wi c i c i c u i c i c
V V V

a A U W A U
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 

b IV 

13

2

3

1u i c u c c
V

a A U
c

β

−

∞

∞

 
 = +
 
 

 

CS 1 1 12 2
2 2 2

2 3

1 1v i c v c c u c c c
V V V

a A U A V U
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 

1 1 12 2

2 2 2

2 3

1 1wi c wc c u c c c
V V V

a A U A W U
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 



126 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

c IV 

12 2

2

3

u i c u c c
V

a a u
c

β β

−

∞

∞

 
 = +
 
 

 

CV 
1 1 12 2 2 2

2 2 2

2 3

v i c v c c u c c c
V V V

a a u a v u
c c c

β β β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 

1 1 12 2 2 2

2 2 2

2 3

wi c wc c u c c c
V V V

a a u a w u
c c c

β β β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = + − +
   
   

 

d CV c c=a A  CS 

e CV c i c=a A  IS 

f CV 

13

2

3

1u c u i c i c
V

a a u
c

β

−

∞

∞

 
 = −
 
 

 

IV 1 1 12 2
2 2 2

2 3

1 1v c v i c i c u i c i c i c
V V V

a a u a v u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 

1 1 12 2

2 2 2

2 3

1 1wc v i c i c u i c i c i c
V V V

a a u a w u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 

g IS 

13

2

3

1u i c u i c i c
V

A a u
c

β

−

∞

∞

 
 = −
 
 

 

IV 
1 1 12 2

2 2 2

2 3

1 1vi c v i c i c u i c i c i c
V V V

A a u a v u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 

1 1 12 2

2 2 2

2 3

1 1wi c wi c i c u i c i c i c
V V V

A a u a w u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 

h IS i c c=A A  CS 

i IS i c c=A a  CV 

j CS c c=A a  CV 

k CS c i c=A A  IS 

l CS 

13

2

3

1u c u i c i c
V

A a u
c

β

−

∞

∞

 
 = −
 
 

 

IV 1 1 12 2
2 2 2

2 3

1 1v c v i c i c u i c i c i c
V V V

A a u a v u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 

1 1 12 2

2 2 2

2 3

1 1wc v i c i c u i c i c i c
V V V

A a u a w u
c c c

β β

− −

∞ ∞ ∞

∞ ∞ ∞

   
   = − + −
   
   

 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 127 
 

 

3.7. Transforms of Momentum and Fluid Mass between 
Reference Frames 

This section describes how the momentum and fluid mass 
components developed in one reference frame can be 
related to those in another using the various transformations 
previously described. Linear momentum vector P  of an 
infinitesimal volume of fluid is defined as the mass m  of 
the infinitesimal fluid volume times the fluid perturbation 
velocity vector U , such that m=P U . 

3.7.1. Vectors Defined for Incompressible Flow Systems 

Define vector set 
1 2 3

{P , P , P }i c i c i c  to represent the 
momentum components when evaluated in a FIS reference 
frame for incompressible flow conditions, such that 

si c i i cm=P U . Term sim  represents the fluid mass of an 
infinitesimal volume of fluid when evaluated with a FIS 
reference frame for incompressible flow conditions. Define 
vector set 

1 2 3
{p , p , p }i c i c i c  to represent the momentum 

components evaluated with a FTV reference frame for 
incompressible flow conditions, such that vi c i i cm= up . 
Term vim  represents the fluid mass of an infinitesimal 
volume of fluid when evaluated with a FTV reference frame 
for incompressible flow conditions. 

3.7.2. Vectors Defined for Compressible Flow Systems 

In a similar manner, define vector set 
1 2 3

P P P{ , , }c c c  to 
represent the momentum components when evaluated with a 
FIS reference frame for compressible flow conditions, such 
that sc c cm=P U . Term scm  represents the fluid mass 
of an infinitesimal volume of fluid when evaluated with a 
FIS reference frame for compressible flow conditions. Let 
vector set 

1 2 3
{p , p , p }c c c  represent the momentum 

components evaluated with a FTV reference frame for 
compressible flow conditions, such that vc c cm=p u . 
Term vcm  represents the fluid mass of an infinitesimal 
volume of fluid when evaluated with a FTV reference frame 
for compressible flow conditions. 

3.7.3. Coordinate System and Reference Frame Transforms 
Assume the momentum terms for incompressible flow in 

the FIS reference frame are related by a transformation to the 
incompressible flow system in a FTV reference frame in the 
following manner when the free-stream flow (with speed 

∞1V ) is parallel to the X-axis: 

1 1 1

2 2

3 3

1 1 1

s

v s

p P

p P

p P

P

i c i c i c i c i

i c i c

i c i c

i i c i c i c i

E F m V

m V G H m V

∞

∞ ∞

= + 


= 


= 
= + 

   (71) 

The four coefficients { , , , }i c i c i c i cE F G H  in (71) are 
unknown constants. The Jacobian determinant of the matrix 
in (71) must equal a value of plus one, such that: 

1
i c i c

i c i c

E F
Det

G H
 

≡ + 
  

    (72) 

Upon examination of the momentum terms in (71) and 
(72), it is clear that they resemble the same expressions used 
in deriving coordinate reference frame transformations. A 
reasonable guess for the unknown coefficients based on an 
incompressible flow system with FTV and FIS reference 
frames is given by the matrix on line a in Table 3 for just the 
X and T coordinates: 

1 1

2
1 1

p 1 1 P1
p P1

i c i c

i c i cMβ∞ ∞∞

    
= ⋅        

    
  (73) 

The above guess will be shown in (81) that the matrix used 
in (73) is in fact the correct one. 

Replace the momentum and Mach terms in the second line 
of (73) with their mass and velocity components, such that: 

2
1

1 12v s s
1 1

i i i c i
V

m V m U m V
cβ β
∞

∞ ∞
∞

= +     (74) 

Divide (74) by the free-stream speed 1V ∞  and rearrange 
terms to solve for the fluid mass vim  in an incompressible 
flow system and a FTV reference frame, such that: 

1

2v s
1

1i i i c
V

m m U
cβ
∞

∞

 
= + 

 
 

  (75) 

Substitute term 21(1 )i cU V c β∞ ∞+  from (53) into (75) 
and rearrange terms, such that: 

2 2

02
2

i v
v s1 1

i c
i im m mc c∞

− = − =

∞

u U
 (76) 

Term 
0

m  is called the rest mass. It represents the fluid’s 
mass when the fluid at a particular point is brought to 
zero-speed conditions in the incompressible flow system. 
Hence, a relationship can be developed for the fluid mass in 
either the FTV or FIS reference frames of incompressible 
flow by rearranging (76), such that: 

2

0 2v 1
i c

im m c∞
= −

u
   (77) 

2

0 2s 1
i c

im m c∞
= −

U
   (78) 



128 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

The above two equations (77) and (78) for fluid mass for 
incompressible flow systems restricted to subsonic flow 
conditions are identical in form to the expression for 
relativistic mass [24, 25]. The denominator terms are called 
the Lorentz factor. It should be obvious that formulas (77) 
and (78) predict fluid mass in an incompressible flow 
system to approach infinity as the magnitude of either of the 
FTV velocity i cu  or FIS velocity i cU  perturbation 

velocity vectors approach the characteristic speed c∞ . 
Clearly predicting an infinite increase in mass as the 
free-stream speed increases is not physically meaningful 
since the linearized expressions used in the derivation for 
potential flow are no longer valid. Modern day jet airplanes 
are quite capable of flying at both transonic and supersonic 
speeds without gaining an infinite mass. 

The momentum vector i cp  in the FTV reference frame 
for an incompressible flow system can now be expressed in 
terms of the rest mass 

0
m  by substituting (77) for mass 

vim  back into the definition (69), such that: 

0

2

2
1

i c
i c i c m c∞

= −
u

p u      (79) 

In a similar manner, the momentum vector i cP  in the 
FIS reference frame for an incompressible flow system can 
now be expressed in terms of the rest mass 

0
m  by 

substituting (78) for mass sim  back into the definition (70), 
such that: 

2

0 2
1

i c
i c i c m c∞

= −
U

P U    (80) 

Equation (80) is in an identical form to that presented by 
[26]. 

3.7.4. Verifying Fluid Mass Formulation 

The approach presented in the previous section that 
derived the fluid mass for an incompressible flow system 
will now be verified. Write out the first line of either (71) or 
(73) for momentum, such that: 

1v s s
1 1

i i c i i c im u m U m Vβ β ∞
= +   (81) 

Replace the mass term vim  in (81) with (77) and mass 
term sim  with (78), such that: 

0 1 0

2

2 2

2

( )1

1 1

i c i c

i c i c

u m U V m

c c

β
∞

∞ ∞

+
=

− −
u U    (82) 

Multiply (82) by the square root term in the denominator 
on the left side of the equal sign in (82), replace the resultant 
ratio of square roots using (53), and rearrange terms, such 
that: 

( ) 11 2
1

1i c i c i c
V

u U V U
c

−

∞
∞

∞

 
= + + 

 
 

    (83) 

Formula (83) for the i cu  velocity component for an 
incompressible flow system with a FTV coordinate frame 
matches exactly that given by the IS IV→  transformation 
matrix listed on line a of Table 4. 

3.7.5. Summary of Subsonic Fluid Mass Transformations 

A summary for the twelve reference frame formulas used 
for fluid mass transformations is given in Table 6. 

4. Summary 
The paper starts by reviewing the wide spread use of 

Prandtl-Glauert correction factors by aeronautical engineers 
before World War II. Correction factors were needed to 
account for the effects of compressibility as a function of 
subsonic air speeds when using theoretical formulas based 
on an incompressible flow theory. This approach fell out of 
favor with the introduction of jet engines that could propel 
aircraft from subsonic to supersonic speeds. Engineers now 
use computational fluid dynamic algorithms to solve the 
complete Navier-Stokes equations that explicitly include all 
of the compressibility effects. 

A more detailed examination of the mathematics behind 
the origin of the correction factors reveals a profound 
analogy to the equations of special relativity. To see this 
connection, a derivation of the classical three dimensional, 
convected wave equation is given in Cartesian coordinates 
representing the disturbance produced by the subsonic 
movement of a slender, solid object through still air with 
velocity ∞V . The velocity potential of the disturbed air is 
solved under the assumptions of compressible flow 
conditions and a fixed-to-vehicle (FTV) coordinate reference 
frame, abbreviated as reference frame CV. Atmospheric air is 
assumed to have the properties of irrotational, inviscid, 
barotropic, isentropic flow conditions; all external forces 
such as gravity are negligible; and the characteristic speed 
c∞  of free-stream air is constant. In order to simplify the 

expressions, the vehicle velocity ∞V  is aligned along the 
X-axis of the coordinate system. Four partial differential 
equations for the transient wave equation of perturbed 
velocity potential are presented to represent alternative 
reference frames labeled as CV (compressible-FTV), CS 
(compressible-FIS), IV (incompressible-FTV), and IS 
(incompressible-FIS). Only the X space and T time 
coordinate components of the partial differential equations 
vary between the different reference frames. A brute force 



 International Journal of Theoretical and Mathematical Physics 2017, 7(5): 113-131 129 
 

 

algorithm is described that searches for the coefficients of 
the 2x2 linear transformation matrices with a unit Jacobian 
determinant to convert the partial differential equation in 
reference frame CV to those of frames CS, IV, and IS. There 
are a total of twelve matrices needed to describe both 
forward and reverse transformations between the four 
reference frames. After trying approximately one million 
combination of terms, only the coefficients for three unique 
transformation matrices are found: the unit, inverse Galilean, 
and Miles matrices. The Lorentz matrix is obtained by 
multiplying the Galilean and Miles matrices together. 
Incremental differences in all of the space and time terms in 
the coordinate transformation matrices are taken. 
Expressions for the twelve 3x3 velocity transformation 
matrices are found by dividing the dX coordinate equation in 
each matrix by the corresponding dT equation. Similar steps 

are used to derive expressions for twelve 3x3 acceleration 
transformation matrices, and finally twelve 1x1 fluid mass 
transformation matrices. 

Every transformation matrix shown in Tables 4 to 6 has an 
inverse form for itself. If iN  represents the i

th matrix in one 
of these three tables, then the dot product of the matrices 
N N•1 7 , N N•2 12 , N N•3 6 , N N•4 10 , N N•5 9 , 
and N N•8 11  equals the identity matrix. In addition, linking 
the dot products of all twelve matrices from a given table into 
a certain order, such as the sequence N N N N• • • •8 10 6 3  
N N N N N N N N• • • • • • •5 7 2 11 9 4 12 1  will produce a 
circular chain returning to the same coordinates that it started 
with. 

Table 6.  Summary of subsonic fluid mass transformations as a function of using FTV versus FIS reference frames and compressible versus incompressible 
flow systems 

Left Side of the Equal Sign Formula Right Side of the Equal Sign 

a IV 
1

2v s
1

1i i i c
V

m m U
cβ
∞

∞

 
 = +
 
 

 IS 

b IV 
1

2v s
1

1i c c
V

m m U
cβ
∞

∞

 
= + 

 
 

 CS 

c IV 
12

2v v
1

i c c
V

m m u
c

β
β

∞

∞

 
= + 

 
 

 CV 

d CV v sc cm m=  CS 

e CV v sc im m=  IS 

f CV 
1

2v v
1

1c i i c
V

m m u
cβ
∞

∞

 
= − 

 
 

 IV 

g IS 
1

2s v
1

1i i i c
V

m m u
cβ
∞

∞

 
 = −
 
 

 IV 

h IS s si cm m=  CS 

i IS s vi cm m=  CV 

j CS s vc cm m=  CV 

k CS s sc im m=  IS 

l CS 
1

2s v
1

1c i i c
V

m m u
cβ
∞

∞

 
= − 

 
 

 IV 

 



130 Michael James Ungs:  Deriving Special Relativity from the Theory of Subsonic Compressible Aerodynamics  
 

 

5. Discussion 
It is worth the effort to compare the form of the convected 

wave equation describing compressible flow in (84) against 
the form of the wave equation for incompressible flow in (85) 
(i.e., the analogous equation used in electromagnetic theory): 

2 2 2

2 2 2

2 2

2

2

2 2

(1 )

2

c c c

c c c

c c

c c c

M
x y z

M M
x

ϕ ϕ ϕ

ϕ ϕ
τ τ

∞

∞ ∞

∂ ∂ ∂
− + + =

∂ ∂ ∂

∂ ∂
+

∂ ∂ ∂

  (84) 

2

2 2 2 2

2 2 2 2
i c i c i c i c

i c i c i c i c

M
x y z
ϕ ϕ ϕ ϕ

τ∞
∂ ∂ ∂ ∂

+ + =
∂ ∂ ∂ ∂

     (85) 

Just based on visual aesthetics, virtually anyone would 
find (85) more appealing than (84). The extra coefficient 

21 M ∞−  and XT cross-derivative in (84) resemble a type of 
rococo styled mathematics. Surely (85) is the true form of the 
equation representing the simplest expression for 
electromagnetic wave propagation. Why would it ever need 
to be more complicated? However, only (84) accounts for the 
effect of fluid compressibility in an absolute coordinate 
system of space and time. Removing the extra coefficient 
and cross-derivative by means of a coordinate transformation 
renders (85) an expression for a fictitious fluid and for space 
and time coordinates that change in magnitude as a function 
of speed. 

It took many years for the majority of the public and 
scientific community to accept both heavier-than-air flight 
and faster-than-sound flight before WWII. Acceptance was 
stymied by arguments put forward to society through the 
popular press that were often based on political or religious 
ideology and parsimonious logic (i.e., if God intended man 
to fly he would have given us wings). 

As another example, the use and interpretation of 
coordinate transformations can be confounded by using 
distance and time measuring devices that are also affected by 
the compressibility of the media itself. If one used an 
acoustic timing device in an inflight vehicle connected to 
outside conditions, its time delay measurements would also 
be affected by changes in air compressibility as speed varied. 
It then follows that if one rejects the hypotheses of speed 
affecting air compressibility, then discrepancies in the time 
delay measurements from the acoustic device would falsely 
be interpreted as space and time coordinates being bent. 

The derivations given herein demonstrate that the Lorentz 
factor is not special in the normal sense of the word. Rather, 
it is a compressibility correction factor arising from the 
consequence of ignoring compressibility in the formulation 
of the convected wave equation. When compressibility is 
ignored, then correction factors are needed to compensate for 
the mathematical artefacts arising from spatial contraction 
and temporal dilation. In addition, velocity, acceleration, and 
mass vary with speed and no longer obey simple addition 
rules. The story of subsonic compressible aerodynamics is 

presented here because some of the same theoretical and 
mathematical developments occurring in manned flight were 
occurring almost simultaneously in the field of physics. The 
difference between the evolution of theory in aerodynamics 
and physics is that the majority of aerodynamists have 
accepted the concept of air compressibility but the majority 
of physicists have rejected the concept of vacuo 
compressibility. 

6. Conclusions 
Twelve coordinate transformations, plus formulas for 

subsonic velocity, acceleration, and mass are derived entirely 
from a brute force iteration routine using the mathematical 
characteristics of the partial differential equations from 
classical fluid dynamics for compressible flow and its 
transformation to incompressible flow. The mathematical 
derivation and interpretation for the twelve sets of transforms 
are apparently new. In addition, it was shown that there was 
no need to introduce additional assumptions concerning the 
positioning and timing between clocks, chord lengths, and 
speeding planes. 

Examination of the formulas for coordinate 
transformation, velocity addition, and mass equations for 
subsonic conditions shows the equations are mathematically 
identical to those used in special relative to describe the 
motion of an electromagnetic wave or a particle, with the 
speed of light in vacuum used in place of the speed of sound 
for air. The convected wave equation for compressible flow 
and the special relativity wave equation only match exactly if 
the special relativity equations are assumed to be based on 
vacuum conditions, where the vacuum is represented by an 
incompressible flow system with a fixed-in-space reference 
frame (IS). 

The proper starting form for using the convected wave 
equation is the form which includes cross derivatives in time 
and space. Any mathematical transform that removes these 
cross derivatives will convert the resultant equation into a 
non-physical form that represents a fictitious fluid. 

 

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	1. Introduction
	2. Convected Wave Equation
	3. Incompressible Flow
	4. Summary
	5. Discussion
	6. Conclusions