1. A course in Gas Dynamics…………………………………. …. …Lecturer: Dr
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
1-4 Speed of Sound
An important parameter in the study of compressible flow is the
speed of sound (or the sonic speed), which is the speed at which an
infinitesimally small pressure wave travels through a medium. Consider a
long constant-area tube filled with fluid and having a piston at one end, as
shown in Figure below. The fluid is initially at rest. At a certain instant the
piston is given an incremental velocity dV to the right. The fluid particles
immediately next to the piston are compressed a very small amount as they
acquire the velocity of the piston. As the piston (and these compressed
particles) continue to move, the next group of fluid particles is compressed
and the wave front is observed to propagate through the fluid at the
characteristic sonic velocity of magnitude a. All particles between the wave
front and the piston are moving with velocity dV to the right and have been
compressed from ρ to (ρ + dρ) and have increased their pressure from p to
(p + dp).

2. pA - (p + dp)A = pAa[(a - dV) - a]
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
• Continuity
For steady one-dimensional flow, we have
m = pVA
But A = const; thus
ρV = const
Application of this to our problem yields
ρa = (ρ + dρ)(a − dV )
Expanding gives us
ρa = ρa − ρ dV + a dρ − dρ dV
Neglecting the higher-order term and solving for dV, we have
adp p
• Momentum
Since the control volume has infinitesimal thickness, we can neglect any
shear stresses along the walls. We shall write the x-component of the
momentum equation, taking forces and velocity as positive if to the right.
For steady one-dimensional flow we may write:
Fx = m(Voutx - Vinx )
pA - (p + dp)A = pAa[(a - dV) - a]
Adp = pAadV
dV =
… … … … … . . (1 - 31)

3. Since we are analyzing an infinitesimal disturbance, we can assume
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
Canceling the area and solving for dV, we have: dp pa
From equations (1-31) and (1-32), we get:
a2 = dp (1 - 33)
Since we are analyzing an infinitesimal disturbance, we can assume
negligible losses and heat transfer as the wave passes through the fluid.
Thus the process is both reversible and adiabatic, which means that it is
isentropic.
a2 = &ap) … … … … . (1 - 34)
This can be expressed in an alternative form by introducing the bulk or
volume modulus of elasticity Ev. This is a relation between volume or
density changes that occurs as a result of pressure fluctuations and is
defined as:
ap ap
s s
Thus
a2 = Ev (1 - 36)
The last two equations are equivalent general relations for sonic velocity
through any medium. The bulk modulus is normally used in connection
with liquids and solids. Table below gives some typical values of this
modulus, the exact value depending on the temperature and pressure of
the medium. For solids it also depends on the type of loading. The
reciprocal of the bulk modulus is called the compressibility.
dV =
… … … … … . . (1 - 32) dp ap *
Ev = -v (av) = p (ap)
(1 - 35) p

4. Bulk Modulus Values for Common Media
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
Bulk Modulus Values for Common Media
Medium Bulk Modulus (psi)
Oil ,000–270,000
Water ,000–400,000
Mercury approx. 4,000,000
Steel approx. 30,000,000
Equation (1-33) is normally used for gases and this can be greatly
simplified for the case of a gas that obeys the perfect gas law. For an
isentropic process, we know that:
p p2 y
p1 s=const p1 p py
p = py C ap
ap s
ap p
ap s py
ap p
ap s p
From equ.(1-33), we get:
a2 = yRT (1 - 38)
a = NOPQ (1 - 39)
Notice that for perfect gases, sonic velocity is a function of the individual
gas and temperature only.
( )
= ( )
(ideal gas)
= const. ( ) =
ypy-1C ( ) =
ypy-1
∴ (
) = y
= yRT
(∵ p = pRT)
(1 - 37)

5. A course in Gas Dynamics…………………………………. …. …Lecturer: Dr
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
The speed of sound changes with temperature and varies with the
fluid.
1.4.1 Mach Number
Mach number, named after the Austrian physicist Ernst Mach( ),
is the ratio of the actual velocity of the fluid ( or an object in still air) to the
speed of sound in the same fluid at the same state.
We define the Mach number as V a
where
V ≡ the velocity of the medium
a ≡ sonic velocity through the medium
It is important to realize that both V and a are computed locally for
conditions that actually exist at the same point. If the velocity at one point
in a flow system is twice that at another point, we cannot say that the Mach
number has doubled.
If the velocity is less than the local speed of sound, M is less than 1 and the
flow is called subsonic. If the velocity is greater than the local speed of
sound, M is greater than 1 and the flow is called supersonic. We shall soon
M =
(1 - 40)