Speed of Sound
• Piston in duct, no heat transfer, no
shaft work, no friction
• Incompressible flow: if piston pushed all
is displaced instantaneously (solid body,
“message velocity” infinite)
• Compressible flow: finite time for
message to travel along the tube.
Speed of Sound
• Layer of air on piston
face vp, p+p, +
• Piston stopped, pulse of
air continues
• Ride on the wave; have
flow come towards you
• Converts to steady flow
V=0
p, 
vp
V=v(x)
P=p(x)
(x)
vp=v
p, 

Euler equation:
v
dv
1 dp

dx
 dx
v
(1)
Continuity:
d
dv
dv
(v)  0  v    0
dx
dx
dx
1 d
1 dv
or

(2)
 dx v dx
p
x

x
x
Equating LHS of (2) & (4)
Intensive variable p(,s)
 2 p  1 dv
0
v    
  sv dx
 p 
 p 
dp  
 d  
 ds
 s  
   s
if isent ropic:
dp  p  
 

dx    s x
(3) int o (1) :

 v
1 d
 
 dx
 p
 
(3)

 dv

 dx
s
(5)
Two solutions:
dv
0
incompressible, v(=c) = 
dx
p  2
dv
2
2. v     c implies
finite
dx
 s
1.

Saw for perfect gas:
(4)

p 
RT
RT
   cc 
 s
Mach Waves
• Point source of sound along a line
normal to the screen and the direction
of a flow where v>c.
v>c
• Zone of action wedge

shaped, half angle
  sin1 1 M  sin1 c v
ct

(Mach angle)
vt
Moving Sound Source
Zone of
silence
Mach cone
Zone of
action
3rd
2nd
1st
Mach Waves, Mach Cones
• For single point source the zone of
action is a cone with same half angle
• Inside wedge/cone is the zone of action
• Outside zone of silence
• Stronger (finite disturbances) move at
speeds > c
• Half angles are larger, different for
wedge and cone.
Velocity - Area Relationship
• Flow model:
– Reversible, adiabatic (isentropic)
– No external work
– Steady, 1-D flow
• Governing Equations:
– Energy equation:
2
v
h0  h 
– Euler Equation
vdv 
– Continuity


2
 0 = dh +vdv (1)
1
dp  0

(2)
d dv dA
vA  const . 
  0
 v A
(3)
Velocity - Area Relationship
– 2nd Law of Thermodynamics: ds  0
(4)
– Thermal equation of state:

Finding the area relationship:
(5)
1
1 dp
1 2
vdv   dp  
d   c d 

 d

pv  0
2
d

v
v
   2 dv   2 dv  M 2 dv

c
c v
v
(6)
Velocity - Area Relationship
(6) into (3)
dv
1 dA

v 1 M 2 A
(7)
• Physical significance:
- for M<1: if A  then v 
dv
v  1 
  
0
2 
dA A 1 M 
- for M>1: if A  then v 
- for M=1: 1-M2 = 0
dvonly finite if dA goes
 to 0 (at throat)

Velocity - Pressure Relation
dp
 v  0
dv
For all values of M:


if v  then p 
Summary
• For subsonic flow an increase in area
decelerates the flow and vice versa. The
increase becomes larger as M increases.
• For supersonic flow an increase in area
accelerates the flow and vice versa.
• Flow can only accelerate from subsonic to
supersonic through a throat; can only
decelerate from M>1 to M<1 isentropically
through a throat.
• When flow decelerates, pressure drops.