Speed of Propagation

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Speed of Propagation
speed of sound is one of the most important
quantities in the study of incompressible
flow - Anderson Moderns Compressible Flow
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SOUND IS A LONGITUDINAL WAVE
dVx
c
c ?
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Speed of Propagation

• sound wave are propagated by
• molecular collisions
• sound wave causes very small
• changes in p, ?, T
• sound wave by definition is weak
• (relative to ambient)
• shock waves are strong
• (relative to ambient)
• and travel faster

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Speed of Propagation Isentropic
I S E N T R O P I C

• changes within wave are small
• gradients are negligible
• particularly for long waves
• implies irreversible
• dissipative effects due to friction and
• conduction are negligible
• no heat transfer through control
• volume
• implies adiabatic

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unsteady
steady
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SOUND SPEED
(1) at any position, no properties are changing
with time
(2) V and ? are only functions of x
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?cA (?d?)(c-dVx)(A) ?cA ?cA- ?d(Vx)A (d?)cA
- (d?)(dVx)A
(?dVxA gtgt d?dVxA) ?(dVx)A (d?)cA dVx (c/?) d?
c

dVx
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dRx
This terms appears only if CV is
accelerating
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dRx represents tangential forces on control
volume because there is no relative motion
along wave (wave is on both sides of top and
bottom of control volume), dRx 0. So FSx
-Adp
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Total forces normal surface forces
Change in momentum flux
From continuity eq.
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Cons. of mass
Cons. of mass
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From momentum eq.
From continuity eq.
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dp/d? c2 c dp/d?1/2 adiabatic? c
dp/d?s1/2 or isothermal? c dp/d?T1/2
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Speed of Propagation
Isentropic Ideal Gas
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Wrong Reasoning
Correct Answer
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For ideal gas, isentropic, constant cp and cv
p/?k const p const ?k const p/ ?k dp/d?s
d(const ?k)/d? kconst?k-1 dp/d?s kp/?
dp/d?s k ?RT/? kRT
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c dp/d?s1/2 kRT1/2
c (kRT)1/2 340 m/s 1120 ft/s, for air at STP
krT1/2 ? ¾ molecular velocity for a perfect
gas 8RT/?1/2
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Note the adiabatic approximation is better at
lower frequencies than higher frequencies
because the heat production due to conduction is
weaker when the wavelengths are longer
(frequencies are lower). The often stated
explanation, that oscillations in a sound wave
are too rapid to allow appreciable conduction of
heat, is wrong. pg 36, Acoustics by Allan
Pierce
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Newton was the first to predict the velocity of
sound waves in air. He used Boyles Law and
assumed constant temperature. c2 dp/d?
?p/??T
FOR IDEAL GAS p ?RT p/? const
if constant temperature Then dp/d? d(?RT)/d?
RT c (RT)1/2 isothermal (k)1/2 too
small or (1/1.18) (340 m/s) 288 m/s
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Speed of sound (m/s) steel 5050 seawater
1540 water 1500 air (sea level) 340
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Moving Sound Source
Shock wave of bullet piercing sheet of
Plexiglass bending of shock due to changes in p
and T
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V 0 M 0
V lt c M lt 1

V gt c M gt 1
V c M 1
.
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As measured by the observer the frequency of
sound coming from the approaching siren is
greater than the frequency of sound from the
receding siren.
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shock increases pressure
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c?t
?
v?t
sin ? c?t/v?t 1/M ? sin-1 (1/M)
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Mach (1838-1916) First to make shock waves
visible. First to take photographs of
projectiles in flight. Turned philosopher
psychophysics all knowledge is based on
sensations
I do not believe in atoms.
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POP QUIZ

• What do you put in a toaster?
• (2) Say silk 5 times,what do cows drink
• (3) What was the first man-made
• object to break the sound barrier?

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Tip speed 1400 ft/s M 1400/1100 1.3
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Sound Propagation Problems
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PROBLEM 1
(faster than a speeding bullet)
Lockheed SR-71 aircraft cruises at around M
3.3 at an altitude of 85,000 feet (25.9 km).
What is flight speed? Table A.3,
pg 719 24km T(K) 220.6 26km T(K)
222.5 25,900m 220.6 1900m (1.9K/2000m)
222 K
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PROBLEM 1
c kRT1/2 1.4287 (N-m)/(kg-K) 222
K1/2 299 m/s V Mc 3.3 299 m/s
987 m/s The velocity of a 30-ob rifle bullet
is about 700 m/s Vplane /
Vbullet 987/700 1.41
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Not really linear, although not apparent at the
scale of this plot.
For standard atm. conditions c 340 m/s at sea
level c 295 m/s at 11 km
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PROBLEM 2
Wind 10 m/s
M 1.35
3 km
T 303 oK

1. What is airspeed of aircraft?
2. What is time between seeing aircraft overhead and
   hearing it?

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PROBLEM 2
M V/c V is airspeed
Wind 10 m/s
M 1.35
3 km
T 303 oK

• What is airspeed of aircraft?
• V (airspeed) Mc 1.35 (kRT)1/2
• 1.35 (1.4287
  N-m/kg-K 303 K)1/2
• 471 m/s (relative to air)

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note if T ? not constant, Mach line would
not be straight
M 1.35
Wind 10 m/s
v is velocity relative to earth 471 10 461
m/s sin ? c/v 1/M
v?t
?
3 km
c?t
T 303 oK
?
?
time to travel this distance distance
/velocity of plane relative to earth
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note if T ? not constant, Mach line would
not be straight
D Vearth t
Wind 10 m/s
M 1.35
?
3000m
3 km
T 303 oK
?

• (b) What is time between seeing aircraft overhead
  and hearing it?
• ? sin-1 (1/M) sin-1 (1/1.35) 47.8o
• Vearth 471m/s 10m/s 461m/s
• D Veartht 461 m/s t 3000m/tan(?)
• t 5.9 s

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Problem 3 Prove that for an ideal
calorically perfect gas that M2 is
proportional to (Kinetic Energy per unit mass
V2/2) (Internal Energy per unit mass u)
Hint Use u cvT cv R/(k-1) c
(kRT)1/2 show that proportionality constant
k(k-1)/2
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Problem 4 Prove that for an ideal calorically
perfect gas that M2 is proportional
to Dynamic Pressure ½ ?V2 Static
Pressure p
Hint Use p ?RT c (kRT)1/2 M V/c
show that proportionality constant k/2
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The End