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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

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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Page 1: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Speed of Propagation

“speed of sound is one of the most important quantities in the study of incompressible flow”

- Anderson : Moderns Compressible Flow

Page 2: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

c

SOUND IS A LONGITUDINAL WAVE

dVx

c = ?

Page 3: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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

Page 4: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Speed of Propagation = Isentropic

• 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

ISENTROPIC

Page 5: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

unsteady

steady

Page 6: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

(1) at any position, no properties are changing with time

(2) V and are only functions of x

SOUNDSPEED

Page 7: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

(dVxA >> ddVxA)(dVx)A = (d)cA

dVx = (c/) d

cA = (+d)(c-dVx)(A)cA = cA- d(Vx)A +(d)cA - (d)(dVx)A

c

dVx

Page 8: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

This terms appears only if CV is accelerating

dRx

Page 9: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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

Page 10: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Total forces = normal surface forces Change in momentum flux

From continuity eq.

Page 11: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Cons. of mass

Cons. of mass

Page 12: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

From momentum eq.

From continuity eq.

Page 13: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

dp/d = c2

c = [dp/d]1/2

adiabatic? c = [dp/d]s1/2

or isothermal? c = [dp/d]T

1/2

Page 14: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Speed of Propagation

Isentropic & Ideal Gas

Page 15: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Correct Answer Wrong Reasoning

Page 16: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

For ideal gas, isentropic, constant cp and cv:

p/k = const p = const k const = p/ k

dp/ds = d(const k)/d = kconstk-1 dp/ds = kp/

dp/ds = k RT/ = kRT

Page 17: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

c = (kRT)1/2 ~ 340 m/s~ 1120 ft/s, for air at STP

c = [dp/ds]1/2 = [kRT]1/2

[krT]1/2 ¾ molecular velocity for a perfect gas = [8RT/]1/2

Page 18: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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

Page 19: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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 temperatureThen: 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

Page 20: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Speed of sound (m/s)steel 5050seawater 1540water 1500air (sea level) 340

Page 21: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Moving Sound Source

Shock wave of bullet piercing sheet of Plexiglass bending of shock due to changes in p and T

Page 22: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

V = 0; M = 0 V < c; M < 1

V = c; M = 1 V > c; M > 1

.

Page 23: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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.

Page 24: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow
Page 25: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

shock increases pressure

Page 26: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow
Page 27: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

vt

ct

sin = ct/vt = 1/M = sin-1

(1/M)

Page 28: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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.”

Page 29: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

(1)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?

POP QUIZ

Page 30: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Tip speed ~ 1400 ft/sM ~ 1400/1100 ~ 1.3

Page 31: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Sound Propagation Problems

Page 32: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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 71924km T(K) = 220.626km T(K) = 222.525,900m ~ 220.6 + 1900m * (1.9K/2000m) ~ 222 K

PROBLEM 1 (faster than a speeding bullet)

Page 33: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

c = {kRT}1/2 = {1.4*287 [(N-m)/(kg-K)] 222 [K]}1/2 = 299 m/s

V = M*c = 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

PROBLEM 1

Page 34: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

Not really linear, althoughnot apparent at the scale

of this plot.

For standard atm. conditionsc= 340 m/s at sea levelc = 295 m/s at 11 km

Page 35: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

3 km

M = 1.35

T = 303 oK

Wind = 10 m/s

(a) What is airspeed of aircraft?(b) What is time between seeing aircraft overhead and hearing it?

PROBLEM 2

Page 36: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

3 km

M = 1.35

T = 303 oK

Wind = 10 m/s

(a) What is airspeed of aircraft?V (airspeed) = Mc = 1.35 * (kRT)1/2 = 1.35 * (1.4*287 [N-m/kg-K] *303 [K])1/2

= 471 m/s (relative to air)

PROBLEM 2

M = V/cV is airspeed

Page 37: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

3 km

M = 1.35

T = 303 oK

Wind = 10 m/s

* note: if T & not constant, Mach line would not be straight

vt

ct

v is velocityrelative to earth= 471 – 10= 461 m/s

sin = c/v = 1/M

time to travel this distance = distance /velocity of plane relative to earth

Page 38: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

3 km

M = 1.35

T = 303 oK

Wind = 10 m/s

(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/sD = Veartht = 461 [m/s] t = 3000[m]/tan() t = 5.9 s

D = Vearth t

3000m

* note: if T & not constant, Mach line would not be straight

Page 39: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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

Page 40: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

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

Page 41: Speed of Propagation “speed of sound is one of the most important quantities in the study of incompressible flow” - Anderson : Moderns Compressible Flow

The End