speed of propagation “speed of sound is one of the most important quantities in the study of...
TRANSCRIPT
![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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/1.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/2.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/3.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/4.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/5.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/6.jpg)
(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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/7.jpg)
(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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/8.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/9.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/10.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/11.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/12.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/13.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/14.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/15.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/16.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/17.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/18.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/19.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/20.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/21.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/22.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/23.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/24.jpg)
![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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/25.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/26.jpg)
![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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/27.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/28.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/29.jpg)
(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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/30.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/31.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/32.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/33.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/34.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/35.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/36.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/37.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/38.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/39.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/40.jpg)
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](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f3e5503460f94c5f3e2/html5/thumbnails/41.jpg)
The End