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## 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**SOUND IS A LONGITUDINAL WAVE**dVx c c = ?**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**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**unsteady**steady**SOUND**SPEED (1) at any position, no properties are changing with time (2) V and are only functions of x**cA = (+d)(c-dVx)(A)**cA = cA- d(Vx)A +(d)cA - (d)(dVx)A (dVxA >> ddVxA) (dVx)A = (d)cA dVx = (c/) d c dVx**dRx**This terms appears only if CV isaccelerating**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**Total forces = normal surface forces**Change in momentum flux From continuity eq.**Cons. of**mass Cons. of mass**From momentum eq.**From continuity eq.**dp/d = c2**c = [dp/d]1/2 adiabatic? c = [dp/d]s1/2 or isothermal? c = [dp/d]T1/2**Speed of Propagation**Isentropic & Ideal Gas**Wrong Reasoning**Correct Answer**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**c = [dp/ds]1/2 = [kRT]1/2**c = (kRT)1/2~ 340 m/s ~ 1120 ft/s, for air at STP [krT]1/2 ¾ molecular velocity for a perfect gas = [8RT/]1/2**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**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**Speed of sound (m/s)**steel 5050 seawater 1540 water 1500 air (sea level) 340**Moving Sound Source**Shock wave of bullet piercing sheet of Plexiglass bending of shock due to changes in p and T**V = 0; M = 0**V < c; M < 1 V > c; M > 1 V = c; M = 1 .**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.**shock increases**pressure**ct** vt sin = ct/vt = 1/M = sin-1(1/M)**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.”**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?**Tip speed ~ 1400 ft/s**M ~ 1400/1100 ~ 1.3**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**PROBLEM 1**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**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**PROBLEM 2**Wind = 10 m/s M = 1.35 3 km T = 303 oK • What is airspeed of aircraft? • What is time between seeing aircraft overhead and hearing it?**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 * (1.4*287 [N-m/kg-K] *303 [K])1/2 • = 471 m/s (relative to air)*** 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 vt 3 km ct T = 303 oK time to travel this distance = distance /velocity of plane relative to earth*** 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 = 3000[m]/tan() t = 5.9 s**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**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