Short Version : 14. Wave Motion

1. Short Version : 14. Wave Motion

2. Wave Properties Wave amplitude Waveform • Pulse • Continuous wave • Wave train Periodicity in space : Wavelength  Wave number k = 2/ Periodicity in time : Period T Frequency  = 2/T

3. Longitudinal & Transverse Waves Longitudinal waves Transverse waves 1-D Vibration Transverse Longitudinal Water Waves Water waves mixed

4. Wave Speed Speed of wave depends only on the medium. Sound in air  340 m/s  1220 km/h. in water  1450 m/s in granite  5000 m/s Small ripples on water  20 cm/s. Earthquake  5 km/s. Wave speed

5. 14.2. Wave Math pk @ x = 0 pk @ x = v t At t = 0, At t , y(0) is displaced to the right by v t.  For a wave moving to the left : For a SHW (sinusoidal): = wave number SHW moving to the right : = phase = wave speed Waves

6. The Wave Equation 1-D waves in many media can be described by the partial differential equation Wave Equation whose solutions are of the form ( towards  x ) v = velocity of wave.  • E.g., • water wave ( y = wave height ) • sound wave ( y = pressure ) • …

7. 14.3. Waves on a String A pulse travels to the right. In the frame moving with the pulse, the entire string moves to the left. Top of pulse is in circular motion with speed v & radius R. Centripedal accel: Tension force F is cancelled out in the x direction: ( small segment )   = mass per unit length [ kg/m ]

8. Wave Power SHO : Segment of length x at fixed x : v = phase velocity of wave

9. Wave Intensity Intensity = power per unit area  direction of propagation [ W / m2 ] Wave front = surface of constant phase. Plane wave : planar wave front. Spherical wave : spherical wave front. Plane wave : Spherical wave :

10. 14.4. Sound Waves Sound waves = longitudinal mechanical waves through matter. Speed of sound in air : P,  = max , x = 0 P = background pressure. = mass density.  = 7/5 for air & diatomic gases.  = 5/3 for monatomic gases, e.g., He. P,  = eqm , |x| = max P,  = min , x = 0

11. Sound & the Human Ear Audible freq: 20 Hz ~ 20 kHz Bats: 100 kHz Ultrasound: 10 MHz db = 0 : Hearing Threshold @ 1k Hz

12. Decibels Sound intensity level : [  ] = decibel (dB)  Threshold of hearing at 1 kHz.   Nonlinear behavior: Above 40dB, the ear percieves  = 10 dB as a doubling of loudness.

13. 14.5. Interference Principle of superposition: tot = 1 + 2 . constructive interference destructive interference Interference

14. Fourier Analysis Fourier analysis: Periodic wave = sum of SHWs. Fourier Series E note from electric guitar

15. Dispersion Dispersion: wave speed is wavelength (or freq) dependent Non-dispersive medium Dispersion Surface wave on deep water: Dispersive medium  long wavelength waves reaches shore 1st. Dispersion of square wave pulses determines max length of wires or optical fibres in computer networks.

16. Beats Beats: interference between 2 waves of nearly equal freq. Constructive Destructive Freq of envelope = 1  2 . smaller freq diff  longer period between beats Beats Applications: Synchronize airplane engines (beat freq  0). Tune musical instruments. High precision measurements (EM waves).

17. Interference in 2-D Destructive Constructive Nodal lines: amplitude  0 path difference = ½ n Water waves from two sources with separation  Interference

18. 14.6. Reflection & Refraction light + heavy ropes A = 0; reflected wave inverted A = max; reflected wave not inverted Partial Reflection Fixed end Rope Free end

19. Partial reflection + normal incidence Partial reflection + oblique incidence  refraction

20. Application: Probing the Earth P wave = longitudinal S wave = transverse S wave shadow  liquid outer core P wave partial reflection  solid inner core Explosive thumps  oil / gas deposits

21. 14.7. Standing Waves Superposition of right- travelling & reflected waves:  B =  A  standing wave String with both ends fixed:  Allowed waves = modes or harmonics n = 1  fundamental mode n > 1  overtones n = mode number Standing Waves y = 0  node y = max  antinode

22. 1 end fixed  node, 1 end free  antinode.  Standing Waves

23. 14.8. The Doppler Effect & Shock Waves Point source at rest in medium radiates uniformly in all directions. When source moves, wave crests bunch up in the direction of motion (   ). Wave speed v is a property of the medium & hence independent of source motion.  Approaching source:   f  Doppler effect

24. t = 0 u T T = period of wave u = speed of source t = T t = 2T 2 uT = uT . Moving Source

25. t = 0 u T T = period of wave u = speed of source t = T t = 2T 2 uT = uT . Moving Source

26. Moving Observers An observer moving towards a point source at rest in medium sees a faster moving wave. Since  is unchanged, observed f increases. For u/v << 1: Prob. 76 • Waves from a stationary source that reflect from a moving object undergo 2 Doppler effects. • A f toward shift at the object. • A f approach shift when received at source.

27. Doppler Effect for Light Doppler shift for EM waves is the same whether the source or the observer moves. correct to 1st order in u/c

28. Shock Waves Shock wave: u > v Mach number = u / v Mach angle = sin1(v/u)  if Source, 1 period ago Moving Source Shock wave front E.g., Bow wave of boat. Sonic booms. Solar wind at ionosphere