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Chapter 14: Wave Motion

Chapter 14: Wave Motion. Mechanical waves. are disturbances that travel through some material or substance called medium for the waves. travel through the medium by displacing particles in the medium . Types of mechanical waves.

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Chapter 14: Wave Motion

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  1. Chapter 14: Wave Motion • Mechanical waves • are disturbances that travel through some material or substance • called medium for the waves. • travel through the medium by displacing particles in the medium Types of mechanical waves • travel in the perpendicular to or along the movement of the • particles or in a combination of both transverse waves: waves in a string etc. longitudinal waves: sound waves etc. waves in water etc.

  2. Longitudinal and transverse waves Types of mechanical waves (cont’d) sound wave = longitudinal wave C = compression R = rarefaction air compressed air rarefied

  3. Longitudinal-transverse waves Types of mechanical waves (cont’d)

  4. Periodic waves • When particles of the medium in a wave undergo periodic • motion as the wave propagates, the wave is called periodic. l wavelength A amplitude t=0 x=0 x Types of mechanical waves (cont’d) t=T/4 t=T period

  5. Wave function • The wave function describes the displacement of particles • in a wave as a function of time and their positions: • A sinusoidal wave is described by the wave function: sinusoidal wave moving in +x direction Mathematical description of a wave angular frequency velocity of wave, NOT of particles of the medium period wavelength sinusoidal wave moving in -x direction v->-v phase velocity

  6. Wave function (cont’d) l wavelength t=0 x x=0 Mathematical description of a wave (cont’d) t=T/4 t=T period

  7. Wave number and phase velocity wave number: phase Mathematical description of a wave (cont’d) The speed of wave is the speed with which we have to move along a point of a given phase. So for a fixed phase, phase velocity

  8. Particle velocity and acceleration in a sinusoidal wave u in textbook velocity Mathematical description of a wave (cont’d) acceleration Also wave equation

  9. General solution to the wave equation wave equation Solutions: such as Mathematical description of a wave (cont’d) The most general form of the solution:

  10. Wave speed on a string • Consider a small segment of string whose • length in the equilibrium position is • The mass of the segment is • The x component of the force (tension) at both • ends have equal in magnitude and opposite in • direction because this is a transverse wave. Speed of a transverse wave • The total y component of the forces is: Newton’s 2nd law mass acceleration

  11. Wave speed on a string (cont’d) • The total y component of the forces is: Speed of a transverse wave (cont’d) wave eq.

  12. Total energy of a short string segment of mass • At point a, the force does work on the string segment right of point a. work done • Power is the rate of work done : a Energy in wave motion Pmax

  13. Maximum power of a sinusoidal wave on a string: • Average power of a sinusoidal wave on a string • The average of over a period: Energy in wave motion (cont’d) • The average power:

  14. Wave intensity for a three dimensional wave from a point source: power/unit area Wave intensity

  15. The principle of superposition • When two waves overlap, the actual displacement of any • point at any time is obtained by adding the displacement • the point would have if only the first wave were present and • the displacement it would have if only the second wave were • present: Wave interference, boundary condition, and superposition

  16. Interference • Constructive interference (positive-positive or negative-negative) Wave interference, boundary condition, and superposition (cont’d) • Destructive interference (positive-negative)

  17. Reflection incident wave reflected wave • Free end For x<xB Wave interference, boundary condition, and superposition (cont’d) At x=xB Vertical component of the force at the boundary is zero.

  18. Reflection (cont’d) • Fixed end For x<xB Wave interference, boundary condition, and superposition (cont’d) At x=xB Displacement at the boundary is zero.

  19. Reflection (cont’d) • At high/low density Wave interference, boundary condition, and superposition (cont’d)

  20. Reflection (cont’d) • At low/high density Wave interference, boundary condition, and superposition (cont’d)

  21. Superposition of two waves moving in the same direction Standing waves on a string • Superposition of two waves moving in the opposite direction

  22. Superposition of two waves moving in the opposite direction creates a standing wave when two waves have the same speed and wavelength. incident reflected Standing waves on a string (cont’d) N=node, AN=antinode

  23. There are infinite numbers of modes of standing waves funda-mental first overtone Normal modes of a string second overtone third overtone L fixed end fixed end

  24. Sound • Sound is a longitudinal wave in a medium • The simplest sound waves are sinusoidal waves which • have definite frequency, amplitude and wavelength. • The audible range of frequency is between 20 and 20,000 Hz. Sound waves

  25. Sound wave (sinusoidal wave) Sinusoidal sound wave function: Change of volume: Sound waves (cont’d) undisturbed cyl. of air disturbed cyl. of air Pressure: pressure bulk modulus S Dx x x+Dx

  26. Pressure amplitude for a sinusoidal sound wave • Pressure: • Pressure amplitude: • Ear Pressure amplitude and ear

  27. Fourier’s theorem and frequency spectrum • Fourier’s theorem: • Any periodic function of period T can be written as fundamental freq. where Perception of sound waves • Implication of Fourier’s theorem:

  28. Timbre or tone color or tone quality Frequency spectrum noise music Perception of sound waves piano piano

  29. velocity of wave • The speed of sound waves in a fluid in a pipe movable piston longitudinal momentum carried by the fluid in motion fluid in equilibrium original volume of the fluid in motion velocity of fluid change in volume of the fluid in motion Speed of sound waves (ref. only) bulk modulus B: -pressure change/frac. vol. change change in pressure in the fluid in motion fluid in motion fluid at rest boundary moves at speed of wave

  30. The speed of sound waves in a fluid in a pipe (cont’d) longitudinal impulse = change in momentum speed of a longitudinal wave in a fluid Speed of sound waves (ref. only) (cont’d) • The speed of sound waves in a solid bar/rod Young’s modulus

  31. The speed of sound waves in gases bulk modulus of a gas ratio of heat capacities equilibrium pressure of gas In textbook - P in textbook (background pressure). - rdensity Speed of sound waves (cont’d) speed of a longitudinal wave in a fluid gas constant 8.314472 J/(mol K) temperature in Kelvin molar mass

  32. Decibel scale As the sensitivity of the ear covers a broad range of intensities, it is best to use logarithmic scale: Definition of sound intensity: ( unit decibel or dB) Intensity (W/m2) Sound intensity in dB Sound level (Decibel scale)

  33. Sound wave in a pipe with two open ends Standing sound waves

  34. Standing sound wave in a pipe with two open ends Standing sound waves

  35. Sound wave in a pipe with one closed and one open end Standing sound waves

  36. Standing wave in a pipe with two closed ends Displacement Standing sound waves

  37. Normal modes in a pipe with two open ends 2nd normal mode Normal modes

  38. Normal modes in a pipe with an open and a closed end (stopped pipe) Normal modes

  39. Resonance • When we apply a periodically varying force to a system that can • oscillate, the system is forced to oscillate with a frequency equal • to the frequency of the applied force (driving frequency): forced • oscillation. When the applied frequency is close to a characteristic • frequency of the system, a phenomenon called resonance occurs. Resonance • Resonance also occurs when a • periodically varying force is applied • to a system with normal modes. • When the frequency of the applied • force is close to one of normal • modes of the system, resonance • occurs.

  40. Two sound waves interfere each other destructive constructive Interference of waves d2 d1

  41. Two interfering sound waves can make beat Two waves with different frequency create a beat because of interference between them. The beat frequency is the difference of the two frequencies. Beats

  42. Two interfering sound waves can make beat (cont’d) Suppose the two waves have frequencies and For simplicity, consider two sinusoidal waves of equal intensity: Then the resulting combined wave will be: Beats (cont’d) As human ears does not distinguish negative and positive amplitude, they hear two max. or min. intensity per cycle, so 2 x (1/2)|fa-fb|= |fa-fb| is the beat frequency fbeat.

  43. Moving listener Doppler effect Source at rest Listener moving left Source at rest Listener moving right

  44. Moving listener (cont’d) • The wavelength of the sound wave does not change whether • the listener is moving or not. • The time that two subsequent wave crests pass the listener • changes when the listener is moving, which effectively changes • the velocity of sound. Doppler effect (cont’d) freq. listener hears freq. source generates velocity of sound at source - for a listener moving away from + for a listener moving towards the source. velocity of listener

  45. Moving source Doppler effect (cont’d) When the source moves

  46. Moving source (cont’d) • The wave velocity relative to the wave medium does not • change even when the source is moving. • The wavelength, however, changes when the source is moving. • This is because, when the source generates the next crest, the • the distance between the previous and next crest i.e. the wave- • length changed by the speed of the source. Doppler effect (cont’d) The source at rest When the source is moving + for a receding source - for a approaching source

  47. Moving source and listener - for a listener moving away from + for a listener moving towards the source. + for a receding source - for a approaching source The signs of vL and vS are measured in the direction from the listener L to the source S. Doppler effect (cont’d) • Effect of change of source speed

  48. Example 1 • A police siren emits a sinusoidal wave with frequency fs=300 Hz. • The speed of sound is 340 m/s. a) Find the wavelength of the waves • if the siren is at rest in the air, b) if the siren is moving at 30 m/s, find • the wavelengths of the waves ahead of and behind the source. • a) • b) In front of the siren: • Behind the siren: Doppler effect (cont’d)

  49. Example 2 • If a listener l is at rest and the siren in Example 1 is moving away • from L at 30 m/s, what frequency does the listener hear? • Example 3 Doppler effect (cont’d) • If the siren is at rest and the listener is moving toward the left at 30 • m/s, what frequency does the listener hear?

  50. Example 4 • If the siren is moving away from the listener with a speed of 45 m/s • relative to the air and the listener is moving toward the siren with a • speed of 15 m/s relative to the air, what frequency does the listener • hear? • Example 5 Doppler effect (cont’d) • The police car with its 300-MHz siren is moving toward a warehouse • at 30 m/s, intending to crash through the door. What frequency does • the driver of the police car hear reflected from the warehouse? Freq. reaching the warehouse Freq. heard by the driver

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