1. 1 Linear Wave Equation The maximum values of the transverse speed and transverse acceleration are v y, max =  A a y, max =  2 A The transverse speed and acceleration do not reach their maximum values simultaneously v is a maximum at y = 0 a is a maximum at y =  A

2. 2 The Linear Wave Equation, cont. The wave functions y (x, t) represent solutions of an equation called the linear wave equation This equation gives a complete description of the wave motion From it you can determine the wave speed The linear wave equation is basic to many forms of wave motion

3. 3 Linear Wave Equation, General The equation can be written as This applies in general to various types of traveling waves y represents various positions For a string, it is the vertical displacement of the elements of the string For a sound wave, it is the longitudinal position of the elements from the equilibrium position For em waves, it is the electric or magnetic field components

4. 4 Linear Wave Equation, General cont The linear wave equation is satisfied by any wave function having the form y (x, t) = f (x  vt) Nonlinear waves are more difficult to analyze A nonlinear wave is one in which the amplitude is not small compared to the wavelength

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8. 8 13.4 Linear Wave Equation Applied to a Wave on a String The string is under tension T Consider one small string element of length  s The net force acting in the y direction is This uses the small-angle approximation

9. 9 Linear Wave Equation and Waves on a String, cont  s is the mass of the element Applying the sinusoidal wave function to the linear wave equation and following the derivatives, we find that This is the speed of a wave on a string It applies to any shape pulse

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16. 16 13.5 Reflection of a Wave, Fixed End When the pulse reaches the support, the pulse moves back along the string in the opposite direction This is the reflection of the pulse The pulse is inverted when it is reflected from a fixed boundary

17. 17 Reflection of a Wave, Free End With a free end, the string is free to move vertically The pulse is reflected The pulse is not inverted when reflected from a free end

18. 18 Transmission of a Wave When the boundary is intermediate between the last two extremes Part of the energy in the incident pulse is reflected and part undergoes transmission Some energy passes through the boundary

19. 19 Transmission of a Wave, 2 Assume a light string is attached to a heavier string The pulse travels through the light string and reaches the boundary The part of the pulse that is reflected is inverted The reflected pulse has a smaller amplitude

20. 20 Transmission of a Wave, 3 Assume a heavier string is attached to a light string Part of the pulse is reflected and part is transmitted The reflected part is not inverted

21. 21 Transmission of a Wave, 4 Conservation of energy governs the pulse When a pulse is broken up into reflected and transmitted parts at a boundary, the sum of the energies of the two pulses must equal the energy of the original pulse

22. 22 13.6 Energy in Waves in a String Waves transport energy when they propagate through a medium We can model each element of a string as a simple harmonic oscillator The oscillation will be in the y-direction Every element has the same total energy

23. 23 Demonstration for energy transfer by wave propagation

24. 24 A sinusoidal wave on a string

25. 25 Energy, cont. Each element can be considered to have a mass of  m Its kinetic energy is  K = 1/2 (  m) v y 2 The mass  m is also equal to  x and  K = 1/2 (  x) v y 2 As the length of the element of the string shrinks to zero, the equation becomes a differential equation: dK =1/2 (  x) v y 2 = 1/2  2 A 2 cos 2 (kx –  t) dx

26. 26 Energy, final Integrating over all the elements, the total kinetic energy in one wavelength is K = 1/4  2 A 2 The total potential energy in one wavelength is U = 1/4  2 A 2 This gives a total energy of E = K + U = 1/2  2 A 2

27. 27 Power Associated with a Wave The power is the rate at which the energy is being transferred: The power transfer by a sinusoidal wave on a string is proportional to the Square of the frequency Square of the amplitude Speed of the wave

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30. 30 13.7 Introduction to Sound Waves Sound waves are longitudinal waves They travel through any material medium The speed of the wave depends on the properties of the medium The mathematical description of sinusoidal sound waves is very similar to sinusoidal waves on a string

31. 31 Speed of Sound Waves Use a compressible gas as an example with a setup as shown at right Before the piston is moved, the gas has uniform density When the piston is suddenly moved to the right, the gas just in front of it is compressed Darker region in the diagram

32. 32 Speed of Sound Waves, cont When the piston comes to rest, the compression region of the gas continues to move This corresponds to a longitudinal pulse traveling through the tube with speed v The speed of the piston is not the same as the speed of the wave The light areas are rarefactions

33. 33 Description of a Sound Wave The distance between two successive compressions (or two successive rarefactions) is the wavelength, As these regions travel along the tube, each element oscillates back and forth in simple harmonic motion Their oscillation is parallel to the direction of the wave

34. 34 Displacement Wave Equation The displacement of a small element is s(x,t) = s max sin (kx –  t) s max is the maximum position relative to equilibrium This is the equation of a displacement wave k is the wave number  is the angular frequency of the piston

35. 35 Pressure Wave Equation The variation  P in the pressure of the gas as measured from its equilibrium value is also sinusoidal  P =  P max cos (kx –  t) The pressure amplitude,  P max is the maximum change in pressure from the equilibrium value The pressure amplitude is proportional to the displacement amplitude  P max =  v  s max V is the speed of the wave.

36. 36 Sound Waves as Displacement or Pressure Wave A sound wave may be considered either a displacement wave or a pressure wave The pressure wave is 90 o out of phase with the displacement wave

37. 37 Speed of Sound Waves, General The speed of sound waves in air depends only on the temperature of the air v = 331 m/s + (0.6 m/s. o C) T C T C is the temperature in Celsius The speed of sound in air at 0 o C is 331 m/s

38. 38 Speed of Sound in Gases, Example Values Note: temperatures given, speeds are in m/s

39. 39 Speed of Sound in Liquids, Example Values Speeds are in m/s

40. 40 Speed of Sound in Solids, Example Values Speeds are in m/s; values are for bulk solids

41. 41 13.8 The Doppler Effect The Doppler effect is the apparent change in frequency (or wavelength) that occurs because the relative motion between the source of a wave and the observer When the motion of the source or the observer moves toward the other, the frequency appears to increase When the motion of the source or the observer moves away from the other, the frequency appears to decrease

42. 42 Doppler Effect, Observer Moving The observer moves with a speed of v o Assume a point source that remains stationary relative to the air It is convenient to represent the waves with a series of circular arcs concentric to the source These surfaces are called a wave front

43. 43 Doppler Effect, Observer Moving, cont The distance between adjacent wave fronts is the wavelength The speed of the sound is v, the frequency is ƒ, and the wavelength is When the observer moves toward the source, the speed of the waves relative to the observer is v rel = v + v o The wavelength is unchanged

44. 44 Doppler Effect, Observer Moving, final The frequency heard by the observer, ƒ ’, appears higher when the observer approaches the source The frequency heard by the observer, ƒ ’, appears lower when the observer moves away from the source

45. 45 Doppler Effect, Source Moving Consider the source being in motion while the observer is at rest As the source moves toward the observer, the wavelength appears shorter As the source moves away, the wavelength appears longer

46. 46 Doppler Effect, Source Moving, cont When the source is moving toward the observer, the apparent frequency is higher When the source is moving away from the observer, the apparent frequency is lower

47. 47 Doppler Effect, General Combining the motions of the observer and the source The signs depend on the direction of the velocity A positive value is used for motion of the observer or the source toward the other A negative sign is used for motion of one away from the other

48. 48 Doppler Effect, final Convenient rule for signs The word toward is associated with an increase in the observed frequency The words away from are associated with a decrease in the observed frequency The Doppler effect is common to all waves The Doppler effect does not depend on distance

49. 49 Shock Wave The speed of the source can exceed the speed of the wave The concentration of energy in front of the source results in a shock wave

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55. 55 13.9 Speed of Sound Waves, General The speed of sound waves in a medium depends on the compressibility and the density of the medium The compressibility can sometimes be expressed in terms of the elastic modulus of the material The speed of all mechanical waves follows a general form:

56. 56 Speed of Transverse Wave in a Bulk Solid The shear modulus of the material is S The density of the material is  The speed of sound in that medium is

57. 57 Speed of Sound in Liquid or Gas The bulk modulus of the material is B The density of the material is  The speed of sound in that medium is

58. 58 Speed of a Longitudinal Wave in a Bulk Solid The bulk modulus of the material is B The shear modulus of the material is S The density of the material is  The speed of sound in that medium is

59. 59 Seismic Waves When an earthquake occurs, a sudden release of energy takes place at its focus or hypocenter. The epicenter is the point on the surface of the Earth radially above the focus The released energy will propagate away from the focus by means of seismic waves

60. 60 Types of Seismic Waves P waves P stands for primary They are longitudinal waves They arrive first at a seismograph S waves S stands for secondary They are transverse waves They arrive next at the seismograph

61. 61 Seismograph Trace

62. 62 Cross-section of the Earth showing paths of waves produced by an earthquake

63. Exercises 6, 13, 18, 23, 27, 33, 42, 46, 59, 60, 61, 68