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Chapter 14 Wave Motion

Chapter 14 Wave Motion. Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic – governed by electricity and magnetism equations, may exist without any medium Matter – governed by quantum mechanical equations.

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Chapter 14 Wave Motion

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  1. Chapter 14 Wave Motion

  2. Types of waves • Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) • Electromagnetic – governed by electricity and magnetism equations, may exist without any medium • Matter – governed by quantum mechanical equations

  3. Types of waves • Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: • Transverse – if the direction of displacement is perpendicular to the direction of propagation • Longitudinal – if the direction of displacement is parallel to the direction of propagation

  4. Types of waves • Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: • Transverse – if the direction of displacement is perpendicular to the direction of propagation • Longitudinal – if the direction of displacement is parallel to the direction of propagation

  5. The linear wave equation • Let us consider transverse waves propagating without change in shape and with a constant wave velocityv • We will describe waves via vertical displacementy(x,t) • For an observer moving with the wave • the wave shape doesn’t depend on time y(x’) = f(x’)

  6. The linear wave equation • For an observer at rest: • the wave shape depends on time y(x,t) • the reference frame linked to the wave is moving with the velocity of the wave v

  7. The linear wave equation • We considered a wave propagating with velocity v • For a medium with isotropic (symmetric) properties, the wave equation should have a symmetric solution for a wave propagating with velocity –v

  8. The linear wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives

  9. The linear wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives

  10. The linear wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives

  11. The linear wave equation • The linear wave equation (not the only one having solutions of the form y(x,t) = f(x ± vt)): • It works for longitudinal waves as well • v is a constant and is determined by the properties of the medium. E.g., for a stretched string with linear density μ = m/l under tension T

  12. Superposition of waves • Let us consider two different solutions of the linear wave equation • Superposition principle – a sum of two solutions of the linear wave equation is a solution of the linear wave equation +

  13. Superposition of waves • Overlapping solutions of the linear wave equation algebraically add to produce a resultant (net) wave • Overlapping solutions of the linear wave equation do not in any way alter the travel of each other

  14. Reflection of waves at boundaries • Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries • Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted • Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

  15. Sinusoidal waves • One of the most characteristic solutions of the linear wave equation is a sinusoidal wave: • A – amplitude, φ – phase constant

  16. Wavelength • “Freezing” the solution at t = 0 we obtain a sinusoidal function of x: • Wavelengthλ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

  17. Wave number • On the other hand: • Angular wave number: k = 2π / λ

  18. Angular frequency • Considering motion of the point at x = 0 • we observe a simple harmonic motion (oscillation) : • For simple harmonic motion (Chapter 13): • Angular frequencyω

  19. Frequency, period • Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion: • Therefore, for the wave velocity

  20. Chapter 14 Problem 24 Ultrasound used in a medical imager has frequency 4.8 MHz and wavelength 0.31 mm. Find (a) the angular frequency, (b) the wave number, and (c) the wave speed.

  21. Interference of waves • Interference – a phenomenon of combining waves, which follows from the superposition principle • Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation • The resultant wave:

  22. Interference of waves • If φ = 0 (Fully constructive) • If φ = π (Fully destructive) • If φ = 2π/3 (Intermediate)

  23. Interference of waves • Considering two sinusoidal waves of the same amplitude, wavelength, but running in opposite directions • The resultant wave:

  24. Nodes Antinodes • Interference of waves • If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave

  25. Standing waves and resonance • For a medium with fixed boundaries (hard reflection) standing waves can be generated because of the reflection from both boundaries: resonance • Depending on the number of antinodes, different resonances can occur

  26. Standing waves and resonance • Resonance wavelengths • Resonance frequencies

  27. Harmonic series • Harmonic series – collection of all possible modes - resonant oscillations (n – harmonic number) • First harmonic (fundamental mode):

  28. More about standing waves • Longitudinal standing waves can also be produced • Standing waves can be produced in 2 and 3 dimensions as well

  29. More about standing waves • Longitudinal standing waves can also be produced • Standing waves can be produced in 2 and 3 dimensions as well

  30. Chapter 14 Problem 40 A 2.0-m-long string is clamped at both ends. (a) Find the longest wavelength standing wave possible on this string. (b) If the wave speed is 56 m/s, what’s the lowest standing-wave frequency?

  31. Rate of energy transmission • As the wave travels it transports energy, even though the particles of the medium don’t propagate with the wave • The average power of energy transmission for the sinusoidal solution of the wave equation • Exact expression depends on the medium or the system through which the wave is propagating

  32. Sound waves • Sound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies detectable by human ears (between ~ 20 Hz and ~ 20 KHz) • Ultrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies higher than detectable by human ears (> 20 KHz) • Infrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies lower than detectable by human ears (< 20 Hz)

  33. Speed of sound • Speed of sound: • ρ – density of a medium, γ – characteristic constant, P – pressure • Traveling sound waves

  34. Intensity of sound • Intensity of sound – average rate of sound energy transmission per unit area • For a sinusoidal traveling wave: • Decibel scale • β – sound level; I0 = 10-12 W/m2 – lower limit of human hearing

  35. Chapter 14 Problem 65 Show that a doubling of sound intensity corresponds to approximately a 3-dB increase in the decibel level.

  36. Sources of musical sound • Music produced by musical instruments is a combination of sound waves with frequencies corresponding to a superposition of harmonics (resonances) of those musical instruments • In a musical instrument, energy of resonant oscillations is transferred to a resonator of a fixed or adjustable geometry

  37. Open pipe resonance • In an open pipe soft reflection of the waves at the ends of the pipe (less effective than form the closed ends) produces standing waves • Fundamental mode (first harmonic): n = 1 • Higher harmonics:

  38. Organ pipes • Organ pipes are open on one end and closed on the other • For such pipes the resonance condition is modified:

  39. Musical instruments • The size of the musical instrument reflects the range of frequencies over which the instrument is designed to function • Smaller size implies higher frequencies, larger size implies lower frequencies

  40. Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances (exaggerated) at low frequencies:

  41. Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances at medium frequencies:

  42. Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances at high frequencies:

  43. Beats • Beats – interference of two waves with close frequencies +

  44. Sound from a point source • Point source – source with size negligible compared to the wavelength • Point sources produce spherical waves • Wavefronts – surfaces over which oscillations have the same value • Rays – lines perpendicular to wavefronts indicating direction of travel of wavefronts

  45. Interference in 2D • Far from the point source wavefronts can be approximated as planes – planar waves • Waves from two sources interfere to produce regions of low and high amplitude: constructive interference and destructive interference

  46. Variation of intensity with distance • A single point emits sound isotropically – with equal intensity in all directions (mechanical energy of the sound wave is conserved) • All the energy emitted by the source must pass through the surface of imaginary sphere of radius r • Sound intensity • (inverse square law)

  47. Andreas Christian Johann Doppler (1803 -1853) • Doppler effect • Doppler effect – change in the frequency due to relative motion of a source and an observer (detector)

  48. Doppler effect • For a moving detector (ear) and a stationary source • In the source (stationary) reference frame: • Speed of detector is –vD • Speed of sound waves is v • In the detector (moving) reference frame: • Speed of detector is 0 • Speed of sound waves is v + vD

  49. Doppler effect • For a moving detector (ear) and a stationary source • If the detector is moving away from the source: • For both cases:

  50. Doppler effect • For a stationarydetector (ear) and a moving source • In the detector (stationary) reference frame: • In the moving (source) frame:

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