1 / 59

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.

rose-dudley
Download Presentation

Chapter 14 Wave Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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:

More Related