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Fundamentals of Physics

Fundamentals of Physics. Chapter 14 Waves - I Waves & Particles Types of Waves Transverse & Longitudinal Waves Wavelength & Frequency Speed of a Traveling Wave Wave Speed on a Stretched String Energy & Power in a Traveling String Wave The Wave Equation

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Fundamentals of Physics

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  1. Fundamentals of Physics Chapter 14 Waves - I • Waves & Particles • Types of Waves • Transverse & Longitudinal Waves • Wavelength & Frequency • Speed of a Traveling Wave • Wave Speed on a Stretched String • Energy & Power in a Traveling String Wave • The Wave Equation • The Principle of Superposition for Waves • Interference of Waves • Phasors • Standing Waves • Standing Waves & Resonance Fundamentals of Physics

  2. Waves & Particles Particles - a material object moves from one place to another. Waves - information and energy move from one point to another, but no material object makes that journey. • Mechanical waves • Newton’s Laws rule! • Requires a material medium • e.g. water, sound, seismic, etc. • Electromagnetic waves • Maxwell’s Equations & 3.0 x 108 m/s • No material medium required • Matter waves • Quantum Mechanics - ~10-13 m • Particles have a wave length - De Broglie (1924) Fundamentals of Physics

  3. A Simple Mechanical Wave A single up-down motion applied to a taut string generates a pulse. The pulse then travels along the string at velocity v. He moves his hand once. Assumptions in this chapter: No friction-like forces within the string to dissipate wave motion. Strings are very long - no need to consider reflected waves from the far end. Fundamentals of Physics

  4. Traveling Waves They oscillate their hand in SHM. Transverse Wave: The displacement (and velocity) of every point along the medium carrying the wave is perpendicular to the direction of the wave. Longitudinal Wave: The displacement (and velocity) of the element of the medium carrying the wave is parallel to the direction of the wave. e.g. a vibrating string e.g. a sound wave Longitudinal, Transverse and Mixed Type Waves Fundamentals of Physics

  5. Transverse Wave Displacement versus position not versus time transverse wave applet Each point along the string just moves up and down. Fundamentals of Physics

  6. Wave Length & Frequency • amplitude - maximum displacement • wavelengthl - distance between repetitions of the shape of the wave. • angular wave number • period - one full oscillation • frequency - oscillations per unit time Fundamentals of Physics

  7. Wave Length & Frequency The phase, kx – wt , changes linearly with x and t, which causes the sine function to oscillate between +1 and –1. space time Fundamentals of Physics

  8. Speed of a Traveling Wave Consider a wave traveling in the positive x direction; the entire wave pattern moves a distance Dx in time Dt: Each point on the wave, e.g. Point A, retains its displacement, y; hence: Note: both x and t are changing! Differentiating: Fundamentals of Physics

  9. Direction of the Wave Consider an unchanging pulse traveling along positive x axis. y = f (x’) = f (x - v t) traveling towards +x y = f (x + v t) traveling towards -x All traveling waves are functions of (kx+wt) = k(x+vt) . Fundamentals of Physics

  10. Descriptions of the phase of a Traveling Wave Fundamentals of Physics

  11. Wave Speed on a Stretched String A single symmetrical pulse moving along a string at speed vwave. In general, the speed of a wave is determined by the properties of the medium through which it travels. Fundamentals of Physics

  12. Wave Speed on a Stretched String Consider a single symmetrical pulse moving along a string at speed v: t = tension in the string m = the string’s linear density (Roughly a circular arc) String moving to the left. Moving along with the pulse on the string. Speed of a wave along a stretched string depends only on the tension and the linear density of the string and not on the frequency of the wave. Fundamentals of Physics

  13. Energy of a Traveling String Wave Driving force imparts energy to a string, stretching it. The wave transports the energy along the string. Energy Fundamentals of Physics

  14. Energy & Power of a Traveling String Wave • Driving force imparts energy to a stretched string. • The wave transports energy along the string. • Kinetic energy - transverse velocity of string mass element, Dm = mDx • Potential energy - the string element Dx stretches as the wave passes. (See Section 14-3) Fundamentals of Physics

  15. The Principle of Superposition for Waves Overlapping waves algebraically add to produce a resultant wave: y1(x,t) y2(x,t) Add the amplitudes: ytotal(x,t) = y1(x,t) + y2(x,t) Overlapping waves do not alter the travel of each other! Fundamentals of Physics

  16. The Principle of Superposition for Waves Interference of waves traveling in opposite directions. Destructive Interference Constructive Interference Fundamentals of Physics

  17. f Interference of Waves Traveling in the Same Direction f = “phase difference” It is easy to show that: Fundamentals of Physics

  18. Interference of Waves The magnitude of the resultant wave depends on therelative phasesof the combining waves - INTERFERENCE. Constructive Interference Destructive Interference Partial Interference Fundamentals of Physics

  19. Standing Waves Two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a string: y1(x,t) = ym sin (k x -w t) positive x direction y2(x,t) = ym sin (k x +w t) negative x direction Their interference with each other produces a standing wave: y’(x,t) = y1(x,t) + y2(x,t) It is easy to show that: For a standing wave, the amplitude, 2ymsin(kx) , varies with position. Fundamentals of Physics

  20. Standing Wave The amplitude of a standing wave equals zero for: minimums @ x = ½ nln = 0, 1, 2, . . . NODES Fundamentals of Physics

  21. Standing Waves minimums @ x = ½ nln = 0, 1, 2, . . . NODES maximums @ x = ½(n + ½) l n = 0, 1, 2, . . . ANTINODES Fundamentals of Physics

  22. Reflections at a Boundary End of the string is free to move Tie the end of the string to the wall “soft” reflection “hard” reflection Node at boundary Antinode at boundary Reflected pulse has opposite sign Reflected pulse has same sign Newton’s 3rd Law Fundamentals of Physics

  23. Reflections at a Boundary From high speed to low speed (low density to high density) From high density to low density Fundamentals of Physics

  24. Standing Waves & Resonance Consider a string with both ends fixed; it has nodes at both ends. This can only be true when: v is the speed of the traveling waves on the string. Resonance for certain frequencies for a string with both ends fixed. Only for these frequencies will the waves reflected back and forth be in phase. Fundamentals of Physics

  25. Standing Waves & Resonance A standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium. Using superposition, the net displacement of the medium is the sum of the two waves. When 180° out-of-phase with each other, they cancel (destructive interference). When in-phase with each other, they add together (constructive interference). Fundamentals of Physics

  26. Standing Waves & Resonance The Harmonic Series both ends fixed Fundamentals of Physics

  27. String Fixed at One End Resonance: Standing wave applet Prenault’s applets fixed end free end Fundamentals of Physics

  28. Fundamentals of Physics Waves - II • Introduction • Sound Waves • The Speed of Sound • Traveling Sound Waves • Interference • Intensity & Sound Level • The Decibel Scale • Sources of Musical Sound • Beats • The Doppler Effect • Detector Moving; Source Stationary • Source moving; Detector Stationary • Bat Navigation • Supersonic Speeds; Shock Waves Fundamentals of Physics

  29. Sound Waves A sound wave is a longitudinal wave of any frequency passing through a medium (solid, liquid or gas). Fundamentals of Physics

  30. The Speed of Sound Elastic property of the medium • Strings - tension (t in N) • Sound - Bulk Modulus (B in N/m2) • Inertial property of the medium • Strings - linear mass density (m in kg/m) • Sound - volume mass density (r in kg/m3) The speed of waves depends on the medium, not on the motion of the source. Fundamentals of Physics

  31. The Speed of Sound For an ideal gas, B/r can be shown to be proportional to absolute temperature; hence, the speed of sound depends on the square root of the absolute temperature. Equation for the speed of sound: T = absolute temperature g = 1.4 for O2 and N2 (~air) R = “universal gas constant” = 8.314 J/mol-K M = molar mass of the gas = 29 x 10-3 kg/mol (for air) Fundamentals of Physics

  32. Traveling Sound Waves Displacement Function: (of the air element about x) SHM Pressure-Variation Function: (pressure change as wave passes x) Fundamentals of Physics

  33. Traveling Sound Waves As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source. Slinky Demo Fundamentals of Physics

  34. Traveling Sound Waves As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source. Fundamentals of Physics

  35. Interference Consider two sources of waves S1 and S2, which are “in phase”: “arrive in phase” f is the “phase difference” @ P1 “Constructive Interference” Fundamentals of Physics

  36. Interference Two sources of sound waves S1 and S2: arrive “out of phase” “Destructive Interference” Fundamentals of Physics

  37. Psychological dimensions of sounds Pitch 150 Hz 300-Hz sound 500-Hz sound 1500 Hz A healthy young ear can hear sounds between 20 - 20,000 Hz. - age reduces our hearing acuity for high frequencies. Loudness 150 Hz with twice the amplitude of 1500 Hz Fundamentals of Physics

  38. Intensity & Sound Level Intensity of a Sound Wave: The power of the wave is time rate of energy transfer. The area of the surface intercepting the sound. All the sound energy from the source spreads out radially and must pass through the surface of a sphere: In terms of the parameters of the source and of the medium carrying the sound, the sound intensity can be shown to be as follows: Fundamentals of Physics

  39. Intensity & Sound Level Alexander Graham Bell The Decibel Scale: b - Sound Level Mammals hear over an enormous range: Sound level (or loudness) is a sensation in the consciousness of a human being. The psychological sensation of loudness varies approximately logarithmically; to produce a sound that seems twice as loud requires about ten times the intensity. (decibel) where I0 is the approximate threshold of human hearing. Fundamentals of Physics

  40. Intensity & Sound Level Human Perception of Sound ~3dB is a factor 2 change in intensity Every 10dB is a factor 10 change in intensity; 20 dB is a factor 100 change in intensity See Table 17-2. Fundamentals of Physics

  41. Intensity & Sound Level Fundamentals of Physics

  42. Sources of Musical Sound Standing Waves in a Pipe Open End • Molecules free to move • Displacement Antinode • Pressure Node Closed End • Molecules cannot move • Displacement node • Pressure Antinode Both ends closed  2 nodes with at least one antinode in between. Both ends open  2 antinodes with at least one node in between. One end closed  1 node and one antinode. Fundamentals of Physics

  43. Sources of Musical Sound nodes or antinodes at the ends of the resonant structure Fundamental Frequency “1st Harmonic” “Fundamental mode” Fundamentals of Physics

  44. Sources of Musical Sound Both Ends Open One End Open Harmonic Number Fundamentals of Physics

  45. Sources of Musical Sound length of an instrument  fundamental frequency Fundamentals of Physics

  46. Sources of Musical Sound Overtones Fundamental & Overtones Fundamentals of Physics

  47. Beats 2 waves with slightly different frequencies are traveling to the right. The superposition of the 2 waves travels in the same direction and with the same speed. The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency. Fundamentals of Physics

  48. Beats Consider two similar sound waves: Superimpose them: “Beat Frequency”: Fundamentals of Physics

  49. Interference: Standing Wavecreated from two traveling waves: 2 sinusoidal waves having the same frequency (wavelength) and the same amplitude are traveling in opposite directions in the same medium. As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left; it stands still. [one dot at an antinode and one at a node] Beats created from two traveling waves: 2 waves with slightly different frequencies are traveling in the same direction. The superposition is a traveling wave, oscillating with the average frequency with its amplitude envelope varying according to the difference frequency. Beats demo Fundamentals of Physics

  50. The Doppler Effect Doppler Effect Applet: Doppler Effect http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/#sound_waves Fundamentals of Physics

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