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Understanding Waves: Motion, Speed, and Reflections

Explore the motion of waves on a string, the speed and frequency of pulses, and the concept of reflections at boundaries. Discover how tension, inertia, and mass per length affect wave behavior. Learn about standing waves, superposition, and the mathematical description of periodic waves.

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Understanding Waves: Motion, Speed, and Reflections

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  1. motion of “pulse” actual motion of string Waves • A pulse on a string (demos) • speed of pulse = wave speed = v • depends upon tension T and inertia (mass per length m) • y = f(x-vt) (animation)

  2. Periodic Waves: coupled harmonic motion (animations) • aka sinusoidal (sine) waves • wave speed v: the speed of the wave, which depends upon the medium only. • wavelength l: the distance over which the wave repeats, • frequency f : the number of oscillations at a given point per unit time. T = 1/f. • distance between crests = wave speed  time for one cycle • l = vT • -> Wavelength, speed and frequency are related by: • v = l f

  3. Mathematical Description of Periodic Waves

  4. The Wave Equation

  5. Transverse Wave Velocity: lifting the end of a string • Tension F • Linear Mass Density (m/L) m • Transverse Force Fy Fnet vt vyt Fy F F l = vt

  6. Reflections at a boundary: fixed end = “hard” boundary Pulse is inverted Reflections at a boundary: free end = “soft” boundary Pulse is notinverted

  7. Reflections at an interface • light string to heavy string = “hard” boundary • faster medium to slower medium • heavy string to light string = “soft” boundary • slower medium to faster medium

  8. Principle of Superposition: When Waves Collide! When pulses pass the same point, add the two displacements (animation)

  9. Standing Waves • vibrations in fixed patterns • effectively produced by the superposition of two traveling waves y(x,t) = (ASW sin kx) coswt • constructive interference: waves add • destructive interference: waves cancel 3l = 2L l = 2L 2l = 2L 4l = 2L node antinode antinode

  10. Example: The A string on a violin has a linear density of 0.60 g/m and an effective length of 330 mm. (a) Find the Tension in the string if its fundamental frequency is to be 440 Hz. (b) where would the string be pressed for a fundamental frequency of 495 Hz?

  11. 5l = 4L l = 4L 3l = 4L 7l = 4L node node antinode antinode Standing Waves II pipe open at one end

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