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Wave Energy and Superposition

Wave Energy and Superposition. Physics 202 Professor Lee Carkner Lecture 7. PAL #5 Waves. A: y=2sin(2x-2t), B: y=4sin(4x-6t) , C: y=6sin(6x-8t). l =2 p /k so l A = p , l B = p /2, l C = p /3 T = 2 p / w so Which wave has largest transverse velocity? Wave C:

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Wave Energy and Superposition

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  1. Wave Energy and Superposition Physics 202 Professor Lee Carkner Lecture 7

  2. PAL #5 Waves • A: y=2sin(2x-2t), B: y=4sin(4x-6t), C: y=6sin(6x-8t) • l =2p/k so lA = p, lB = p/2, lC = p/3 • T = 2p/w so • Which wave has largest transverse velocity? • Wave C: • Largest wave speed? • v = lf = l/T, vA = 1, vB = 1.5, vC = 1.3

  3. PAL #5 Waves (cont) • Wave with y = 2 sin (2x-2t), find time when x= 5.2 cm has max a • Happens when y = ym = 2 • 1 = sin (2x-2t) • p/2 = (2x - 2t) • t = [2x-(p/2)]/2 • t = • Maximum velocity when y = 0 • 0 = • 2x -2t = arcsin 0 = 0 • t = x • t =

  4. Velocity and the Medium • The speed at which a wave travels depends only on the medium • Tension (t) • If you force the string up, tension brings it back down • Linear density (m = m/l =mass/length) • You have to convert the PE to KE to have the string move

  5. String Properties • How does wave speed depend on the string? v = (t/m)½ = lf • Wave speed is solely a property of the medium • The wavelength then comes from the equation above • The wavelength of a wave on a string depends on how fast you move it and the string properties

  6. Tension and Frequency

  7. Energy • A wave on a string has both kinetic and elastic potential energy • We input this energy when we start the wave by stretching the string • This energy is transmitted down the string • The energy of a given piece of string changes with time as the string stretches and relaxes • Assuming no energy dissipation

  8. Power Dependency • The average power (energy per unit time) is thus: P=½mvw2ym2 • v and m depend on the string • ym and w depend on the wave generation process

  9. Superposition yr = y1 +y2 • Traveling waves only add up as they overlap and then continue on • Waves can pass right through each other with no lasting effect

  10. Interference • The waves may be offset by a phase constant f y1 = ym sin (kx - wt) y2 = ym sin (kx - wt +f) yr = ymr sin (kx - wt +½f) • What is ymr (the resulting amplitude)? • Is it greater or less than ym?

  11. Interference and Phase ymr = 2 ym cos (½f) • The phase constant can be expressed in degrees, radians or wavelengths • Example: 180 degrees = p radians = 0.5 wavelengths

  12. Resultant Equation

  13. Combining Waves

  14. Types of Interference • Constructive Interference -- when the resultant has a larger amplitude than the originals • Fully constructive -- • No offset or offset by a full wavelength • Destructive Interference -- when the resultant has a smaller amplitude than the originals • Fully destructive -- • Offset by 1/2 wavelength

  15. Next Time • Read: 16.11-16.13

  16. Consider mark made on a piece of string with a wave traveling down it. At what point does the mark have the largest velocity? : At what point does the mark have the largest acceleration? • In the middle : At the top • At the top : In the middle • In the middle : In the middle • At the top: At the top • Velocity and acceleration are constant

  17. Suppose you are producing a wave on a string by shaking. What properties of the wave do you directly control? • Amplitude • Wavelength • Frequency • Propagation velocity • a and c only

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