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Harmonic Waves

Harmonic Waves. An harmonically oscillating point is described by a sine wave. y = A cos w t An object can take a sinusoidal shape in space. y = A cos kx. 1 wavelength. 1 period. Sinusoidal Behavior. y. t. y. x. Two Variables.

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Harmonic Waves

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  1. Harmonic Waves

  2. An harmonically oscillating point is described by a sine wave. y = A cos wt An object can take a sinusoidal shape in space. y = A cos kx 1 wavelength 1 period Sinusoidal Behavior y t y x

  3. Two Variables • To describe a complete wave requires both x and t. • This harmonic motion is for a harmonic wave.

  4. Wave Speed • The speed is related to the wavenumber • v = l/T • v = (2p/k) / (2p/w) • v = w/k • The wavenumber is related to the speed • k = 2p/l = w/v

  5. While boating on the ocean you see wave crests 14 m apart and 3.6 m deep. It takes 1.5 s for a float to rise from trough to crest. What is the wave speed? The time from trough to crest is half a period: T = 3.0 s. The wavelength is l = 14 m. The speed can be found directly: v = l/T = 4.7 m/s. Seasick

  6. Wave Power • Wave energy is proportional to amplitude squared. • E = ½ mv2 = ½ mL(wA)2 • Power is the time rate of change of energy. • Proportional to the speed • Proportional to the amplitude squared

  7. Intensity • Intensity of a wave is the rate energy is carried across a surface area. • This is true for linear and other waves. • For a spherical wave, the intensity I = P/A = P/4pr2

  8. Rope Snake • A garden hose has 0.44 kg/m. A child pulls it with a tension of 12 N, then shakes it side to side to make waves with 25 cm amplitude at 2.0 cycles per second. • What is the power supplied by the child? • Find the power from the speed and frequency. • Now use the equation for power • P = 11 W

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