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Understanding Harmonic Waves: Concepts and Applications

Explore the nature of harmonic waves, sinusoidal behavior, wave speed calculations, wave power, intensity, and practical examples like boat waves and rope vibrations. Learn about the dynamics of waves in various contexts.

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Understanding Harmonic Waves: Concepts and Applications

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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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