Harmonic Waves Sinusoidal Behavior An harmonically oscillating point

Harmonic Waves

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

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

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

Seasick ] ] 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.

Wave Power ] Wave energy is proportional to amplitude squared. • E = ½ mv 2 = ½ m. L(w. A)2 ] Power is the time rate of change of energy. • Proportional to the speed • Proportional to the amplitude squared

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/4 pr 2

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 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