Harmonic Waves

Sinusoidal Behavior  An harmonically oscillating point is described by a sine wave. y = A cos  ty = A cos  t  An object can take a sinusoidal shape in space. y = A cos kxy = A cos kx 1 period1 wavelength y y t 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 = /T v = (2  k  2  v =  /k  The wavenumber is related to the speed k = 2  =  /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 = 14 m.  The speed can be found directly: v = / T = 4.7 m/s.

Wave Power  Wave energy is proportional to amplitude squared. E = ½ mv 2 = ½  L(  A) 2E = ½ mv 2 = ½  L(  A) 2  Power is the time rate of change of energy. Proportional to the speedProportional to the speed Proportional to the amplitude squaredProportional 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  r 2

 Find the power from the speed and frequency.  Now use the equation for power P = 11 WP = 11 W 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? next