Energy carried by a travelling wave

• Sinusoidal waves have equations like

           y(x, t) = A sin(k*x - omega*t)
  

• The speed of a wave may be expressed as

            v = wavelength/period = wavelength*frequency
  

• If one holds time fixed, a sinosoidal wave has a repeating, sinusoidal shape as a function of position
• If one looks at a fixed position, a sinosoidal wave moves in a repeating, sinusoidal fashion as a function of time
• The angular wave number k is defined as

                       k = 2*pi/wavelength
  

• Partial derivatives involve holding all but a single variable fixed, and then looking at the effect of small changes in that one variable
• Taking the partial derivative of a sinusoidal wave equation with respect to time yields simple harmonic motion
• The power transmitted to a medium of linear mass density "mu" as a wave passes through it is

              Power = 0.5 * mu * omega^2 * A^2 * velocity
  

• Viewgraph 14

• Viewgraph 15

Note that the energy deposited by this wave depends on several of its properties, and in different ways.

• the energy increases as the amplitude squared
• the energy increases as the frequency cubed

        Joe sings into a long cardboard tube of radius r = 0.2 m.
        He hits a perfect middle C (256 Hz), causing the air
        molecules to shake back-and-forth.  At the far end of the
        tube, the sound wave carries a power of 0.01 Watt = 0.01 Joule/sec .

        How large is the amplitude of motion of the air molecules
        in the tube?