Question

When a plane wave traverses a medium, individual particles execute periodic motion given by equation: y = 8sin[$\frac{ \pi }{2}( \frac{x}{4} + 4t)$]
where lengths are expressed i cm and time in seconds. Calculate
(i) the amplitude, wavelength, velocity, and frequency of waves.
(ii) phase difference for two positions of the same particle which are occupied at time interval 0.4 sec apart.

Answer 1

Plane wave equation of motion:  (progressive wave it is called). It is progressing in the negative x direction.

  y = 8 Sin [π/2 * (x/4 + 4 t) ]   ,  where x & y are in cm.   t is in sec.
     = 8 * Sin [ 2π t + π/8 * x ]

General equation for the displacement of particles in a plane wave, progressing in positive x direction, and particles oscillating in y direction:

    y = A * Sin [ ω t - k x]  =  A Sin [2π (t/T - x/λ)]
           where ω = 2π f = angular frequency.   f = frequency
                      T = time period,  v = velocity.   λ = wavelength

Comparing the equations, we get:
     ω = 2π rad/sec.   f = 1 Hz.
     λ = wavelength = 16 cm.
     v = λ * f = 16 cm/sec.

ii)  Same particle means that  x = same.  Let us say t and t+0.4 sec. are the time instants we find the phases.

 Phases are:  π/2 * [x/4 + 4 t]   radians
           and :  π/2 * [x/4 + 4*(t+0.4) ] = π/2 * [x/4 + 4 t + 1.6 ]  radians.

Difference:  π/2 * 1.6 = 0.80 π  or  4π/5 radians.