WAVE PHENOMENA

Wave Motion In wave motion energy is transferred with no net mass transport. There are two basic types of wave. Transverse Direction of oscillations Direction of wave The direction of the oscillation is at right angles to the direction of the wave. Water waves and electromagnetic radiation are examples of transverse waves.

Longitudinal Direction of oscillations Direction of wave Oscillations take place in the same direction as the wave motion. Sound waves are longitudinal.

The travelling wave equation The simplest mathematical form of a wave is a sine or cosine function. cosine For a wave travelling from left to right with speed v, the particle will be performing SHM in the y-direction. y = a sin  t y = a cos  t. If y=0 at t=o then If y=a at t=0 sine

The y displacement is also a function of the x, displacement, i.e. y= f (x,t) The displacement at x=0 is given by : y = a sin  t (or y = a cos  t if y=a at t=0) The time, t, for the wave disturbance to travel from A (x = 0) to B (x = x) is: t = x/v So, at A, the time is: (t - x/v) If we substitute into y = a sin  t Then: y = a sin  (t – x/v) However, we know that:  =2  f and v = f λ So: y = a sin 2  f (t – x /f λ) Therefore: x v

We can therefore find the displacement of a medium at positions, x, and time, t, if we know the wavelength and frequency. Example A periodic wave travelling in the x direction is described by the equation: y = 0.2 sin (4  t-0.1x) Find(a) the amplitude (b) the frequency (c) the wavelength (d) the speed (e) the displacement of the medium in the y direction at the point x = 25m when the time is 0.3s

(a)We first need to re-arrange, y = 0.2 sin (4  t-0.1x) into the form (a)a = 0.2m (b)f = 2Hz (c) λ = 20  m (d)v = f λ = 2 x 20  = 2 x 20  = 40  ms -1 = 40  ms -1 (e) y =0.2sin2  (2x0.3- (0.1x2.5)/2  ) = 19.1cm = 19.1cm

Note: For a wave travelling in the negative x-direction the displacement is given by: Intensity of a wave The intensity of a wave is directly proportional to the square of its amplitude.