# Refraction of waves at plane boundaries

Refraction of waves at plane boundaries

When straight waves pass from deep to shallow water, their wavelength becomes shorter. Both the long and the short waves, however, appear at rest when viewed simultaneously through the stroboscope. This shows that, although the wavelength
}, has altered, the frequency, f, has remained the same. Now, since the velocity, v = fi., it means that the waves travel more slowly in shallow water than in deep.

This can be illustrated by placing a rectangular piece of perspex of suitable thickness in the tank to reduce the local water depth.
Furthermore, when the angle of incidence is anything other than zero (i.e., perpendicular incidence), the change in wavelength and speed automatically brings about a change in the direction of travel of the waves when they cross the boundary (see Fig.
26.7). In other words, refraction occurs. Now the direction in which the waves are travelling is at right angles to the wavefront, so in accordance with the usual convention we have drawn a normal and marked the angle of incidence, i, and the angle of refraction, r. This may be compared with the refraction of light when it passes from one medium to another. It is clear from Fig. 26.7 that the wavelength has changed from i’l to ;’2′ From the geometry of the diagram we note that there are two right-angled triangles with
angles i and r, and sides AI and A2 respectively, together with a common hypotenuse AB.

Also, we have already seen on page 250 that the refractive index, n, is defined by, sin i Therefore, the refractive index for water waves passing from deep to shallow water Now the stroboscope tells us that the frequency,/, of the waves remains unaltered:
hence, using the wave equation v = fA, velocity in deep water = VI = fAI velocity in shallow water