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
Comparison with light waves. Using a diagram similar to Fig. 26.7, Huygens considered the case of plane light waves being refracted from one medium to a more optically dense medium and obtained the same result, namely, velocity of light in first medium refractive Index = velocity of light in second medium Now, the mean refractive index of water (as far as light is concerned) is 1.33. Therefore, if Huygens was right it meant that light ought to travel 1.33 times faster in air than in water. In contrast, Newton’s corpuscular theory gave a theoretical result which was the exact opposite to this.
When Huygens and Newton put forward their two different theories in the sixteenth century it was not possible to test them experimentally, since no method was available for measuring the velocity of light in anything other than free space. Nearly 150 years later, it was a triumph for the wave theory when Jean Foucault in France devised a method for measuring the velocity of light in both air and water, and found that it actually was less in water.