Refraction at a plane boundary

What refraction is

• Refraction is the change of direction of a wave when it crosses into a medium where it travels at a different speed.
• The direction change happens only at the boundary; inside each medium the wave travels in a straight line.
• Refraction happens to all waves — water waves entering shallow water, light entering glass, sound entering warm air.

Why the direction changes

• Consider a set of wavefronts arriving at an angle.

• One end of each wavefront reaches the slow medium first and slows down, while the other end is still travelling at the original speed.

• The wavefront therefore pivots — like a car whose left wheels hit sand while the right wheels are still on tarmac.

• Since rays are perpendicular to wavefronts, the ray direction turns as well.

• Entering a slower medium: the wave bends toward the normal.

• Entering a faster medium: the wave bends away from the normal.

• Along the normal ($\theta_i = 0°$): the wave slows or speeds up but does not change direction — both ends of the wavefront cross the boundary at the same instant, so there is nothing to pivot.

What happens to each wave quantity

Quantity	Entering a slower medium
Speed	decreases
Wavelength	decreases in the same proportion
Frequency	unchanged
Direction	bends toward the normal
Amplitude	usually decreases a little (energy is also reflected)

• The relationships on the resource sheet tie the ratios together:

$$\frac{n_1}{n_2} = \frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1}$$

• $n_1, n_2$ — refractive indices of the two media
• $v_1, v_2$ — wave speeds in each medium
• $\lambda_1, \lambda_2$ — wavelengths in each medium
• Note the order: $n_1$ pairs with $v_2$ and $\lambda_2$. A larger refractive index goes with a smaller speed and a shorter wavelength.

Reading a wavefront diagram

• Closer wavefronts = shorter wavelength = slower medium.
• Wider wavefronts = longer wavelength = faster medium.
• The wavefronts must stay continuous across the boundary — they cannot break or double up, because the frequency is the same on both sides.
• Count the wavefronts in a given distance on each side if you are asked to compare wavelengths.

Water waves over a shallow region

• Water waves travel more slowly in shallow water.
• A ramp or submerged shelf therefore refracts them:
  • entering the shallow region at an angle, they bend toward the normal,
  • their wavelength shortens — visible as the wavefronts crowding together,
  • their frequency stays the same.
• This is why waves at a beach line up nearly parallel to the shore no matter which direction they came from: the end of each wavefront that reaches shallow water first slows, and the wave pivots round.