5.4 - Refraction of Waves

Refraction and Wave Speed

Refraction is a change in the direction of wave propagation caused by a change in velocity as a wave enters another medium.
Water-wave speed depends on water depth; sound-wave speed is influenced by air density; light-wave speed depends on the optical density of the medium.
A perspex sheet in a ripple tank creates a shallow region for comparison with the surrounding deeper water.
Wave frequency remains unchanged because it is determined by the source. The same stroboscope setting can make incident and refracted waves appear stationary.

Direction of travel	Changes in the water wave
Deep to shallow	Speed and wavelength decrease. The direction bends towards the normal, so $i > r$ .
Shallow to deep	Speed and wavelength increase. The direction bends away from the normal, so $i < r$ .

The incident direction is perpendicular to the incident wavefronts.
The normal is perpendicular to the boundary between the two depths at the point of incidence.
The refracted direction bends towards the normal on entering shallow water, or away from the normal on entering deeper water.
Refracted wavefronts are perpendicular to the refracted direction. Their spacing is smaller in shallow water and larger in deep water.

Sound travels faster in hot air than in cold air.
During the day, air near the ground is hotter than air above. Sound bends away from the ground, giving a weaker sound to an observer at ground level.
At night, air near the ground is colder than air above. Sound bends towards the ground and is heard more clearly.
Sea waves approaching a shoreline:
- Offshore wavefronts are nearly straight and parallel because the waves travel at a uniform speed.
- Near a cape, shallower water reduces speed and wavelength. Wave energy converges and amplitude becomes larger.
- In a bay, deeper water permits higher speed and longer wavelength. Energy spreads over a wider region, giving smaller amplitude and calmer water.
- These speed differences curve the wavefronts to follow the shoreline.

v_{1} = f λ_{1}, v_{2} = f λ_{2}

\frac{v _{1}}{v _{2}} = \frac{λ _{1}}{λ _{2}}, \frac{v _{1}}{λ _{1}} = \frac{v _{2}}{λ _{2}}

Example 1: in shallow water, $λ_{1} = 2 cm$ and $v_{1} = 8 cm s^{- 1}$ . In deep water, $v_{2} = 12 cm s^{- 1}$ :

λ_{2} = \frac{v _{2} λ _{1}}{v _{1}} = \frac{12 ( 2 )}{8} = 3 cm

Example 2: a $12 cm$ distance contains four wavelengths in region P and eight in region Q. Thus $λ_{1} = 12/4 = 3 cm$ and $λ_{2} = 12/8 = 1.5 cm$ . With $v_{1} = 18 cm s^{- 1}$ :

v_{2} = \frac{v _{1} λ _{2}}{λ _{1}} = \frac{18 ( 1.5 )}{3} = 9 cm s^{- 1}