Wave Behaviour: Reflection Refraction and the Doppler Effect

Reflection at a Surface

When a wave hits a surface, it may reflect, meaning all or part of its energy bounces back into the original medium. The direction of this reflection is governed by the law of reflection.

The incident angle ($i$) is the angle between the incoming wave direction and the normal, an imaginary line perpendicular to the surface at the point of impact. The reflected angle ($r$) is the angle between the normal and the direction of the reflected wave.

The Law of Reflection

For any wave reflecting from a surface:

Incident angle = Reflected angle ($i = r$)

The incident ray, the reflected ray, and the normal must all lie in the same plane.

Reflection of Crests and Troughs

In diagrams showing wavefronts (crests or troughs), such as those seen in a water ripple tank, the crests are always perpendicular to the direction of wave travel (the rays). When ripples hit a straight barrier, the reflected crests maintain their shape but change direction according to the law of reflection.

Smooth and Rough Surfaces

The law of reflection applies at every point on a surface. On a smooth surface, all normals are parallel, leading to orderly reflection (specular reflection) that can form clear images. On a rough surface, the normals point in various directions, causing rays to reflect in random directions (diffuse reflection), which is why we can see objects like paper from many angles but cannot see an image in them.

Worked Example: Measuring the distance to the Moon

Astronomers measure the distance to the Moon by timing a laser pulse ($v = 3.00 \times 10^8 \text{ ms}^{-1}$) reflecting back to Earth. If the average round-trip time is $2.57 \text{ s}$, calculate the distance in km.

1. Calculate total distance: $\text{Total distance} = v \times t = 3.00 \times 10^8 \times 2.57 = 7.71 \times 10^8 \text{ m}$.
2. Halve the distance for a one-way trip: $3.855 \times 10^8 \text{ m}$.
3. Convert to km: $385,500 \text{ km}$ (approx $386,000 \text{ km}$).

Worked Example: Echoes from a Cliff

A man stands $170 \text{ m}$ from a cliff. He bangs wood together so that the echoes arrive exactly halfway between hits. At what frequency should he bang them? (Speed of sound $v = 340 \text{ ms}^{-1}$).

1. Time for one echo: $d = 170 \times 2 = 340 \text{ m}$. $t = d/v = 340/340 = 1.0 \text{ s}$.
2. For the echo to be halfway between hits, the time between hits ($T$) must be double the echo time: $T = 2.0 \text{ s}$.
3. Frequency $f = 1/T = 1/2.0 = 0.50 \text{ Hz}$.

Refraction at a Boundary

Refraction occurs when waves cross a boundary between two different media where their speeds differ. This change in speed causes a change in direction, unless the wave is travelling along the normal.

1. If a wave slows down, it refracts towards the normal (e.g. air to glass).
2. If a wave speeds up, it refracts away from the normal (e.g. glass to air).

Refraction of Wavefronts

Wavefronts (crests) are perpendicular to the direction of travel. When light enters glass from air, it slows down, causing the wavefronts to bunch closer together (shorter wavelength) and tilt towards the normal.

Conservation of Energy and Partial Reflection

It is rare for $100\%$ of energy to reflect or refract. At most boundaries, some energy is reflected, some is transmitted (refracted), and some is absorbed.

Energy is conserved: $\text{Incident energy} = \text{reflected} + \text{transmitted} + \text{absorbed}$.

Worked Example: Coastal Refraction

Wave crests approaching a shore change direction and wavelength as the water gets shallower.

In this scenario, waves travel more slowly in shallower water. Because frequency cannot change, the speed decrease causes a decrease in wavelength. This change in speed causes the waves to refract toward the coastline.

Effects on Wave Properties

Reflection Properties

Reflection only changes the direction of a wave. Because the wave stays in the same medium, its speed, frequency, and wavelength remain constant. Every crest that hits the surface reflects from it, so the number of waves per second (frequency) remains the same.

Refraction Properties

Refraction changes the speed, wavelength, and direction, but the frequency always remains constant.

Frequency is constant because every crest that reaches the boundary must enter the second medium. Since $v = f\lambda$, if the speed $v$ decreases, the wavelength $\lambda$ must decrease by the same factor.

Effect on crossing boundary	Frequency	Wavelength	Direction
Speed Increase	No change	Increases	Away from normal
Speed Decrease	No change	Decreases	Toward normal

Worked Example: Atmospheric Refraction

As light from a star enters the Earth's atmosphere, it slows down. This causes the light to refract towards the normal. This makes the star appear higher in the sky than its actual position, unless the star is directly overhead (along the normal), where no direction change occurs.

Analogy between Water Waves and Light

A ripple tank is used to show wave behaviour (using crest shadows), while a ray box shows light behavior (using rays). Both types of waves follow the same laws of reflection and refraction, allowing us to use water waves to model light behaviour.

However, there are fundamental differences:

1. Light is electromagnetic; water waves are mechanical.
2. Light travels through a vacuum; water waves require a medium.
3. Light consists of vibrating fields; water waves consist of vibrating particles.

The Doppler Effect

The Doppler effect is the change in observed wavelength and frequency due to the relative motion between a wave source and an observer.

1. Approaching: As the source moves towards an observer, successive wave crests are emitted from positions closer to the observer. This 'bunches up' the waves, resulting in a shorter wavelength and a higher frequency.
2. Receding: As the source moves away, the waves are 'stretched out', resulting in a longer wavelength and a lower frequency.

Key points of the Doppler effect:

• The faster the relative motion, the greater the shift.
• Only motion along the line between the source and observer causes a shift. Motion perpendicular to this line (e.g. flying in a perfect circle around a listener) produces no Doppler shift.

Worked Example: Galaxies and Redshift

Hubble and Slipher found that light from distant galaxies had longer wavelengths than light from sources at rest. This increase in wavelength implies the galaxies are moving away from Earth. Furthermore, the shift was greater for more distant galaxies, proving they are moving away at higher speeds.