Waves: Properties & Behaviour

What is a wave?

A wave is a disturbance that transfers energy from one place to another without transferring matter. When you drop a stone into a pond, ripples spread outwards carrying energy across the surface, but the water itself does not travel to the edge: a floating leaf simply bobs up and down on the spot. The water particles oscillate about a fixed position while the energy passes through them.

Key terms

Oscillation — a repeated back-and-forth (or up-and-down) motion about a central point.

Energy transfer — waves move energy; they do not carry the medium with them.

This single idea explains a lot. Sound lets you hear a distant voice without air rushing from the speaker's mouth into your ear. Light from the Sun warms your skin without any "stuff" arriving. In every case the source vibrates, and that vibration ripples outward as a wave.

Transverse and longitudinal waves

Waves come in two types, defined by the direction of vibration relative to the direction the energy travels.

In a transverse wave the oscillations are at right angles (90°) to the direction of energy transfer. The wave moves forward while particles move up and down across it.

In a longitudinal wave the oscillations are parallel to the direction of energy transfer. The medium is squashed and stretched, creating regions of compression (particles close together) and rarefaction (particles spread apart).

Watch out

A common exam trap is to say sound is transverse. It is longitudinal — air molecules vibrate back and forth along the same line the sound travels. Only the energy moves forward; the air molecules stay roughly in place.

Describing a wave

A few quantities let us describe any wave precisely.

Amplitude ($A$) — the maximum displacement of a point from its rest (undisturbed) position, measured in metres. A bigger amplitude means more energy.
Wavelength ($\lambda$) — the distance between two adjacent points in phase, e.g. crest to crest or compression to compression, measured in metres.
Frequency ($f$) — the number of complete waves passing a point each second, measured in hertz (Hz). $1\ \text{Hz} = 1$ wave per second.
Period ($T$) — the time taken for one complete wave to pass a point, measured in seconds.

The wave equation

Two equations tie these quantities together and you must know both.

The relationship between period and frequency is:

$$T = \frac{1}{f}$$

The wave equation links wave speed to frequency and wavelength:

$$v = f\lambda$$

where $v$ is the wave speed in m/s, $f$ is the frequency in Hz, and $\lambda$ is the wavelength in m.

Worked example

A radio station broadcasts at a frequency of $f = 200\ \text{kHz}$. The radio waves travel at the speed of light, $v = 3.0 \times 10^{8}\ \text{m/s}$. Find the wavelength.

First convert: $200\ \text{kHz} = 200\,000\ \text{Hz} = 2.0 \times 10^{5}\ \text{Hz}$.

Exam tip

Rearrange $v = f\lambda$ confidently in both directions: $f = \dfrac{v}{\lambda}$ and $\lambda = \dfrac{v}{f}$. Watch for units — kHz, MHz, and km must become Hz and m first, or you lose marks.

Reflection

When a wave hits a barrier it can be reflected — it bounces off. We measure angles from the normal, an imaginary line drawn at 90° to the surface at the point where the wave strikes.

The law of reflection states:

the angle of incidence equals the angle of reflection ($i = r$).

The reflected wave keeps the same speed, frequency and wavelength — only its direction changes. This law applies to all waves: light bouncing off a mirror, water waves off a wall, and the echo of sound off a cliff.

Refraction

When a wave passes from one material into another it changes speed, and this can make it bend. This bending is called refraction.

The cause is the change of speed at the boundary:

A wave slowing down (e.g. light going from air into glass) bends towards the normal.
A wave speeding up (e.g. light going from glass into air) bends away from the normal.
If the wave hits the boundary along the normal ($i = 0°$), it slows but does not bend.

We measure how strongly a material bends light using its refractive index ($n$), found from the angles of incidence and refraction:

$$n = \frac{\sin i}{\sin r}$$

Here $i$ is the angle of incidence (in the first material, usually air) and $r$ is the angle of refraction (in the denser material). A larger $n$ means the light slows more and bends more. The refractive index has no units because it is a ratio.

Watch out

Both $i$ and $r$ are always measured from the normal, never from the surface. Measuring from the surface is one of the most common errors in refraction questions.

Total internal reflection and the critical angle

When light travels from a dense material (like glass) towards a less dense one (like air), it bends away from the normal. As you increase the angle of incidence, the refracted ray bends further until, at one special angle, it travels right along the boundary. This is the critical angle ($c$).

If the angle of incidence is greater than the critical angle, no light escapes — it is all reflected back inside. This is total internal reflection (TIR).

Two conditions are needed for TIR:

1. Light must be travelling from a denser material into a less dense material.
2. The angle of incidence must be greater than the critical angle.

The critical angle links to the refractive index by $\sin c = \dfrac{1}{n}$, so a higher refractive index gives a smaller critical angle.

Real world

Optical fibres use total internal reflection to carry data and light. A thin glass core is surrounded by cladding of lower refractive index. Light hits the boundary at an angle greater than the critical angle, so it reflects perfectly again and again, zig-zagging along the fibre with almost no loss. This carries broadband internet and lets endoscopes see inside the body.

The ripple tank experiment

A ripple tank is a shallow, transparent tray of water used to study wave behaviour. A motor vibrates a bar to make straight wavefronts, or a dipper makes circular ones. A lamp above shines through the water, projecting bright and dark bands of the wavefronts onto a screen below, so the waves are easy to see and measure.

With a ripple tank you can demonstrate:

Reflection — place a straight barrier in the water; the wavefronts bounce off, obeying $i = r$.
Refraction — a sheet of glass makes part of the tank shallower. Water waves travel slower in shallow water, so the wavefronts slow down, their wavelength shortens, and they change direction at the boundary, just like light refracting.

To measure frequency, you can use a stroboscope (a flashing light): when the flash rate matches the wave frequency, the wavefronts appear to freeze. You can then measure the wavelength on the screen and use $v = f\lambda$ to find the wave speed.

Exam tip

When waves refract, the frequency never changes — it is fixed by the source. Because $v = f\lambda$, if the speed drops then the wavelength must drop too. Reach for this whenever a question asks what happens to a wave at a boundary.