Wave motion and features

A wave is a travelling disturbance that transfers energy without an overall transfer of matter along with it.

For example, when one end of a rope is disturbed, the disturbance travels along the rope. The parts of the rope vibrate around their equilibrium positions—their positions when undisturbed—rather than travelling along with the disturbance. Similarly, a water wave travels across the surface while the water moves locally rather than travelling across the surface with the wave.

Wave features

A crest or peak is a point of maximum displacement above equilibrium. A trough is a point of maximum displacement below equilibrium. The amplitude is the maximum displacement from the equilibrium position, measured to a crest or a trough.

The wavelength, $$\lambda$$, is the distance between successive points along the direction of travel that are at the same stage of vibration. For example, it is the distance from one crest to the next crest.

A wavefront is a line joining points on a wave that are at the same stage of vibration. Points on the same wavefront move through their vibrations together.

Frequency and wave speed

Frequency, $$f$$, is the number of complete vibrations passing a point each second. It is measured in hertz ($$\text{Hz}$$): a frequency of one hertz means one complete vibration passes the point each second.

Wave speed is the distance the disturbance travels per unit time. It describes how quickly the wave moves, not how far the material itself travels.

Wave-speed equation

A wave travels one wavelength during each complete cycle. Its frequency is the number of cycles passing a point each second, so the distance the wave travels each second is its frequency multiplied by its wavelength.

The relationship is:

Here, $$v$$ is wave speed in metres per second ($$\mathrm{m/s}$$), $$f$$ is frequency in hertz ($$\mathrm{Hz}$$), and $$\lambda$$ is wavelength in metres ($$\mathrm{m}$$). One hertz means one cycle per second.

For example, a wave with frequency $$4\ \mathrm{Hz}$$ and wavelength $$3\ \mathrm{m}$$ travels at $$v = 4 \times 3 = 12\ \mathrm{m/s}$$.

Rearranging the wave-speed equation

If wave speed and wavelength are known, divide speed by wavelength to find frequency. If wave speed and frequency are known, divide speed by frequency to find wavelength:

Transverse waves

In a transverse wave, the vibrations are at right angles to the direction in which the wave travels, or propagates. For example, if a wave travels horizontally, its vibrations may be vertical. The vibrations do not travel along with the wave; they occur across its direction of travel.

The following waves can be modelled as transverse:

• Electromagnetic radiation: electric and magnetic fields oscillate at right angles to the direction of propagation.
• Water waves: the up-and-down movement of the water surface can be modelled as a vibration perpendicular to the direction the wave travels.
• Seismic S-waves, or secondary waves: the material through which the wave travels vibrates at right angles to the direction of propagation.

Longitudinal waves

In a longitudinal wave, the material vibrates back and forth parallel to the direction in which the wave travels. For example, if the wave travels to the right, the material vibrates left and right.

As the material vibrates, it produces alternating regions:

• Compressions, where the material is more compressed.
• Rarefactions, where the material is less compressed.

These compressions and rarefactions travel through the material. The material itself vibrates back and forth rather than travelling along with the wave.

Sound waves and seismic P-waves, also called primary waves, can be modelled as longitudinal waves.

Wave reflection at a plane surface

When a wave meets a surface and returns into the region it came from, it has been reflected. A plane surface is a flat surface. Reflection at a plane surface changes the wave’s direction according to a simple rule.

Angles of incidence and reflection

The normal is an imaginary line drawn perpendicular to the surface at the point where the wave meets it. Both angles are measured from this line, not from the surface:

• The angle of incidence is between the incoming wave’s direction and the normal.
• The angle of reflection is between the reflected wave’s direction and the normal.

The angle of incidence equals the angle of reflection. For example, if a wave approaches at $$30^\circ$$ to the normal, it leaves at $$30^\circ$$ to the normal on the other side. A wave travelling along the normal returns along the same path.

Reflection in a ripple tank

In a ripple tank, straight water wavefronts can be directed towards a straight barrier. A wavefront is a line joining points at the same stage of a wave, such as points along a crest. The wave travels perpendicular to its wavefronts.

After the waves meet the barrier, reflected straight wavefronts travel away from it. Their orientation shows that the direction of travel has changed. Measuring the incoming and reflected directions from the normal shows that the two angles are equal.

Wave refraction

A wave can change speed when it enters a different region. If it crosses the boundary at an angle, the change in speed also changes its direction. This change in direction is called refraction.

A wavefront is a line joining points at the same stage of a wave, such as points along a crest. When a wavefront reaches a boundary at an angle, one part enters the new region before the rest. That part changes speed first, causing the wavefront to turn. The direction of travel, which is perpendicular to the wavefront, turns with it.

The normal is an imaginary line perpendicular to the boundary. A wave that slows down bends towards the normal; a wave that speeds up bends away from it. At normal incidence, the wave travels along the normal and its wavefront reaches the boundary all at once. Its speed changes, but its direction does not.

Refraction: Frequency and wavelength

The wave’s frequency remains unchanged as it crosses the boundary. Wave speed, frequency and wavelength are related by:

Here $$v$$ is wave speed, $$f$$ is frequency and $$\lambda$$ is wavelength. Since frequency stays the same, a wave that slows down has a shorter wavelength; a wave that speeds up has a longer wavelength. This change in wavelength occurs even at normal incidence, when the direction does not change.

Refraction in a ripple tank

In a ripple tank, water waves travel more slowly in shallow water than in deep water. As waves enter a shallower region, their frequency stays the same but their wavelength decreases, so successive wavefronts are closer together.

If the wavefronts meet the change in depth at an angle, the part entering the shallow water first slows first. The wavefronts turn, and the waves bend towards the normal. If they meet the boundary at normal incidence, they slow down and their wavelength decreases, but they continue in the same direction.

Wave diffraction

When waves pass through a gap or around the edge of an obstacle, they spread into regions they would not reach if they continued only in straight lines. This spreading is called diffraction.

The amount of spreading depends on the wavelength relative to the size of the gap or obstacle. For waves passing through a gap, diffraction is especially noticeable when the gap width is comparable to the wavelength. The waves then spread widely beyond the gap. If the gap is much wider than the wavelength, the waves spread mainly near its edges and continue mostly straight ahead through the middle.

Diffraction in a ripple tank

In a ripple tank, the wavefronts show the positions of corresponding points on the water waves, such as their crests. Straight wavefronts approaching a gap between two barriers can be seen spreading out beyond it. When the gap is comparable to the wavelength, the emerging wavefronts are strongly curved.

Wavefronts can also be seen spreading around the edge of a barrier, reaching water behind the barrier that lies outside their original straight-ahead path.