Progressive and Stationary Waves

Progressive waves transfer energy from one place to another. They require a medium to move through (e.g. water or air).

Amplitude (A): the distance between the maximum displacement of the wave from the equilibrium position.

Wave speed (c): measured in ms^-1

Frequency (f): the number of waves that pass through a point in one second, measured in Hz.

Time period (T): the time taken for one whole wave to pass through a point. f = 1/T

Wavelength (λ): the length of one whole wave (from peak to peak or trough to trough) measured in metres (m).

Wave speed can be calculated using the following equation:

c = fλ
wave speed = frequency x wavelength

• Particles oscillate parallel to the direction of energy transfer/direction of travel of the wave
• Compressions and rarefactions
• Sound waves and seismic p-waves are longitudinal
• Cannot be polarised

The phase of a wave is the fraction of the wave that has passed. It can be measured in degrees or radians where 1° = 2π/360 radians. One complete cycle = 360° = 2π rad.

Depending on the shape of the wave, you can think of it as a sine or cosine wave and label 90°, 180°, 270° and 360° (or π/2, π, 3π/2 and 2π) to work out the phase difference.

Two different waves or two points on a wave can be compared by their phase difference. If two points on a wave are exactly half a cycle apart = π rad = 180° they are in anti-phase. If the points have exactly the same phase difference, they are in phase. Otherwise, they are out of phase.

Electromagnetic waves are made up of oscillating electric and magnetic fields, these fields act in all planes. A polarising filter can be used to absorb components of fields acting in the direction of the polarising filter. This allows for fields acting perpendicular to the polariser to pass through – the wave is now polarised as it has one plane of polarisation.

As only transverse waves (electromagnetic) can be polarised, we can use this property to distinguish between transverse and longitudinal waves.

Applications of polarisers:

• Polaroid filters found in sunglasses allow one plane of light to pass through the lenses. This reduces the glare from surfaces (e.g. water and snow) and reduces eye strain.
• TV or radio aerials can be aligned to receive signals (waves) transmitted in a specific plane. For example, if a transmitter transmits vertically plane-polarised waves, the aerials can be positioned vertically to receive these waves.

If two or more waves of the same type (e.g. all electromagnetic waves) meet, their displacements can be added to produce a resultant wave.

Constructive superposition occurs when waves meet in phase, their amplitudes are added to produce a new wave.

Destructive superposition occurs when the waves meet out of phase, this can result in a new wave with a lower amplitude.

Coherent waves have the same frequency, wavelength and a constant phase difference. When two coherent waves travelling in opposite directions meet, they superimpose and form a stationary wave.

Stationary waves appear as though they are not moving. They consist of nodes and antinodes. Antinodes have maximum amplitude and nodes have zero amplitude (on the equilibrium line).

These antinodes are formed by constructive interference of the two waves – the waves arrived in phase. The nodes are formed by destructive interference (the waves have arrived out of phase).

Stationary waves can be formed by microwaves reflecting off a surface (e.g. a metal plate) and superimposing. An aerial can be moved between the transmitting source and the surface to detect nodes and antinodes. The same can be done with sound waves, however, a microphone is used to receive the waves.

Stationary waves can also be formed on a string. The string is fixed at one end and attached to a signal generator on the other. At specific frequencies (and fixed tension in the string), stationary waves known as harmonics are formed. 

The first harmonic is the lowest frequency at which a stationary wave forms.

Factors affecting the frequency at which the harmonics form include:

• String tension,
• Mass per unit length of string
• String length.

These properties are summarised in the equation for the first harmonic.

f = 1/2L x (T/µ)^1/2

f is the frequency of the first harmonic (Hz)
L is the length of the string (m)
T is the tension in the string (N)
µ is the mass per unit length of the string (kgm^-1)

The length of the sting for the first harmonic = half the wavelength of the wave.

L = λ/2

For the second harmonic: L = λ

For the third harmonic: L = 3λ/2

This can be seen by counting from peak to peak or trough to trough.