Waves: displacement-time and distance-time graphs

In a nutshell

All waves have a displacement, amplitude wavelength, period, frequency and wave speed. A displacement-distance graph is a snapshot of all the oscillating particles 'frozen' in time. A displacement-time graph shows one particle and how its displacement changes with time. Phase difference is how much an oscillating particle lags behind another oscillating particle.

Equations

Description	Equation
Frequency formula	$f = \frac{1}{T}$

Variable definitions

Quantity name	Symbol	Derived unit	Alternate unit	SI Unit
$frequency$	$f$	$Hz$		$s^{-1}$
$period$	$T$	$s$		$s$
$displacement$	$s$	$m$		$m$
$wavelength$	$\lambda$	$m$		$m$
$amplitude$	$A$	$m$		$m$
$wave\space speed$	$v$	$ms^{-1}$		$ms^{-1}$
$phase \space difference$	$\phi$	$rad$	$\degree, \space \lambda$	$rad$

Note: Knowing the symbol for phase difference is not required!

Properties of a wave

All waves have these following properties.

Definitions table

Displacement	The distance from the equilibrium position to an oscillating particle.
Amplitude	The maximum displacement of an oscillating particle
Wavelength	The distance between two identical adjacent points on a wave. For example, from one crest to the next crest, or one rarefaction to the next rarefaction.
Period	The time taken for on wavelength to pass a certain point.
Frequency	The amount of wavelengths passing through a certain point per second.
Wave speed	The distance travelled by a wave per second.

The frequency formula

The definitions of period and frequency show that they are reciprocals of each other. This means that the period and frequency can be related.

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

Displacement-distance graphs

A displacement-distance graph shows the displacement of the oscillating particles against the distance along the wave's path. A displacement-distance graph is often referred to as a wave profile, or a 'snapshot' of a wave (i.e. at a fixed time).

Wave profiles can be used to find the wavelength of the wave. The distance of one full cycle of a wave is equal to a wavelength. The wavelength can be measured between any two identical adjacent points on a wave.

Physics; Waves; KS5 Year 12; Waves: displacement-time and distance-time graphs

A.	displacement ( $m$ )
B.	distance ( $cm$ )
1.	Amplitude
2.	Wavelength
3.	Trough
4.	Crest

The amplitude is given by the maximum displacement of a particle.

Phase difference

Phase difference can be thought of as how far a particle on a wave lags behind another particle. This could be a particle on the same wave, or a different wave.

Phase difference is measured as an angle, and is given in units of degrees, radians or wavelengths.

Note: A complete cycle of a wave is represented by $360 \degree$ or $2\pi \,rads$ .

If two particles oscillate in-step they are in phase. In-step means that they are at the same point in the cycle of the wave, and have the same displacement and velocity. These particles have a phase difference of an integer multiple of $360 \degree$ or $2\pi \,rads$ (e.g. $2 \pi, 4\pi, 6\pi,...$ )

If two particles oscillate completely out of step they are antiphase. The phase difference between these particles is an odd number multiple of $180 \degree$ or $\pi \,rads$ (e.g. $\pi, 3\pi, 5\pi,...$ )

Example

Physics; Waves; KS5 Year 12; Waves: displacement-time and distance-time graphs

A.	displacement ( $m$ )
B.	distance ( $m$ )

A and E have a phase difference of $360 \degree$ or $2\pi \,rads$ . A and E are in phase.

A and C have a phase difference of half of A and E which is $180 \degree$ or $\pi \,rads$ . A and C are antiphase. This is the same for C and E.

A and B have a phase difference of half of A and C which is $90\degree$ or $\frac{\pi}{2}\,rads$ . This is the same for B and C, C and D, and D and E.

A and D have a phase difference of three-quarters of A and E which is $270 \degree$ or $\frac{3}{2}\pi \,rads$ . This is the same for B and E.

Displacement-time graphs

A displacement-time graph shows one particle and how its displacement changes with time (i.e. at a fixed distance).

A displacement-time graph can be used to determine the period of a wave. The time it takes a particle to complete a full oscillation is the period of the wave. This can be measured between any two identical adjacent points on the graph.

Physics; Waves; KS5 Year 12; Waves: displacement-time and distance-time graphs

A.	displacement ( $m$ )
B.	time ( $s$ )
T	period

The amplitude is given by the maximum displacement of the particle.