K326 / K327 / 2027
General properties of waves overview

Topic 2 of 5

Wave quantities and graphs

A snapshot shows different points at one instant. A time trace shows one point at different instants. Their horizontal axes answer different questions.

Displacement describes a point's position relative to equilibrium, with a chosen positive direction. For the rope below, upward displacement is positive and downward displacement is negative. This is the local displacement of the rope, not the distance travelled by the wave.

Five quantities to identify

Wave speed, v
The distance travelled by the wave pattern per unit time, commonly in m/s. Follow a recognisable feature such as a crest. This is different from the changing speed of an individual vibrating point.
Frequency, f
The number of complete oscillations per second, measured in hertz (Hz). A frequency of 2.5 Hz means 2.5 complete cycles each second.
Wavelength, λ
The distance between adjacent points at the same stage of the wave cycle, measured in a length unit such as m. Examples are crest to next crest, trough to next trough, or one compression centre to the next.
Period, T
The time taken for one complete oscillation, in seconds. A point returns to the same stage of its motion after one period.
Amplitude, A
The maximum displacement from equilibrium. Its unit is a length unit such as m or cm. Measure from the equilibrium line to a crest or trough, not from crest to trough.

A complete cycle links period and frequency

If a point completes five oscillations in 2.0 s, its period is 2.0/5 = 0.40 s, and its frequency is 5/2.0 = 2.5 Hz. Taking longer for each cycle means completing fewer cycles per second.

f = 1/T   and   T = 1/fUse T in seconds to obtain f in Hz. Count complete cycles, not every passage through equilibrium.

A point crosses equilibrium twice in a cycle, once in each direction. When timing from an equilibrium crossing, finish at an equivalent crossing moving in the same direction. For a longitudinal wave, a compression centre to the neighbouring rarefaction centre is half a wavelength.

Read the same rope wave in two ways

The following wave travels to the right. Its amplitude is 2.0 cm, wavelength is 0.80 m and period is 0.40 s. The first graph is a snapshot at t = 0. The second follows the marked material point at x = 0.20 m.

Two graphs describe the same right-moving rope wave

Amplitude is 2.0 cm, wavelength is 0.80 m and period is 0.40 s. The wave speed is 2.0 m/s. The orange point in the snapshot is the material point followed by the time graph.

Snapshot at t = 0 s

Snapshot at t = 0 s: signed displacement in centimetresHorizontal position ranges from zero to 1.60 metres. At time zero, crests are at 0.20 and 1.00 metres and troughs at 0.60 and 1.40 metres. A bracket between adjacent crests marks the 0.80 metre wavelength. The orange marker is at position 0.20 metres and displacement plus 2.0 centimetres. Both graphs use the same vertical scale from minus 2.0 to plus 2.0 centimetres. The bracket A on the right spans zero to plus 2.0 centimetres, so amplitude is 2.0 centimetres, not the 4.0 centimetre peak-to-peak span. These are different horizontal quantities, not two wave shapes at different instants.Displacement / cmWavelength = 0.80 m-20+20.00.40.81.21.6APosition along rope / m

Different points on the rope, all at one instant: read wavelength from the position axis.

One point at x = 0.20 m

One point at x = 0.20 m: signed displacement in centimetresTime ranges from zero to 0.80 seconds for the one material point at position 0.20 metres. Its displacement starts at plus 2.0 centimetres, reaches zero at 0.10 seconds, minus 2.0 at 0.20, zero at 0.30 and plus 2.0 again at 0.40. The cycle repeats. A bracket marks the 0.40 second period. Both graphs use the same vertical scale from minus 2.0 to plus 2.0 centimetres. The bracket A on the right spans zero to plus 2.0 centimetres, so amplitude is 2.0 centimetres, not the 4.0 centimetre peak-to-peak span. These are different horizontal quantities, not two wave shapes at different instants.Displacement / cmPeriod = 0.40 s-20+20.00.20.40.60.8ATime / s

One point on the rope, followed through time: read period from the time axis.

The distance graph compares different rope points at one instant. The time graph follows the single point at x = 0.20 m. Both use displacement vertically, but only the snapshot gives wavelength directly and only the time trace gives period directly.

Reading the snapshot

Displacement against distance

The crests are at x = 0.20 m and x = 1.00 m. Their separation is λ = 1.00 - 0.20 = 0.80 m. The troughs at 0.60 m and 1.40 m are also 0.80 m apart.

The maximum displacement from zero is A = 2.0 cm. From +2.0 cm at a crest to -2.0 cm at a trough is 4.0 cm, twice the amplitude.

The horizontal axis is position, so this graph alone gives no time for a cycle. Frequency cannot be found from its crest spacing alone.

Reading the time trace

Displacement against time

The marked point starts at +2.0 cm. It reaches equilibrium at 0.10 s, -2.0 cm at 0.20 s, equilibrium at 0.30 s, and +2.0 cm again at 0.40 s.

It has then completed one cycle: T = 0.40 s and f = 1/0.40 = 2.5 Hz. Its amplitude is still 2.0 cm.

This horizontal axis is time. The distance between two peaks along the drawn page represents a time interval, not a wavelength.

A graph of displacement is not a graph of wave speed

On the time trace, a rising section means that the marked point's displacement is increasing in the chosen positive direction. It does not mean that the travelling wave is speeding up. A negative displacement means the point is below equilibrium, not that the wave must travel to the left.

Read values from the axes and their scales. The graph's physical width on a phone or a printed page is not the represented wavelength or period.

Check the horizontal axis first. A distance snapshot supplies spacing. A fixed-point time trace supplies timing. They can describe the same wave while carrying different information.

Optional check A displacement-distance snapshot has crests at 0.20 m and 1.00 m, with maximum displacement +2.0 cm and minimum -2.0 cm. What can be read from this snapshot alone?
A displacement-distance snapshot has crests at 0.20 m and 1.00 m, with maximum displacement +2.0 cm and minimum -2.0 cm. What can be read from this snapshot alone?