Coaching Centre
Wave graphs and the wave equation
Amplitude, wavelength, period and frequency, displacement-distance and displacement-time graphs and how to tell them apart, predicting particle motion from a snapshot, and the wave equation v = fλ with T = 1/f.
A surfer at Bondi counts the waves passing the end of the rock pool: one every 8 seconds. A lifeguard on the tower estimates the crests are about 120 m apart out past the break. From just those two numbers you can work out that the swell is travelling at 15 m/s, about 54 km/h, and that the storm that made it, far out in the Tasman, might be a day and a half away. This page gives you the tools.
The words that describe a wave
| Quantity | Symbol and unit | Meaning |
|---|---|---|
| Displacement | $y$ (m) | how far a particle is from its rest (equilibrium) position at a given moment; can be positive or negative |
| Amplitude | $A$ (m) | the maximum displacement, measured from the rest position (not crest to trough) |
| Wavelength | $\lambda$ (m) | the distance between two neighbouring points in phase, e.g. crest to crest or compression to compression |
| Period | $T$ (s) | the time for one complete oscillation of a particle, or for one full wave to pass a point |
| Frequency | $f$ (Hz) | the number of complete oscillations per second; 1 Hz = 1 s$^{-1}$ |
| Wave speed | $v$ (m s$^{-1}$) | the speed at which the wave pattern (and its energy) travels |
Period and frequency are reciprocals:
$T = \dfrac{1}{f}$
Two points are in phase when they are doing exactly the same thing at the same time (both at a crest and about to fall, say). Points half a wavelength apart are out of phase: one is at a crest while the other is at a trough.
Two different graphs
A wave changes in both space and time, so there are two ways to graph it. They look almost identical, which is exactly why students mix them up. Always read the horizontal axis first.
A displacement–distance graph ($y$ against $x$) is a photograph of the whole wave at one instant. It shows the shape of the wave, so you read the wavelength and the amplitude from it. It cannot tell you the period on its own.
A displacement–time graph ($y$ against $t$) follows a single particle as the wave passes. You read the period (and so the frequency) and the amplitude from it. It cannot tell you the wavelength on its own.
To find the wave speed you need one of each: $\lambda$ from the first and $T$ (or $f$) from the second.
Longitudinal waves use the same graphs. The displacement $y$ is now measured along the direction of travel (positive to the right, say), so a "crest" on the graph is not a compression. Compressions sit where the graph crosses the axis with particles on either side displaced towards it. At HSC level you mostly need to recognise that a longitudinal wave can be graphed this way and read $\lambda$, $T$ and $A$ from the graph exactly as for a transverse wave.
Which way is a particle moving?
A snapshot does not show motion, but if you know which way the wave travels you can work it out. Imagine the whole curve shifted a tiny distance in the direction of travel. Each particle must move vertically to its new position on the shifted curve.
A quick check: for a wave moving to the right, the "uphill" side of each crest (the side facing the direction of travel) is moving up next. Particles at a crest or trough are momentarily at rest; particles at the equilibrium line are moving fastest.
The wave equation
In one period $T$ a source makes one complete wave, and the front of the wave moves forward one wavelength $\lambda$. Speed is distance over time, so
$v = \dfrac{\lambda}{T} = f\lambda$
This is true for every kind of wave. Remember what controls each quantity: the source sets the frequency, the medium sets the speed, and the wavelength adjusts to fit, $\lambda = v/f$. When a wave passes into a new medium its frequency stays the same and its speed and wavelength change together.
A displacement–distance graph for a water wave shows crests 0.40 m apart and a height of 0.05 m above the rest level. A displacement–time graph for a cork on the water shows it rises and falls with a period of 0.25 s. Find the amplitude, frequency and wave speed.
Ocean swell has a period of 8.0 s and a wavelength of 120 m in deep water. (a) Find its speed. (b) A storm is 2000 km away. How long does the swell take to arrive?
Find (a) the wavelength of the ABC Classic FM signal in Sydney, broadcast at 92.9 MHz, and (b) the frequency of green light with a wavelength of 530 nm.
- At the start, f = 1.0 Hz and λ = 2.0 m, so the readouts show v = 2.00 m/s and T = 1.00 s. Set f = 2.0 Hz and λ = 1.5 m: check that v = 3.00 m/s and T = 0.50 s.
- Count the crests on the top graph between 0 and 4 m, then count the cycles on the bottom graph in 4 s. Which graph changes when you move the wavelength slider, and which when you move the frequency slider?
- Press Freeze. Predict whether the orange particle is moving up or down, then press Play and check against the readout.
- Switch the wave to moves left without changing anything else. What happens to the particle's direction?
Test yourself
A displacement–time graph for a particle on a rope shows a sine curve that repeats every 0.20 s with a maximum displacement of 3.0 cm. Which quantities can be found from this graph alone?
The time axis gives $T = 0.20$ s, so $f = 5.0$ Hz, and the vertical axis gives $A = 3.0$ cm. The wavelength (and so the speed) needs a displacement–distance graph.
A wave of frequency 250 Hz travels at 340 m/s. Its wavelength is
$\lambda = v/f = 340/250 = 1.36$ m.
A transverse wave moves to the right. At one instant a particle is at the equilibrium position on the right-hand (leading) side of a crest. The particle is
The crest is about to arrive at that point, so the particle must be rising. At the equilibrium position its speed is greatest.
A source in a ripple tank is changed from 10 Hz to 20 Hz. The water depth is not changed. Compared with before, the waves have
The depth (medium) sets the speed. With $v$ fixed, doubling $f$ halves $\lambda$.
Points P and Q on a wave are 1.5 wavelengths apart. When P is at a crest, Q is
Every whole wavelength brings you back in phase; the extra half wavelength puts Q exactly out of phase, at a trough.
Label the displacement–distance graph.
Which graph do you read each quantity from?
The time for one complete oscillation is the , which is the reciprocal of the . Wave speed equals frequency multiplied by . When a wave enters a new medium, its stays the same, because it is set by the .
A student sends waves along a rope. A displacement–distance graph taken from a video shows 3 complete waves in 4.8 m with crests 12 cm above the rest position. A displacement–time graph for one point shows 10 complete oscillations in 4.0 s. (a) Determine the amplitude, wavelength, period and frequency. (b) Calculate the wave speed. (c) The student shakes the rope twice as often without changing the tension. Predict the new wavelength, with a reason.
(a) Amplitude is measured from the rest position: $A = 12$ cm $= 0.12$ m. Three waves in 4.8 m gives $\lambda = 4.8/3 = 1.6$ m. Ten oscillations in 4.0 s gives $T = 0.40$ s, so $f = 1/T = 2.5$ Hz.
(b) $v = f\lambda = 2.5 \times 1.6 = 4.0\ \text{m s}^{-1}$.
(c) The speed depends on the rope and its tension, which have not changed, so $v$ stays at 4.0 m/s. Shaking twice as often makes $f = 5.0$ Hz, so $\lambda = v/f = 4.0/5.0 = 0.80$ m: half the original wavelength.