Describing a wave: amplitude, frequency, period, speed and wavelength

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This unit opens the waves half of Unit 2, Waves and Electricity, and everything in it is examined on that written paper, WPH12.

The two core practicals in it (the speed of sound and the vibrating string) are also examined as practical skills on the Unit 3 paper, WPH13.

Waves carry about two thirds of the WPH12 marks, and this unit's stationary-wave questions turn up in every series, so the vocabulary in this first section is the base for all of it.

The idea: energy moves, the medium stays

Drop a stone in a pond and a ring of ripples spreads out, yet a leaf floating on the water only bobs up and down where it is.

That is what a progressive wave is: a disturbance that transfers energy from one place to another without transferring the material it travels through.

Each part of the medium (a water molecule, a coil of a spring, an air molecule) oscillates about its own fixed equilibrium position, and passes the disturbance on to its neighbour.

To describe a wave we need numbers for two separate things: how each particle moves (how far, how often) and how the pattern moves (how far apart the repeats are, how fast it travels).

The five terms in this section do exactly that, and two graphs show them.

The snapshot: displacement against distance

Imagine photographing a long stretched spring while a transverse wave runs along it.

Plotting each coil's displacement (its distance from its equilibrium position, positive on one side and negative on the other) against its position along the spring gives a graph like a photograph of the wave.

Graph of displacement in centimetres against distance along a stretched spring in metres at one instant: a sine curve of amplitude 2.0 cm with crests at 0.15 m, 0.75 m and 1.35 m

Two quantities come straight off this graph:

Amplitude is the maximum displacement from the equilibrium position: here 2.0 cm, measured from the zero line to a crest. It is not the crest-to-trough height, which is twice the amplitude.

Wavelength $\lambda$ is the distance between two adjacent points that are oscillating in phase, such as two neighbouring crests: here from 0.15 m to 0.75 m, so $\lambda = 0.60$ m. Two neighbouring troughs, or two neighbouring points where the curve crosses zero going the same way, give the same answer.

The clock: displacement against time for one point

Now watch one painted coil of the same spring and plot its displacement against time.

The graph has the same shape, but its horizontal axis is time, so it gives different information.

Graph of displacement in centimetres against time in seconds for one marked coil of the spring: a sine curve of amplitude 2.0 cm that repeats every 0.40 s, starting at zero and rising

Period $T$ is the time for one complete oscillation of a point: the time between two neighbouring crests on this graph, 0.40 s.

Frequency $f$ is the number of complete oscillations per unit time (per second), measured in hertz (Hz): $1\ \text{Hz} = 1\ \text{s}^{-1}$ . One oscillation takes $T$ , so in one second there are $1/T$ of them:

f = \frac{1}{T}

This relation is learn it: not on the list; it is used on almost every WPH12 paper but is not printed.

For the coil above, $f = 1/0.40 = 2.5$ Hz.

The amplitude appears on both graphs; the wavelength only on the snapshot; the period only on the time graph.

A common slip is to read a "wavelength" off a time axis.

Speed: how fast the pattern travels

The speed of a wave is the distance travelled by the wave (by a crest, or by the energy it carries) per unit time.

It is not the speed of any particle: the coil above never goes anywhere, while the crests run along the spring.

If a crest is timed travelling 3.6 m along the spring in 2.4 s, the wave speed is $3.6/2.4 = 1.5\ \text{m s}^{-1}$ , while each coil just moves up and down 2.0 cm either side of its rest position.

Why it matters: the frequency is set by the source (the hand or vibrator shaking the spring), and the speed is set by the medium (how tight and how heavy the spring is).

The wavelength is the result of both.

The next section turns that into the wave equation.

Reading an oscilloscope trace

Electrical signals from a microphone or a signal generator are displayed on an oscilloscope as a displacement-time graph.

The screen has a grid of large squares (divisions), each split into five small ones on the centre lines.

The time-base setting gives the time per horizontal division, and the Y-gain gives the voltage per vertical division.

Oscilloscope screen of 10 by 8 divisions showing a sine wave trace of amplitude 2.0 divisions; exactly three complete cycles fit into 8.0 divisions across the screen

To get an accurate period, never measure one cycle: measure across as many complete cycles as fit on the screen, then divide.

Reading one cycle to the nearest small division (0.2 division) on a period of about 2.7 divisions is a 7% uncertainty; reading three cycles over 8.0 divisions cuts it to under 3%.

Where this leads

These five terms are used in every later wave section: in 2.3b for refraction and diffraction (where the frequency stays fixed as the speed changes), in 2.3c where frequency sets a photon's energy, and in Unit 5 where simple harmonic motion describes the motion of each particle in a wave.

In medicine, the ultrasound frequency chosen for a scan sets the wavelength and so the smallest detail the scan can show.

How the marks work

Every marking point is worth one mark, written "(1)"; there are no letter codes. A calculation typically earns "(1) use of the equation" for the substituted equation and "(1) answer with unit".

Show the equation with your numbers in: the "use of" mark is given for a correct substitution even if a power of ten slips later.

A missing or wrong unit on a final answer loses the last mark, once per clip; there is no unit penalty on a "show that".

A "show that" answer must reach at least one more significant figure than the printed value; a bare printed value scores nothing.

Errors carry forward (ecf) between parts. A dependent mark in an explanation needs the reason before it.

The one asterisked 6-mark item is marked on content points (up to 4) plus linkage (up to 2): six points joined in a logical chain for full marks.

On the list and learn it (2.3a)

On the list at the end of WPH12: $v = f\lambda$ and $v = \sqrt{\dfrac{T}{\mu}}$ ; data: $g = 9.81\ \text{m s}^{-2}$ (needed for the tension from hanging masses, $W = mg$ , also on the Unit 1 list).

Learn it: not on the list: $f = \dfrac{1}{T}$ ; phase difference $= \dfrac{2\pi \times \text{path difference}}{\lambda}$ (in radians, or $360°$ in place of $2\pi$ for degrees); a string fixed at both ends vibrating in its fundamental mode has $\lambda = 2l$ ; a pipe closed at one end has $\lambda = 4L$ in its fundamental mode; adjacent nodes are $\lambda/2$ apart; speed = distance ÷ time.

Key terms

Amplitude: the maximum displacement of a point on the wave from its equilibrium position.

Frequency: the number of complete oscillations per unit time; unit hertz (Hz).

Period: the time taken for one complete oscillation; $T = 1/f$ .

Wavelength: the distance between two adjacent points on the wave that are oscillating in phase (for example, two adjacent crests).

Wave speed: the distance travelled by the wave per unit time.

Worked example 1: amplitude and wavelength from a snapshot

Use the graph of displacement against distance for the spring to determine the amplitude and the wavelength of the wave.

(3)

1. Amplitude: from the zero line to a crest, $A = 2.0$ cm (1) (not 4.0 cm, the crest-to-trough height)

2. Use more than one wavelength: the curve crosses zero going upwards at 0 and again at 1.20 m, which is two complete wavelengths, so $2\lambda = 1.20$ m (1) reading over several wavelengths

3. $\lambda = 0.60$ m (1) answer with unit

Answer: amplitude 2.0 cm, wavelength 0.60 m.

Measuring across two wavelengths halves the effect of any reading error.

Worked example 2: period and frequency from a displacement-time graph

Use the graph of displacement against time for the painted coil to calculate the frequency of the wave.

(2)

1. Two complete cycles take 0.80 s, so $T = 0.80/2 = 0.40$ s; $f = \dfrac{1}{T} = \dfrac{1}{0.40}$ (1) use of $f = 1/T$ with $T$ from the graph

2. $f = 2.5$ Hz (1) answer with unit

Answer: 2.5 Hz.

The answer comes from the time axis: no wavelength can be read from this graph.

Worked example 3: frequency from an oscilloscope trace

The trace on the oscilloscope screen is the signal from a microphone.

The time-base is set at 0.25 ms per division.

Determine the frequency of the sound.

(3)

1. Three complete cycles occupy 8.0 divisions (1) measured over several cycles

2. $3T = 8.0 \times 0.25 = 2.0$ ms, so $T = 0.667$ ms $= 6.67 \times 10^{-4}$ s (1) time from the time-base, converted to seconds

3. $f = \dfrac{1}{6.67 \times 10^{-4}} = 1.5 \times 10^3$ Hz (1) answer with unit

Answer: $1.5 \times 10^3$ Hz (1.5 kHz).

Counting large divisions, not small ones, and leaving the time in ms (which gives 1.5 "Hz") are the two slips that lose these marks.

Worked example 4: show that, from a quarter of a cycle

A marked coil on a long spring moves from its equilibrium position to its maximum displacement in 0.15 s.

Show that the frequency of the wave is about 1.7 Hz.

(3)

1. Equilibrium to maximum is a quarter of an oscillation, so $T = 4 \times 0.15 = 0.60$ s (1)

2. $f = \dfrac{1}{T} = \dfrac{1}{0.60}$ (1) use of $f = 1/T$

3. $f = 1.67$ Hz (1) value given to more figures than the printed 1.7

Answer: 1.67 Hz, which is about 1.7 Hz.

Treating 0.15 s as half a period gives 3.3 Hz, a common trap: equilibrium to maximum is a quarter, maximum to minimum is a half.

Exam tip

When you read a period from an oscilloscope trace or a printed graph, measure across as many complete cycles as the screen shows and divide, and say so in your working ("3 cycles = 8.0 div").

Check whether the time-base is per large division, and convert ms or μs to s before using $f = 1/T$ .

Common mistakes

Giving the crest-to-trough height as the amplitude. It is tempting because it is the whole "size" of the wave, but amplitude is measured from the equilibrium (zero) line to a crest, so it is half that height.

Measuring wavelength from a crest to the next trough. That is half a wavelength. A wavelength joins two adjacent points in phase: crest to crest, or trough to trough.

Reading one cycle from a trace. Recurring in reports: one cycle gives a large reading uncertainty. Use several cycles, and do not confuse large and small divisions.

Confusing the wave speed with the speed of the particles. The particles oscillate about fixed positions with a speed that keeps changing; the wave speed is the steady speed at which the pattern, and the energy, moves along.

Check your understanding

1. A sound wave has a frequency of 50 Hz. Which of the following is the time taken for an air molecule to move from its maximum displacement on one side to its maximum displacement on the other side? (1)

A 0.0050 s
B 0.010 s
C 0.020 s
D 0.040 s Total [1]

2. Which of the following pairs of quantities can both be read from a graph of displacement against time for one point on a wave? (1)

A amplitude and wavelength
B wavelength and period
C amplitude and period
D wave speed and wavelength Total [1]

3. The graph shows the displacement against distance for a water wave at one instant.

Graph of displacement in centimetres against distance in metres for a water wave at one instant: a sine curve starting at zero, drawn from 0 to 1.0 m

(a) Determine the amplitude of the wave. (1)
(b) Determine the wavelength of the wave. (2)
(c) A small float on the water makes 12 complete oscillations in 15.0 s. Calculate the frequency of the wave. (2) Total [5]

4. On an oscilloscope with a time-base of 2.0 ms per division, five complete cycles of a trace occupy 7.5 divisions. Show that the frequency of the signal is about 330 Hz. (3) Total [3]

5. A student watching a float on a lake says that the float moves along with the wave at the wave speed. Explain why the student is wrong. (2) Total [2]

6. State what is meant by
(a) the amplitude of a wave (1)
(b) the wavelength of a wave. (1) Total [2]

Answers

1. B (1). The period is $1/50 = 0.020$ s, and one extreme to the other is half a cycle, 0.010 s. A is a quarter of a period (equilibrium to one extreme). C is a whole period. D is two periods.

2. C (1). A time graph gives the amplitude (largest displacement) and the period (time for one cycle). A and B need the wavelength, which is a distance and comes only from a displacement-distance graph. D needs a wavelength too, and the speed cannot be read from either graph alone.

3. (a) 1.5 cm (1). (b) Three complete wavelengths end at 0.96 m (zero crossing going upwards) (1); $\lambda = 0.96/3 = 0.32$ m (1) (accept 0.31 to 0.33 m). (c) $f = 12/15.0$ (1); $f = 0.80$ Hz (1).

4. $5T = 7.5 \times 2.0 = 15$ ms (1); $T = 3.0 \times 10^{-3}$ s and $f = 1/T$ (1); $f = 333$ Hz (1), given to more figures than the printed 330.

5. The water (and the float) only oscillates up and down about a fixed position (1); the wave transfers energy along the surface but not the water, so the float stays in the same place on average (1).

6. (a) The maximum displacement of a point (on the wave) from its equilibrium position (1). (b) The distance between two adjacent points (on the wave) oscillating in phase, for example crest to crest (1).

Describing a wave: amplitude, frequency, period, speed and wavelength

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