Grade by Grade
E. Waves and Sound · Waves and Sound

Describing Periodic Waves: Amplitude, Wavelength, Frequency and Period

E2.1 E3.1 A1.12 A1.13About 70 minutes
Learning goals
  • I can identify amplitude, wavelength, crest, trough, compression and rarefaction on a diagram.
  • I can calculate frequency and period and convert between them using f = 1/T.
  • I can read wavelength from a displacement–position graph and period from a displacement–time graph.
  • I can identify points on a wave that are in phase or out of phase.

Words to know

amplitude (A)
The maximum distance a particle moves from its rest position. Measured in metres (or cm, mm).
wavelength (λ)
The length of one complete cycle of a wave: the distance from crest to crest, trough to trough, or from the centre of one compression to the next. Measured in metres.
frequency (f)
The number of complete cycles per second. Unit: hertz (Hz), where 1 Hz = 1 cycle per second = 1 s⁻¹.
period (T)
The time for one complete cycle. Unit: seconds. T = 1/f.
cycle
One complete repetition of the motion: for example, from a crest, down through a trough, and back to a crest.
in phase
Two points on a wave are in phase when they are at the same point in their cycle (same displacement and moving the same way). They are a whole number of wavelengths apart.
phase shift
How far one wave is ahead of or behind another, often given as a fraction of a wavelength or cycle.

Counting the beat

Put your hand on a washing machine during the spin cycle, or watch a maple branch sway in the wind. Both repeat the same motion over and over. Physicists describe any repeating (periodic) motion with a few numbers: how big the motion is, how often it repeats and, for a wave, how long each repeat is in space. In this lesson you will learn to read those numbers from diagrams and graphs, a skill you will use in every lesson that follows.

Amplitude and wavelength

The amplitude (A) is the greatest distance a particle moves from its rest position. It is measured from the rest line to a crest (or to a trough), not from crest to trough. Bigger amplitude means more energy: louder sound, taller water waves.

The wavelength (λ), the Greek letter lambda, is the length of one complete cycle. Measure it between any two matching points: crest to crest, trough to trough, or the centre of one compression to the centre of the next. A common mistake is measuring crest to trough, which is only half a wavelength.

00.40.81.21.622.4position (m)-0.150.15displacement (m)wavelength λ = 0.80 mamplitude A = 0.15 mcresttrough
A displacement–position graph is a snapshot of the whole wave at one instant. Read the wavelength along the position axis (crest to crest = 0.80 m) and the amplitude from the rest line to a crest (0.15 m).

Frequency and period

The frequency (f) is the number of complete cycles each second. Its unit is the hertz (Hz): 1 Hz = 1 cycle per second. The period (T) is the time for one cycle, in seconds.

They are reciprocals:

f = 1/T and T = 1/f

If a swing takes 2.0 s to go back and forth once, T = 2.0 s and f = 1/2.0 = 0.50 Hz. If a tuning fork vibrates 440 times a second, f = 440 Hz and T = 1/440 = 0.00227 s = 2.27 ms.

From data you can find frequency as f = number of cycles ÷ time.

Useful prefixes: 1 kHz = 10³ Hz, 1 MHz = 10⁶ Hz, 1 ms = 10⁻³ s. Keep significant figures sensible: a period measured as 1.5 s gives a frequency of 0.67 Hz (two significant figures), not 0.666666 Hz.

0246810time (ms)-22displacement (mm)period T = 4.0 ms
A displacement–time graph follows one particle as time passes. The distance along the time axis for one cycle is the period: here T = 4.0 ms, so f = 1/0.0040 s = 250 Hz. Amplitude is 2.0 mm.

Two graphs that look the same

A displacement–position graph and a displacement–time graph can look identical. Always read the horizontal axis first. Position axis → wavelength. Time axis → period.

Phase

Two points on a wave are in phase if they are doing exactly the same thing at the same moment: same displacement, moving in the same direction. Points that are exactly one wavelength apart (or 2λ, 3λ…) are in phase. Points that are half a wavelength apart are out of phase: when one is at a crest, the other is at a trough. This idea is the key to interference, which comes later in this unit.

PQRS1λ1λ
P, Q and R are crests, each one wavelength apart, so they are in phase. S is at a trough, half a wavelength from P and Q, so S is out of phase with them.
Worked example

Example 1: Reading a snapshot

On a displacement–position graph of a wave on a rope, the first crest is at 0.30 m and the fourth crest is at 2.70 m. A crest is 6.0 cm above the rest line. Find the wavelength and amplitude.

  1. From the first crest to the fourth crest is 3 complete wavelengths (1st→2nd, 2nd→3rd, 3rd→4th). Counting gaps, not crests, avoids an off-by-one error.
  2. 3λ = 2.70 m − 0.30 m = 2.40 m, so λ = 2.40 ÷ 3 = 0.80 m.
  3. The amplitude is measured from the rest line: A = 6.0 cm = 0.060 m.

λ = 0.80 m and A = 0.060 m.

Worked example

Example 2: Frequency and period from data

A child on a swing at a park completes 15 full swings (back and forth) in 36 s. Find the period and the frequency.

  1. Period = time ÷ number of cycles = 36 s ÷ 15 = 2.4 s.
  2. Frequency = cycles ÷ time = 15 ÷ 36 s = 0.42 Hz (two significant figures, since 15 and 36 limit the precision).
  3. Check: 1 ÷ 2.4 s = 0.4166… Hz ≈ 0.42 Hz. ✓

T = 2.4 s and f ≈ 0.42 Hz.

Worked example

Example 3: Converting units

A dog whistle vibrates at 23 kHz. Find its period in microseconds (1 μs = 10⁻⁶ s).

  1. Convert to hertz: 23 kHz = 23 × 10³ Hz = 2.3 × 10⁴ Hz.
  2. T = 1/f = 1 ÷ (2.3 × 10⁴ Hz) = 4.35 × 10⁻⁵ s.
  3. In microseconds: 4.35 × 10⁻⁵ s = 43.5 × 10⁻⁶ s ≈ 43 μs (two significant figures).

T ≈ 4.3 × 10⁻⁵ s, about 43 μs.

Just for fun

A physics teacher asked her class, "What's the period of a wave?"

A student answered, "The time it takes to finish one sentence about it." Not quite, but she did say the class was on the right wavelength.

Hands-on activity

Measure the period of a pendulum and a ruler

You need: a string about 1 m long, a small weight such as a key or a washer, a ruler, a table, and a phone stopwatch.

  1. Tie the key to the string and hold the other end still (or tape it to the edge of a table). Pull the key a little to one side and let go.
  2. Time 20 complete swings (back and forth = 1 swing). Divide by 20 to get the period. Timing many cycles makes your reaction-time error much smaller per cycle.
  3. Calculate the frequency with f = 1/T.
  4. Repeat with a string half as long. What happens to T and f?
  5. Hold a ruler flat on a table with about 20 cm sticking out over the edge. Twang it gently and listen. Slide it so only 10 cm sticks out and twang again. Which is higher pitched? Which has the higher frequency?

What to look for: a 1.0 m pendulum has a period of about 2.0 s (f ≈ 0.50 Hz). Halving the length makes the period shorter (about 1.4 s) and the frequency higher. The shorter ruler vibrates faster: higher frequency, higher pitch.

Your turn

1

A crest is 12 cm above the rest position and a trough is 12 cm below it. What is the amplitude?

2

The distance from a crest to the next trough is 0.45 m. What is the wavelength?

3

A speaker cone vibrates 300 times in 1.5 s. Find its frequency.

4

A wave has a period of 0.025 s. Find its frequency.

5

A radio station broadcasts at 99.9 MHz. Write this frequency in hertz using scientific notation.

6

On a displacement–time graph, one cycle spans 8.0 ms. Find T and f.

7

Two points on a rope are 1.5 wavelengths apart. Are they in phase or out of phase?

More practice: warm-up

1

What is the unit of frequency?

2

If the frequency doubles, what happens to the period?

3

What symbol is used for wavelength?

4

How many hertz are in 2.5 kHz?

5

On which kind of graph do you read the wavelength: displacement–position or displacement–time?

More practice: on your own

1

Five complete wavelengths fit in 3.0 m of a Slinky. Find λ.

2

A hummingbird beats its wings about 50 times per second. Find the period of one wingbeat in milliseconds.

3

On a longitudinal wave, the centre of the first compression is at 10 cm and the centre of the fifth compression is at 90 cm. Find λ.

4

A student times 10 swings of a pendulum as 13.6 s. Another times 1 swing as 1.2 s. Whose period is more reliable, and why?

5

Point A is at a crest. Point B is 0.25 λ farther along the wave. Describe where B is.

Challenge

1

A displacement–position graph shows a wave with crests at 0.15 m and 0.75 m (consecutive). A displacement–time graph for one point on the same wave shows a period of 0.20 s. How far does the wave travel in one period, and what is its speed?

2

A wave has frequency 1.5 kHz. How many complete cycles happen in 0.040 s? How many in 1.0 minute?

3

Explain why it is wrong to say "the amplitude of a wave is the height from trough to crest". If the trough-to-crest height of an ocean swell is 3.0 m, what is its amplitude, and how many times more energy would a 6.0 m swell carry (same wavelength)?

Check your understanding

Quiz

1.

The distance from one crest to the next crest is the…

2.

A wave has a frequency of 50 Hz. What is its period?

3.

The amplitude of a transverse wave is measured from…

4.

On a displacement–time graph, the horizontal length of one cycle is the…

5.

Two points on a wave are 3λ apart. They are…

6.

A guitar string vibrates 1320 times in 3.0 s. Its frequency is…

For parents and teachers

Big idea: students must use wave vocabulary precisely (E2.1) and handle units and significant figures (A1.12, A1.13). Everything later in the unit depends on reading λ, A, f and T correctly.

How to help: Ask your teen to explain the difference between the two graph types using the horizontal axis. Time a pendulum together and check the arithmetic f = 1/T.

Common difficulties:

  • Measuring amplitude crest-to-trough (that is 2A).
  • Measuring wavelength crest-to-trough (that is λ/2).
  • Counting crests instead of gaps (from crest 1 to crest 4 is 3 wavelengths).
  • Unit prefixes: kHz, MHz, ms and μs. Have them write the power of ten every time.
  • Reporting too many digits. Match the significant figures of the data.