🔢 Frequency
frequency = number of waves ÷ time taken
This shares the waves out between the seconds, to give waves each second.
Properties of Waves » Frequency and Time Period
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 3.6, 3.7
- The link between frequency and time period: f = 1 / T
- How to change milliseconds into seconds before you calculate
- How to find a frequency by counting waves in a given time
- How to solve two-step problems on sound and electromagnetic waves
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Unlock This CourseTwo ways to describe the timing of a wave
In Describing Waves you met two quantities that are about time. Remember, the frequency is the number of complete waves each second, and the period is the time for one complete wave. The spec calls the period the time period, with the symbol T. They are two ways of saying the same thing, so if you know one you can always work out the other.
Frequency and time period
frequency = 1 / time period
f = 1 / T
f is the frequency in hertz (Hz) and T is the time period in seconds (s).
This equation is given on the formulae page of the exam paper. You do not have to memorise it, but you must be able to use it quickly and correctly.
Here is why it works. Suppose each wave takes 0.05 s to pass. How many of them fit into one second? The answer is 1 ÷ 0.05 = 20. So 20 waves pass each second, and the frequency is 20 Hz. Dividing 1 second by the time for one wave always gives the number of waves in a second.
Key terms:
- Time period (T): the spec's name for the period of a wave, measured in seconds.
- Millisecond (ms): one thousandth of a second. 1 ms = 0.001 s.
Using the equation both ways
The equation can be turned round to find the time period from the frequency. The two versions look almost the same: in each one, you divide 1 by the quantity you know.
| To find | Use | In words |
|---|---|---|
| Frequency | f = 1 ÷ T | 1 ÷ time period |
| Time period | T = 1 ÷ f | 1 ÷ frequency |
The time period must be in seconds. Short time periods are often given in milliseconds, so convert first.
| Given in | Change to | How | Example |
|---|---|---|---|
| milliseconds (ms) | s | divide by 1000 | 4 ms = 0.004 s |
| seconds (s) | ms | multiply by 1000 | 0.005 s = 5 ms |
| minutes | s | multiply by 60 | 2 minutes = 120 s |
Worked example 1: finding frequency
Question: A wave on a rope has a time period of 0.4 s. Calculate its frequency.
f = 1 ÷ T
f = 1 ÷ 0.4
f = 2.5 Hz
Worked example 2: milliseconds
Question: The sound from a tuning fork has a time period of 4 ms. Calculate the frequency of the sound.
Convert: 4 ms = 0.004 s
f = 1 ÷ T
f = 1 ÷ 0.004
f = 250 Hz
Worked example 3: finding time period
Question: A guitar string makes a sound of frequency 200 Hz. Calculate the time period of the sound. Give your answer in milliseconds.
T = 1 ÷ f
T = 1 ÷ 200
T = 0.005 s
T = 0.005 × 1000 = 5 ms
Big time period, small frequency
Because you divide 1 by the time period, a long time period gives a low frequency and a short time period gives a high frequency.
| Time period | Frequency (1 ÷ T) |
|---|---|
| 2 s | 0.5 Hz |
| 1 s | 1 Hz |
| 0.1 s | 10 Hz |
| 0.01 s (10 ms) | 100 Hz |
| 0.001 s (1 ms) | 1000 Hz |
If the time period doubles, the frequency halves. If the time period halves, the frequency doubles. Use this to check your answers: slow waves such as waves on the sea have time periods of several seconds and frequencies of less than 1 Hz. Sounds have much shorter time periods, usually measured in milliseconds, and much higher frequencies.
Finding a frequency by counting waves
Often you are not told the time period. Instead you are told how many waves pass a point in a given time. You can get both quantities from this.
⏰ Time period
time period = time taken ÷ number of waves
This shares the time out between the waves, to give seconds for each wave.
Worked example 4: counting waves
Question: A girl stands on a pier and counts 15 waves passing a post in 60 s. Calculate the frequency and the time period of the waves.
frequency = number of waves ÷ time taken = 15 ÷ 60 = 0.25 Hz
T = 1 ÷ f = 1 ÷ 0.25 = 4 s
Check: time period = time taken ÷ number of waves = 60 ÷ 15 = 4 s. Both methods agree.
Counting many waves is better than timing one. Timing a single wave with a stopwatch is badly affected by your reaction time. Timing 15 waves and dividing spreads that error over all 15.
Two-step problems
The spec asks you to use f = 1 / T together with the wave equation from The Wave Equation, v = f × λ. Frequency is in both equations, so it is the link between them. If a question gives you a time period and asks for a wavelength or a speed:
- Use f = 1 ÷ T to find the frequency.
- Put that frequency into v = f × λ.
Worked example 5: a sound wave
Question: A loudspeaker makes a sound with a time period of 2 ms. The speed of sound in air is 340 m/s. Calculate the wavelength of the sound.
Convert: 2 ms = 0.002 s
Step 1: f = 1 ÷ T = 1 ÷ 0.002 = 500 Hz
Step 2: λ = v ÷ f = 340 ÷ 500
λ = 0.68 m
Worked example 6: an electromagnetic wave
Question: A radio wave has a time period of 1 × 10−6 s. Radio waves travel at 3 × 108 m/s. Calculate the wavelength of the radio wave.
Step 1: f = 1 ÷ T = 1 ÷ (1 × 10−6) = 1 × 106 Hz
Step 2: λ = v ÷ f = (3 × 108) ÷ (1 × 106)
λ = 300 m
Worked example 7: working backwards
Question: Waves in a harbour travel at 2 m/s and have a wavelength of 16 m. Calculate the time period of the waves.
Step 1: f = v ÷ λ = 2 ÷ 16 = 0.125 Hz
Step 2: T = 1 ÷ f = 1 ÷ 0.125
T = 8 s
Common mistakes
- Leaving the time period in milliseconds. 1 ÷ 4 gives 0.25, but 4 ms is 0.004 s and the frequency is 250 Hz. Divide by 1000 first.
- Dividing the wrong way round when counting waves. Frequency is waves ÷ time. Time ÷ waves gives the time period.
- Putting the time period into the wave equation. v = f × λ needs the frequency. Find f first.
- Mixing up the units. Frequency is in Hz and time period is in s. Write the unit on every answer.
- Not checking the size of the answer. A short time period must give a high frequency. If both are small, something has gone wrong.
Exam-style question
(a) A student watches water waves pass a wooden post in a lake. 20 waves pass the post in 50 s. Calculate the frequency of the waves. [2 marks]
(b) Calculate the time period of the water waves. [2 marks]
(c) A flute makes a sound with a time period of 2.5 ms. The speed of sound in air is 340 m/s. Calculate the wavelength of the sound. [4 marks]
(d) Microwaves are electromagnetic waves. They travel at 3 × 108 m/s. A microwave has a time period of 4 × 10−10 s. Calculate the wavelength of the microwave. [3 marks]
Model answer
(a) frequency = number of waves ÷ time = 20 ÷ 50 (1). f = 0.4 Hz (1).
(b) T = 1 ÷ f = 1 ÷ 0.4 (1). T = 2.5 s (1). (50 ÷ 20 = 2.5 s also gains both marks.)
(c) 2.5 ms = 0.0025 s (1). f = 1 ÷ T = 1 ÷ 0.0025 = 400 Hz (1). λ = v ÷ f = 340 ÷ 400 (1). λ = 0.85 m (1).
(d) f = 1 ÷ T = 1 ÷ (4 × 10−10) = 2.5 × 109 Hz (1). λ = v ÷ f = (3 × 108) ÷ (2.5 × 109) (1). λ = 0.12 m (1).
Exam tip
Parts (c) and (d) never mention frequency, but you cannot get to the wavelength without it. When a wave question gives you a time period, write f = 1 ÷ T as your first line. In (c), change the 2.5 ms into seconds before you touch the calculator.