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4.1.4Waves

The Wave Equation — Edexcel GCSE Combined Science 1SC0 - Physics Revision Notes & Topical Questions

5 min read 9 parts Lesson 5 of 10

📚 Edexcel GCSE Combined Science 1SC0 - Physics Hub / Topical Revision
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Part 1 of 7The wave equation

The speed of a wave is the distance a crest travels each second. In one period T a crest travels one wavelength λ, so the speed is v = λ / T. Since f = 1 / T, this becomes the wave equation:

v = f λ

What the symbols mean: v = wave speed (m s−1) · f = frequency (Hz) · λ = wavelength (m)
The wave equation
v = wave speed in metres per second (m s−1)
f = frequency in hertz (Hz)
λ = wavelength in metres (m)

It applies to all wavessound, water, light and the rest of the electromagnetic spectrum.

Part 2 of 7Rearranging

f = v / λ and λ = v / f.

1
Use a triangle with v on top and f and λ at the bottom if it helps, but always write the working.
2
Make sure the units are consistent.
3
If the wavelength is in centimetres, convert it to metres first.
4
If the frequency is in kHz or MHz, convert it to Hz.

The speed of a wave is the distance divided by the time as for any speed, so v = d / t also applies to wave pulses, for example finding the speed of sound from an echo.

Part 3 of 7What decides the speed, frequency and wavelength

The frequency of a wave is set by the source and does not change when the wave moves into a new medium.

Medium
The speed is set by the medium.
Since v = f λ and f is fixed, the wavelength changes in proportion to the speed.
So when a wave slows down on entering a new medium, its wavelength is smaller.
This idea explains refraction (see Reflection and Refraction).

Typical speedssound in air about 340 m s−1, sound in water about 1500 m s−1, sound in steel about 5000 m s−1, and electromagnetic waves in a vacuum 3.0 × 108 m s−1.

Part 4 of 7Using the equation on graphs

For a fixed wave speed, a graph of wavelength against 1/frequency is a straight line through the origin with gradient v.

Using the equation on graphs
A graph of frequency against wavelength for a fixed speed is a curve that falls as wavelength increases (an inverse relationship): doubling the frequency halves the wavelength.
This pattern is why the electromagnetic spectrum can be drawn in a single line with long-wavelength waves having low frequency and short-wavelength waves high frequency.

Part 5 of 7Measuring wave speed in practice

In a ripple tank, you can count the waves passing a mark in a known time, or use a stroboscope to freeze the pattern, measure the wavelength over several waves for accuracy, and calculate v = f λ.

Measuring wave speed in practice
Measuring across 5 or 10 wavelengths and dividing reduces the uncertainty compared with measuring one.
The same relation is used in the experiment on the speed of sound (see Investigating the Speed of Sound).

Part 6 of 7Common exam problem types

Wave equation problems come in a few standard forms.

Common exam problem types
First, the direct calculation: given two of v, f and λ, find the third.
Second, the two-medium problem: the frequency stays the same and v and λ change together, so v1/λ1 = v2/λ2.
Third, the echo problem,
  • which combines v = d / t with the fact that the sound travels there and back,
  • so the distance to the reflector is half the total distance travelled.

An example of the third typea ship's sonar sends a pulse and receives the echo 0.60 s later.

Common exam problem types
The speed of sound in sea water is 1500 m s−1.
The total distance is 1500 × 0.60 = 900 m, so the sea bed is 450 m below the ship.
Many mistakes come from forgetting the factor of two.

Fourth, the dependence on a graph

  • if you plot v against f for a fixed wavelength, you get a straight line through the origin with gradient λ.
  • Check that you can read the gradient and give it a unit of metres.

Part 7 of 7Worked examples

  1. A ripple tank wave has frequency 8.0 Hz and wavelength 2.5 cm. Calculate its speed.

  2. A radio station broadcasts at 98 MHz. The speed of radio waves is 3.0 × 108 m s−1. Find the wavelength.

  3. A sound wave of frequency 680 Hz travels at 340 m s−1 in air and then enters water where its speed is 1500 m s−1. Find the wavelength in each.

Key terms

TermMeaning
Wave equationv = f λ; links speed, frequency, wavelength
Wave speedDistance travelled by the wave each second
HertzUnit of frequency
MediumMaterial that sets the wave speed
Inverse relationshipAs one quantity doubles, the other halves

Avoid these mistakes

1
Using the wavelength in cm without converting.
2
Believing the frequency changes when a wave changes medium – it stays the same.
3
Thinking a bigger amplitude means a faster wave.
4
Using the wave speed in air for light.
5
Rearranging wrongly to λ = f / v.

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