Using the wave equation

Definition

Wave speed

The distance travelled by a wave each second.

Every wave speed calculation uses the wave equation, $f\lambda$ , which you must be able to recall.
Before you start, pick out the two quantities you have been given and the one you need to find.
To find the speed, multiply the frequency by the wavelength: $f\lambda$ .
To find the frequency, divide the speed by the wavelength: $\dfrac{v}{\lambda}$ .
To find the wavelength, divide the speed by the frequency: $λ=vf\lambda = \dfrac{v}{f}$ .
You can check a rearrangement with simple numbers, such as $v = 10$ , $f = 2$ and $λ=5\lambda = 5$ , which fit all three forms.

Units and prefixes

The equation only gives the right answer when every quantity is in its standard unit: $m/s\text{m/s}$ for speed, $Hz\text{Hz}$ for frequency and $m\text{m}$ for wavelength.
Convert centimetres to metres by dividing by $100$ , so $m45\ \text{cm} = 0.45\ \text{m}$ .
Convert millimetres to metres by dividing by $1000$ , so $m8\ \text{mm} = 0.008\ \text{m}$ .
Convert kilohertz to hertz by multiplying by $1000$ , so $Hz2.5\ \text{kHz} = 2500\ \text{Hz}$ .
Convert megahertz to hertz by multiplying by $0001\,000\,000$ , so $Hz3.0\ \text{MHz} = 3.0 \times 10^{6}\ \text{Hz}$ .
Very large or very small answers are clearest in standard form, so $m0.0015\ \text{m}$ is written as $1.5 \times 10^{-3}\ \text{m}$ .
Give the final answer to a sensible number of significant figures, usually the same as the least precise value in the question.

Multi-step problems

Harder questions do not give you $f$ and $λ\lambda$ directly, so you must work them out from other information first.
If the period is given, find the frequency first using $\dfrac{1}{T}$ .
If a distance covering several waves is given, divide it by the number of waves to get one wavelength.
If you are asked how long a wave takes to travel a distance, find its speed and then use $time=distancespeed\text{time} = \dfrac{\text{distance}}{\text{speed}}$ .
Writing down each intermediate value with its unit stops you losing track, and lets the examiner award marks for each correct step.

Example

Speed of a sound wave

A sound wave has a frequency of $kHz2.0\ \text{kHz}$ and a wavelength of $cm17\ \text{cm}$ .
Both values are converted to standard units first, giving $2000\ \text{Hz}$ and $m\lambda = 0.17\ \text{m}$ .

f\lambda

\times 0.17

340\ \text{m/s}

Example

Wavelength from a period

Waves on a lake have a period of $s0.50\ \text{s}$ and travel at $m/s1.2\ \text{m/s}$ .
The frequency is found from the period first.

\frac{1}{T} = \frac{1}{0.50} = 2.0\ \text{Hz}

The wave equation is then rearranged to make wavelength the subject.

\lambda = \frac{v}{f} = \frac{1.2}{2.0} = 0.60\ \text{m}

Example

Frequency and travel time on a spring

A wave on a long spring travels at $m/s4.5\ \text{m/s}$ , and six complete waves fit along $m3.0\ \text{m}$ of the spring.
One wavelength is $m3.0\ \text{m} \div 6 = 0.50\ \text{m}$ .

\frac{v}{\lambda} = \frac{4.5}{0.50} = 9.0\ \text{Hz}

The time for a wave to travel the full $m3.0\ \text{m}$ of spring comes from the distance and the speed.

\frac{d}{v} = \frac{3.0}{4.5} = 0.67\ \text{s}

Common Mistake

Do not substitute a wavelength in centimetres or a frequency in kilohertz without converting it, because the answer will be wrong by a factor of $100$ or $1000$ .
Do not multiply when you should divide, and check that a higher frequency gives a shorter wavelength in your answer.
Do not round intermediate values too early, because rounding errors build up in multi-step questions.

Exam technique

Setting out a calculation

Write the equation first, either as $f\lambda$ or already rearranged, because this alone often earns a mark.
Show the substitution with the numbers in standard units, so that a slip on the calculator still leaves you credit for your method.
Write the final answer on the answer line with its unit, such as $m/s\text{m/s}$ , $Hz\text{Hz}$ or $m\text{m}$ .
Higher tier questions expect you to rearrange the equation yourself, so practise all three forms until they are automatic.

Self review

What are the three rearranged forms of the wave equation?
What is $mm750\ \text{mm}$ in metres?
What is the frequency of a wave with a period of $s0.20\ \text{s}$ ?
What is the wavelength of a $Hz500\ \text{Hz}$ sound wave travelling at $m/s340\ \text{m/s}$ ?
Why should you write down the equation before substituting any numbers?

Using the wave equation

Definition

Wave speed

The distance travelled by a wave each second.

Every wave speed calculation uses the wave equation, $f\lambda$ , which you must be able to recall.
Before you start, pick out the two quantities you have been given and the one you need to find.
To find the speed, multiply the frequency by the wavelength: $f\lambda$ .
To find the frequency, divide the speed by the wavelength: $\dfrac{v}{\lambda}$ .
To find the wavelength, divide the speed by the frequency: $λ=vf\lambda = \dfrac{v}{f}$ .
You can check a rearrangement with simple numbers, such as $v = 10$ , $f = 2$ and $λ=5\lambda = 5$ , which fit all three forms.

Units and prefixes

The equation only gives the right answer when every quantity is in its standard unit: $m/s\text{m/s}$ for speed, $Hz\text{Hz}$ for frequency and $m\text{m}$ for wavelength.
Convert centimetres to metres by dividing by $100$ , so $m45\ \text{cm} = 0.45\ \text{m}$ .
Convert millimetres to metres by dividing by $1000$ , so $m8\ \text{mm} = 0.008\ \text{m}$ .
Convert kilohertz to hertz by multiplying by $1000$ , so $Hz2.5\ \text{kHz} = 2500\ \text{Hz}$ .
Convert megahertz to hertz by multiplying by $0001\,000\,000$ , so $Hz3.0\ \text{MHz} = 3.0 \times 10^{6}\ \text{Hz}$ .
Very large or very small answers are clearest in standard form, so $m0.0015\ \text{m}$ is written as $1.5 \times 10^{-3}\ \text{m}$ .
Give the final answer to a sensible number of significant figures, usually the same as the least precise value in the question.

Multi-step problems

Harder questions do not give you $f$ and $λ\lambda$ directly, so you must work them out from other information first.
If the period is given, find the frequency first using $\dfrac{1}{T}$ .
If a distance covering several waves is given, divide it by the number of waves to get one wavelength.
If you are asked how long a wave takes to travel a distance, find its speed and then use $time=distancespeed\text{time} = \dfrac{\text{distance}}{\text{speed}}$ .
Writing down each intermediate value with its unit stops you losing track, and lets the examiner award marks for each correct step.

Example

Speed of a sound wave

A sound wave has a frequency of $kHz2.0\ \text{kHz}$ and a wavelength of $cm17\ \text{cm}$ .
Both values are converted to standard units first, giving $2000\ \text{Hz}$ and $m\lambda = 0.17\ \text{m}$ .

f\lambda

\times 0.17

340\ \text{m/s}

Example

Wavelength from a period

Waves on a lake have a period of $s0.50\ \text{s}$ and travel at $m/s1.2\ \text{m/s}$ .
The frequency is found from the period first.

\frac{1}{T} = \frac{1}{0.50} = 2.0\ \text{Hz}

The wave equation is then rearranged to make wavelength the subject.

\lambda = \frac{v}{f} = \frac{1.2}{2.0} = 0.60\ \text{m}

Example

Frequency and travel time on a spring

A wave on a long spring travels at $m/s4.5\ \text{m/s}$ , and six complete waves fit along $m3.0\ \text{m}$ of the spring.
One wavelength is $m3.0\ \text{m} \div 6 = 0.50\ \text{m}$ .

\frac{v}{\lambda} = \frac{4.5}{0.50} = 9.0\ \text{Hz}

The time for a wave to travel the full $m3.0\ \text{m}$ of spring comes from the distance and the speed.

\frac{d}{v} = \frac{3.0}{4.5} = 0.67\ \text{s}

Common Mistake

Do not substitute a wavelength in centimetres or a frequency in kilohertz without converting it, because the answer will be wrong by a factor of $100$ or $1000$ .
Do not multiply when you should divide, and check that a higher frequency gives a shorter wavelength in your answer.
Do not round intermediate values too early, because rounding errors build up in multi-step questions.

Exam technique

Setting out a calculation

Write the equation first, either as $f\lambda$ or already rearranged, because this alone often earns a mark.
Show the substitution with the numbers in standard units, so that a slip on the calculator still leaves you credit for your method.
Write the final answer on the answer line with its unit, such as $m/s\text{m/s}$ , $Hz\text{Hz}$ or $m\text{m}$ .
Higher tier questions expect you to rearrange the equation yourself, so practise all three forms until they are automatic.

Self review

What are the three rearranged forms of the wave equation?
What is $mm750\ \text{mm}$ in metres?
What is the frequency of a wave with a period of $s0.20\ \text{s}$ ?
What is the wavelength of a $Hz500\ \text{Hz}$ sound wave travelling at $m/s340\ \text{m/s}$ ?
Why should you write down the equation before substituting any numbers?

5.1.4 Wave speed problems

Using the wave equation

Units and prefixes

Multi-step problems

Speed of a sound wave

Wavelength from a period

Frequency and travel time on a spring

Setting out a calculation

Review 5.1.4 Wave speed problems by teaching Genie

5.1.4 Wave speed problems

Using the wave equation

Units and prefixes

Multi-step problems

Speed of a sound wave

Wavelength from a period

Frequency and travel time on a spring

Setting out a calculation

Review 5.1.4 Wave speed problems by teaching Genie

5.1.4 Wave speed problems

Using the wave equation

Units and prefixes

Multi-step problems

Speed of a sound wave

Wavelength from a period

Frequency and travel time on a spring

Setting out a calculation

Review 5.1.4 Wave speed problems by teaching Genie

5.1.4 Wave speed problems Revision Guide

5.1.4 Wave speed problems

Using the wave equation

Units and prefixes

Multi-step problems

Speed of a sound wave

Wavelength from a period

Frequency and travel time on a spring

Setting out a calculation

Review 5.1.4 Wave speed problems by teaching Genie

5.1.4 Wave speed problems Revision Guide