The wave equation
Definition
Wavelength
The distance from one point on a wave to the equivalent point on the next wave.
Definition
Frequency
The number of complete waves passing a point each second.
- Every electromagnetic wave obeys the wave equation, v=fλv=f\lambdav=fλ.
- In words, wave speed=frequency×wavelength\text{wave speed}=\text{frequency}\times\text{wavelength}wave speed=frequency×wavelength.
- vvv is the wave speed in metres per second (m/s\text{m/s}m/s), fff is the frequency in hertz (Hz\text{Hz}Hz) and λ\lambdaλ is the wavelength in metres (m\text{m}m).
- A frequency of 1 Hz1\ \text{Hz}1 Hz means that one complete wave passes a point every second.
- For any electromagnetic wave travelling in a vacuum or in air, vvv is 3.0×108 m/s3.0\times10^{8}\ \text{m/s}3.0×108 m/s.
- You need to recall this equation and rearrange it as f=vλf=\dfrac{v}{\lambda}f=λv or λ=vf\lambda=\dfrac{v}{f}λ=fv.
Frequency and wavelength
Definition
Inverse proportion
A relationship between two quantities in which multiplying one by a factor divides the other by the same factor.
- Because vvv is the same for every electromagnetic wave, the product fλf\lambdafλ always equals 3.0×108 m/s3.0\times10^{8}\ \text{m/s}3.0×108 m/s.
- This makes the frequency and wavelength of electromagnetic waves inversely proportional to each other.
- A wave with a higher frequency must therefore have a shorter wavelength.
- A wave with a lower frequency must have a longer wavelength.
- If the frequency doubles, the wavelength halves, and if the frequency is ten times larger, the wavelength is ten times smaller.
- This is why radio waves, which have the lowest frequencies, have the longest wavelengths, while gamma rays, which have the highest frequencies, have the shortest wavelengths.
- The relationship only holds when the waves are compared in the same medium, because both waves must have the same speed.
Large and small numbers
- Electromagnetic frequencies and wavelengths cover an enormous range, so they are written in standard form or with unit prefixes.
- The frequency prefixes you need are 1 kHz=1×103 Hz1\ \text{kHz}=1\times10^{3}\ \text{Hz}1 kHz=1×103 Hz, 1 MHz=1×106 Hz1\ \text{MHz}=1\times10^{6}\ \text{Hz}1 MHz=1×106 Hz and 1 GHz=1×109 Hz1\ \text{GHz}=1\times10^{9}\ \text{Hz}1 GHz=1×109 Hz.
- The wavelength prefixes you need are 1 cm=1×10−2 m1\ \text{cm}=1\times10^{-2}\ \text{m}1 cm=1×10−2 m, 1 mm=1×10−3 m1\ \text{mm}=1\times10^{-3}\ \text{m}1 mm=1×10−3 m, 1 μm=1×10−6 m1\ \mu\text{m}=1\times10^{-6}\ \text{m}1 μm=1×10−6 m and 1 nm=1×10−9 m1\ \text{nm}=1\times10^{-9}\ \text{m}1 nm=1×10−9 m.
- Convert every value into hertz and metres before you substitute it into v=fλv=f\lambdav=fλ.
- A quick check is that your frequency multiplied by your wavelength should come back to 3×1083\times10^{8}3×108.
Example
Wavelength of a radio station
- A radio station broadcasts at 100 MHz100\ \text{MHz}100 MHz.
- Convert the frequency into hertz: 100 MHz=1.0×108 Hz100\ \text{MHz}=1.0\times10^{8}\ \text{Hz}100 MHz=1.0×108 Hz.
- Rearrange the wave equation and substitute:
Example
Frequency of green light
- Green light has a wavelength of 500 nm500\ \text{nm}500 nm.
- Convert the wavelength into metres: 500 nm=5.0×10−7 m500\ \text{nm}=5.0\times10^{-7}\ \text{m}500 nm=5.0×10−7 m.
- Rearrange the wave equation and substitute:
- This frequency is 666 million times higher than the radio station's, and the wavelength is 666 million times shorter.
Example
Comparing two waves by ratio
- An X-ray has a frequency 100010001000 times higher than an ultraviolet wave, and the ultraviolet wavelength is 1.0×10−8 m1.0\times10^{-8}\ \text{m}1.0×10−8 m.
- Both waves have the same speed, so the X-ray wavelength must be 100010001000 times shorter:
Practical
Measuring the wavelength of microwaves
- Aim: to measure the wavelength of the microwaves inside a microwave oven, and then use the oven's frequency to estimate the speed of electromagnetic waves.
- Apparatus: a microwave oven with its frequency printed on the label (usually 2.45 GHz2.45\ \text{GHz}2.45 GHz), a microwave-safe plate, a large flat bar of chocolate or a thin even layer of grated cheese, a ruler with a millimetre scale and oven gloves.
- Method:
- Remove the turntable so that the plate stays perfectly still while the oven runs.
- Spread the chocolate or cheese in a thin, even layer that covers most of the plate.
- Put the plate in the oven and heat it on a low power setting for about 151515 to 202020 seconds.
- Stop as soon as a few small, glossy melted patches appear, because if heating continues the patches spread and merge.
- Lift the plate out with oven gloves and look for a row of melted spots with firm, unmelted food between them.
- Measure from the centre of one melted spot to the centre of a spot several spots further along the row, then divide by the number of gaps to find the mean gap, ddd.
- Repeat with a fresh layer of food along a different direction across the plate, and take the mean of your results.
- How it works: the microwaves reflect off the metal walls of the oven and form a fixed pattern of hot spots, and neighbouring hot spots are half a wavelength apart.
- Results: in a typical oven the melted spots are about 6 cm6\ \text{cm}6 cm apart, which gives a wavelength of about 12 cm12\ \text{cm}12 cm.
- Maths: double the mean gap to find the wavelength, λ=2d\lambda=2dλ=2d, then multiply by the oven frequency to find the speed, so a gap of 6.1 cm6.1\ \text{cm}6.1 cm gives λ=0.122 m\lambda=0.122\ \text{m}λ=0.122 m and v=2.45×109×0.122≈3.0×108 m/sv=2.45\times10^{9}\times0.122\approx3.0\times10^{8}\ \text{m/s}v=2.45×109×0.122≈3.0×108 m/s.
- Watch out: measuring across several gaps and dividing reduces the effect of the uncertainty in judging the centre of each spot, and a rotating plate smears the pattern so that no clear spots form.
- Safety: never run the oven empty or with anything metal inside, let melted chocolate or cheese cool before touching it because it can burn, and never eat food used in the laboratory.
Common Mistake
- Do not take the gap between two neighbouring melted spots as the wavelength, because it is only half a wavelength.
- Do not substitute values in MHz\text{MHz}MHz, GHz\text{GHz}GHz or nm\text{nm}nm, because the answer will be wrong by a power of ten unless you convert to hertz and metres first.
- Do not say that frequency and wavelength increase together, because a higher frequency always means a shorter wavelength for electromagnetic waves.
Exam technique
Wave equation questions
- Write v=fλv=f\lambdav=fλ before substituting, then rearrange it, so your method is clear even if the arithmetic slips.
- Give very large or very small answers in standard form, and always include the unit.
- In a "compare" question, link the two quantities with a reason, for example: the frequency is higher, so the wavelength is shorter, because all electromagnetic waves travel at the same speed.
- Check the size of your answer against where the wave sits in the spectrum, since a visible-light wavelength should be a few hundred nanometres.
Self review
- Write down the equation that links wave speed, frequency and wavelength.
- If the frequency of an electromagnetic wave is tripled, what happens to its wavelength?
- Calculate the wavelength of a radio wave with a frequency of 200 kHz200\ \text{kHz}200 kHz.
- Convert 650 nm650\ \text{nm}650 nm into metres in standard form.
- Why is the gap between melted spots in the microwave oven doubled to find the wavelength?
