An Equation for all Waves

Solving the Equation

Light, sound and water all travel as waves. The equation that describes this wave motion is the same for all three types of wave. This is it:
The frequency, f, of a wave is the number of times a wave's crests pass a point in a second. If you watch a water wave in the bath pass over one of your toes twice every second the frequency of the wave is 2 Hz. The unit "Hz" is short for hertz, named after the German physicist Heinrich Hertz (1857 - 94). A more mathematically useful way to write 2 Hz is 2s -1 . That is, "2 per second". The speed of light has been given the letter c. In fact, as with all mathematical letters it is just a label and any letter would do providing we state what we mean when we use it. The speed of light is close to 300,000 km/s (186,300 miles per second). Another way of writing this is:
This means a three with eight zeros behind it, i.e. 300,000,000 metres per second. The Greek letter lambda, which looks like an upside-down letter y, is used to denote the wavelength of the wave. Again this is just a label, or shorthand, in order to allow us to work quickly with the equation. Wavelengths are usually measured in metres. Providing we know any two of the three quantities we can find the other one, either directly or by rearranging the equation. Let's try solving the equation as it is.
In this example we will consider the frequency of radio waves. Radio waves are just another form of "light", i.e. part of the electromagnetic spectrum, and so travel at the speed of light. Let's say we have a radio with a dial that is only marked in MHz. This is a measurement of frequency and we note that 1 MHz is the same as 1 million hertz (the M in MHz stands for "mega", which means million). We are told of a radio broadcast we want to hear but we are only given the wavelength of the station and not the frequency. The wavelength we are given is 3.26 metres. We know the speed of light and we know the wavelength so it's now an easy matter to plug these numbers into the equation and find the frequency of the radio station:
This gives us a frequency of 92 MHz, which is found in the FM range of most domestic radios.

Visible Light

The wavelengths of visible light are measured in nanometres, nm (billionths of a metre) but the equation works just the same. For example, red light has a wavelength of around 620 - 740 nm and blue light has a wavelength of around 445 - 500 nm. When we look at a light source the colours we see are dictated by the frequency of the light. These frequencies are very high by everyday standards. Have a look around the room and find something that's the colour red. How many times are the tiny crests of the light waves coming from that red object passing through the front of your eyes every second? In other words, what is the frequency of red light? Well, we know the speed of light and can take an average figure for the wavelength of red light. Let's say it's 670 nm, that is:
We now have everything we need to work out the frequency:
So when we look at something that is medium red about 448,000,000,000,000 tiny little wave crests pass through our eyes every second! This is certainly a very large number but still measurable using modern equipment. Indeed, the screen you are using had to have this number taken into account when it was being designed. If it hadn't you wouldn't be able to see this:

An Equation for all Waves

Light, sound and water all travel as waves. The equation that describes this wave motion is the same for all three types of wave. This is it:
The frequency, f, of a wave is the number of times a wave's crests pass a point in a second. If you watch a water wave in the bath pass over one of your toes twice every second the frequency of the wave is 2 Hz. The unit "Hz" is short for hertz, named after the German physicist Heinrich Hertz (1857 - 94). A more mathematically useful way to write 2 Hz is 2s -1 . That is, "2 per second". The speed of light has been given the letter c. In fact, as with all mathematical letters it is just a label and any letter would do providing we state what we mean when we use it. The speed of light is close to 300,000 km/s (186,300 miles per second). Another way of writing this is:
This means a three with eight zeros behind it, i.e. 300,000,000 metres per second. The Greek letter lambda, which looks like an upside-down letter y, is used to denote the wavelength of the wave. Again this is just a label, or shorthand, in order to allow us to work quickly with the equation. Wavelengths are usually measured in metres. Providing we know any two of the three quantities we can find the other one, either directly or by rearranging the equation. Let's try solving the equation as it is.

Solving the Equation

In this example we will consider the frequency of radio waves. Radio waves are just another form of "light", i.e. part of the electromagnetic spectrum, and so travel at the speed of light. Let's say we have a radio with a dial that is only marked in MHz. This is a measurement of frequency and we note that 1 MHz is the same as 1 million hertz (the M in MHz stands for "mega", which means million). We are told of a radio broadcast we want to hear but we are only given the wavelength of the station and not the frequency. The wavelength we are given is 3.26 metres. We know the speed of light and we know the wavelength so it's now an easy matter to plug these numbers into the equation and find the frequency of the radio station:
This gives us a frequency of 92 MHz, which is found in the FM range of most domestic radios.

Visible Light

The wavelengths of visible light are measured in nanometres, nm (billionths of a metre) but the equation works just the same. For example, red light has a wavelength of around 620 - 740 nm and blue light has a wavelength of around 445 -500 nm. When we look at a light source the colours we see are dictated by the frequency of the light. These frequencies are very high by everyday standards. Have a look around the room and find something that's the colour red. How many times are the tiny crests of the light waves coming from that red object passing through the front of your eyes every second? In other words, what is the frequency of red light? Well, we know the speed of light and can take an average figure for the wavelength of red light. Let's say it's 670 nm, that is:
We now have everything we need to work out the frequency:
So when we look at something that is medium red about 448,000,000,000,000 tiny little wave crests pass through our eyes every second! This is certainly a very large number but still measurable using modern equipment. Indeed, the screen you are using had to have this number taken into account when it was being designed. If it hadn't you wouldn't be able to see this: