Saturday, 9 January 2016

Frequency, wavelength, amplitude, period...

Above: The anatomy of a simple wave - I'm going to keep putting this up, its a good memory jerker for all kind of waves!

Waves are nature's way of moving energy around (see the last post) - in the form of moving oscillations - without having to move matter. Nature's way of moving energy around is something scientists are naturally interested in, so they've come up with ways of measure them. Those measurements can be useful, which is why we get students to learn them. No, really, science teachers don't want to make you learn useless things - measuring waves helps with everything from sea defences to radio. Because all scientists are evil…. Ahem, I mean; 'cause it’s useful we’ll call each of these things we measure by a letter, and arrange them into equations. The first wave measurement is... 


 Above: A better understanding of wavespeed, and how the wave's forwards speed adds to the boat's speed and turns it into a giant ramp, would have saved this man a lot of bruises and an expensive boat repair bill. 
Unless he's crazy, and just doesn't care. Which seems likely. Video courtesy of Pete Koff.

This is an easy one – in this post we described a wave as an up-down disturbance in the water (or, for other kinds of waves, an oscillation in whatever the wave medium is),  that moves through the water without the water travelling anywhere? Well how fast the up/down disturbance –the wave – moves through the water is the wave’s speed, and it just gets represented by the letter ‘v’. The formal definition is: 

'How far the wave travels in 1 second'. 

See, that wasn't so painful (unless you're the guy in that boat). Next measurement for waves is……


This is literally ‘how long one wave is’ and gets the letter lambda (λ). The easiest way to measure one wavelength is the distance between a wave peak and the next one along. Sometimes two peaks aren't visible next to each other, so it’s worth pointing out that really the wavelength is the distance between two identical points on the wave. The formal definition (and you might get asked for it in a test, so I’ll put it in bold: 

'The wavelength  is the distance between two identical points in the wave cycle, such as the distance between two adjacent crests'.

By identical we mean  ‘at the same height, and moving in the same direction up or down, with the same speed and the same acceleration’, not just identical in appearance, or height. On a still picture of a wave this can be hard to see. One way to check is that the end point should be on a slope with the same gradient as the start point. Another way to check is that the whole pattern of the wave has been passed along. So, for a simple wave, you trace along the wave, past one peak and one trough (in any order). The next point after  that, which is on the same height as your starting point and on a slope going the same direction, is one wavelength away from your starting point. Like this:

It's important to note that this rule only applies to a simple wave, like the one shown above. For a more complex wave the rule is that the whole pattern of the wave (which might include multiple peaks and troughs) has to repeat before you get to the point where one wavelength has been passed. More complex waves shouldn't come up too much until you're at a level where you already know all of these things, but it's important to remember.

Frequency and period:

The idea of frequency is closely related to the idea of wavelength: Frequency is basically a measure of how quickly one wavelength of a wave passes a given point.
Imagine you’re on a beach (or, if you’re near the coast, actually go to one), and standing knee deep in the water, facing the direction that the waves are coming from (If you live near the beach but the water’s cold you can still just imagine this by the way,  getting pneumonia doesn't aid learning much). However many  complete waves – so one peak to the next, one trough to the next, or whatever point on the wave you like- pass by you in one second is the frequency. It gets represented by the letter f (in an unusually commonsense move by science), and it’s formal definition (the one you might be asked to write in a test) is: 

'The frequency f is the number of complete waves passing a point on the x-axis in a given time period'.

Frequency is measured in hertz, so one wave each second equal 1 Hertz, two waves per second equals two Hertz, and so on. 

A very closely related measure is the period of the wave - this is the amount of time it takes for one complete wave to pass you (or any fixed point). the formal definition is : 

'The time for a particle on a medium to make one complete vibrational cycle'.

Above: My first ever gif!

If you think about those two definitions, something that will probably seem too obvious to mention will pop out of you: The period of the wave is the same as the period of whatever is creating it. The same goes for the frequency. So, for example, a tuning fork with a frequency of 200 Hertz will produce soundwaves with a frequency of 200 hertz. Later we'll see how things can happen to waves that will change their frequency and period, but they all start off inheriting those things from whatever created them.

Amplitude is 'the maximum displacement of the wave from the x - axis' (formal definition). What that means is the size of the oscillation, measured from the zero point - so the distance from zero to the top of a peak, or zero to the bottom of a trough, not bottom of peak to top of trough - like this:

Above: Amplitude is measured from the  axis up or down, never from the bottom of the wave to the top.
Amplitude determines how much energy a wave of a given frequency carries - if you think it over that's kind of obvious in fact: A big, tall wave, like the one that the boat ran over in the video at the top, obviously has more energy in it than a lower wave with the same frequency, because it takes more energy to lift all that water higher.

Two equations for waves:
Imagine waves made by one duckling hitting another duckling: 

OK this video isn't very informative, since the waves the ducklings make aren't very clear to see, but it's a good excuse to include a video with fluffy ducklings. Video courtesy of Horses and More, who has more fluffy duckling and animal videos on their channel.

If the first duckling starts making waves with the same wave speed, but which are twice as close together as when it started – so their wavelengths are halved  – then the second duckling will be passed by twice as many waves per second. If the wavelength stays the same but they’re moving twice as fast, then the second duckling will again get hit by twice as many waves per second. In both cases the frequency has gone up. This tells us that the frequency gets bigger with wavespeed increasing, and smaller with wavelength increasing, and from that we can get our first useful equation –please don’t run away – which goes:

“Frequency equals wavespeed divided by wavelength” 

So if you were asked " what's the frequency of a wave with speed of 700 meters / second and a wavelength of 2 meters you could do :
700 divided by 2 equals 350 
700 /  2 = 350

and we know the unit is hertz, so the answer is 350 Hertz.

One more equation ( and then we can go back to Netflix)

Like we said above, the amount of time a wave takes to pass you - so, to go from one peak to the next, or  one trough to the next - is called the period. The period is easiest found by dividing one by the frequency, so the equation is:

'Period equals one divided by the frequency'

So, if we have a wave that has length of three meters, and a speed of 100 meters per second we can find the frequency....

F = 100/3 = 30 Hertz

then we can find the period....

P = 1/30 = 0.0333333... seconds.

Now, the truth is that none of the above (well, not much) will stick with you unless you practise with it -  so here're some practice questions. Yes, I'm evil. No I never promised I wouldn't be, and practice really does work. To get the answers follow this link.

Practice questions:

Light (and other electromagnetic radiation) is a wave travelling through electric and magnetic fields rather than water, and it travels at 300,000,000 meters per second. What are the frequencies of light with wavelengths of :

450 nanometers*

570 nanometers

900 nanometers

* 1 nano meter = 0.000000001 meters

What are the periods of these waves?

Waves travelling through another planet's ocean have a period of 2 seconds and a wavelength of 20 meters. How fast are these waves travelling?

A radio wave ( which is an electromagnetic wave) has a wavelength of ten meters. What is it's frequency?


The equation relating wave speed, frequency, and wavelength is:

   a) What does each symbol mean?

   b) Can you make wavelength the subject of the equation?

That's all guys!

For more practice questions you can go to here or here. 

For past exam papers head to the SQA website, here.

No comments:

Post a Comment