Physicopoeia

φυσικοποιΐα
THINGS I WISH I'D KNOWN WHEN I STARTED TEACHING PHYSICS

Are water waves Transverse or Longitudinal?

Instinctively, most pupils (and not a few teachers, textbooks, syllabuses and national curriculum documents) will be tempted to say Transverse. Obviously, right? Surely ripples on water couldn't look more like a classic transverse wave sketch if they tried. Unfortunately, this is a case of something that looks like a duck, walks like a duck and quacks like a duck turning out to be a camel in disguise. Well, sort of. The late great Feynman puts it very well, as usual:

'Now, the next waves of interest, that are easily seen by everyone and which are usually used as an example of waves in elementary courses, are water waves. As we shall soon see, they are the worst possible example, because they are in no respect like sound or light; they have all the complications waves can have.'

The Feynman Lectures on Physics, section 51-4

As Feynman suggests, the actual movement of water molecules in a water wave turns out to be extremely complex. On the surface of the water, the particles move in a nearly circular pattern which is therefore a mixture of transverse and longitudinal behaviour. There are many animations of this which make it much clearer, such as this one.

[Amusingly (or tragically, depending on your point of view), when I first found this, the first Up Next video suggested was this one. Aimed explicitly at GCSE pupils, it contains exactly the wrong animation at 2:01]

In fact, many people might already have experienced this first-hand when standing in deep water (up to about your chest) at the coast or perched on an inflatable something in the sea. As each wave approaches you, you feel pulled towards it before riding up and over the crest of the wave. Over time, if you don't do anything, you will find yourself being taken in the direction of the waves: this wouldn't happen if the waves were purely transverse.

A moment's thought will soon make it clear that pure transverse wave motion would be impossible in water: since water is, to all intents and purposes, incompressible, how could the water possibly move downwards to form a trough without also moving sideways as well? As you get deeper in the water, the up and down motion becomes increasingly suppressed so the molecule motion becomes more and more like pure longitudinal motion.

So, what do I teach my pupils? When I'm giving them a list of transverse waves for their notes, I miss out water waves entirely. In fact, since someone nearly always suggests it as an example, I usually get them to append 'NOT water waves!' to their list, just in case. There are so many other better examples of transverse waves (any electromagnetic wave, for example) that there shouldn't be any real need to worry too much about water waves in this context.

WHAT ABOUT THEIR SPEED?

(and why is refraction so difficult to show convincingly in a ripple tank?)

If you've ever tried to show refraction of water waves in an actual ripple tank, you'll probably be aware that a change in depth of water affects the waves' speed. You'll probably also have found it very difficult to demonstrate refraction convincingly because depth only actually makes a difference to speed when the water is extremely shallow.

 

The speed of shallow waves ('shallow', in this context, meaning waves in a depth of water d ≤ λ/4), is quite well approximated by:

[Water wave formula 1]

Yes, that's a hyperbolic tangent function. Since a typical ripple tank will produce waves of only a few mm wavelength, you would only be able to show convincing refraction effects with a water depth of less than a millimetre which is almost impossible to achieve.

Once the water gets deeper than about half a wavelength, the tanh function becomes very nearly equal to 1, so:

[Water wave formula 2]

Hyperphysics has a very helpful page with diagrams and graphs that goes into much more detail.