4.7 A partial solution
Derivation and validation of the wave equation: a second order partial differential equation which must be satisfied by any 1-dimensional wave.
Now that we've got our hands on this new tool - partial differentiation - we're in a position to continue our search for a general expression for a wave, without being intimidated by the problem of dependence on both distance and time.
So, that gets us an interesting expression:
This seems like a potentially important result: a general wave equation, which says that all that is required to be a wave function is compatibility with this simple relationship between the partial derivatives. There is a complication, however: by starting from the equation y = f(x – vt), we’ve assumed the wave is moving in a specific direction (to the right). We need to check what happens if it is moving instead to the left:
Question 4-8 reveals an uncomfortable issue: the partial differential equation obtained to describe a wave moving to the left is similar but not the same as that for a curve moving right. We can combine the relationships as:
where a wave has to satisfy one or other of these not quite identical equations. This is, at very least, a bit messy. More disturbingly, we have already introduced the idea that waves can be superposed to make a different wave – but what if one of the component waves satisfies one of these differential equations and the other satisfies the other one? Let’s investigate with an example:
The superposition of two waves is itself always a wave (we’ll verify this in general terms a little bit later), so this is a problem because we now have an example of a wave that does not comply with our initial attempt at a general wave equation. Now let’s see if we can avoid this problem by going a step further and differentiating again.
This exercise establishes that the relationship between the second partial derivatives with respect to time and displacement is the same, regardless of which direction the wave is travelling in. We have, therefore, an equation that will apply to any 1-dimensional wave. This equation – because of its importance – is usually just called the wave equation:
You can build your confidence in this new equation by verifying that it holds for the problematic superposition of two waves travelling in opposite directions, which we introduced in Question 4-9:
What we have arrived at is a second order differential equation whose solutions are the equations that will describe a one-dimensional travelling wave. This is one of those premier league equations that have huge importance in physics. It is used in modelling and predicting the properties of a great variety of waves: water waves, seismic waves, sound waves, electromagnetic waves ..... For us, it will be a key component in constructing an equation that embodies wave-particle duality.
We got here in a slightly strange way by assuming what we thought a solution should look like and then working backwards to find the differential equation whose solution would have such a form. Interestingly, it is possible to derive the exact same equation in a more conventional (but less straightforward) way in specific physical contexts: for a simple mechanical wave it can be constructed starting from Hooke’s law, while for an electromagnetic wave it can be derived from Maxwell’s equations.
We can verify that our general equation for a sinusoidal wave is, indeed, a solution to the wave equation:
Taking this general result, because A, v and λ can have any values, and we can have different phases and directions of propagation, we already have an infinity of possibilities when it comes to finding solutions to the wave equation. Moreover, we have already seen an example (Question 4-11) of a superposition of sinusoids which is a solution. The specific solution that applies in a particular case may be dictated by lots of different factors – depending on what type of wave it is - including the frequency and magnitude of the disturbance that creates the wave and the medium through which it propagates. To find the form of a particular wave, we need to take account of additional restrictions – boundary conditions – which are imposed by the physical reality of the wave. This is exactly how we will proceed when we apply this equation to matter waves in chapter 6.
