Understanding the Wave Equation

August 10, 2024

My previous article, titled Understanding the Heat Equation, focused on building intuition for this differential equation:^[1]

\frac{\partial F}{\partial t} = \frac{\partial ^{2} F}{\partial x ^{2}}

F_{t} = F_{xx}

This equally important differential equation is called the Wave Equation:

\frac{\partial ^{2} F}{\partial t ^{2}} = \frac{\partial ^{2} F}{\partial x ^{2}}

F_{tt} = F_{xx}

Don’t the two equations look similar? Well, as might be expected, the extra differentiation on the left-hand side makes the solutions come out very different, but even so, there must be some connection between these two equations, don’t you think? Perhaps the insights we already picked up on the Heat Equation might even help to make sense of the Wave Equation?

Well, first thing: It is not a coincidence that I am calling the two function parameters in either equation $x$ and $t$ . When either equation is used to describe real, physical phenomena, in both cases $x$ represents position in space and $t$ , position in time. So again, it seems that the meanings of the two equations must be closely linked.

Let’s review the meaning of $F_{xx}$ , which appears on the right-hand side of both Heat and Wave Equations. The second derivative measures curvature, and because it’s an $x$ derivative, it’s the curvature of a contour in space. More to the point, curvature tells us whether each point along the contour is higher or lower than its neighbours (on average), and by how much.

When the second derivative is positive, we have a concave curve, one which opens upwards. That means each point along the curve is (on average) lower than its neighbours. Conversely, when the second derivative is positive, we have a convex curve, which opens downwards, and each point along the curve is (on average) higher than its neighbours. The magnitude of the second derivative tells us how much each point is higher or lower than the average of its neighbours.

The Heat Equation relates that quantity to $F_{t}$ ; the rate of change in temperature (with the passing of time). And that makes sense! Because heat energy always conducts from hot to cold, when the temperature at a point is higher than its neighbours, heat will flow away from that point, and its temperature will go down. Conversely, when the temperature at a point is lower than its neighbours, heat will flow toward that point, and its temperature will go up.

Now we are ready to tackle the Wave Equation. It’s much like the Heat Equation, but instead of $F_{t}$ , the left-hand side is $F_{tt}$ . Instead of rate of change (or velocity), that is the acceleration of change.

In other words, THIS is what the Wave Equation is saying: We have some quantity which varies over space. Just like temperature, this quantity tends to go up where it’s lower than at neighbouring points, and go down where it’s higher than at neighbouring points. But unlike temperature, this quantity doesn’t just move towards the average value at neighbouring points, it accelerates towards the average value at neighbouring points.

(Read the preceding paragraph as many times as needed to fully grasp every word! It is the most important paragraph in this article!)

Some Applications of the Wave Equation

Now, this is the next big question: what kind of physical “quantity” actually behaves as described above? Here is an example:

Imagine taking a slice through a body of water. The contour of the water surface along that slice will form some curve. The water molecules at the surface are subject to surface tension, so where they are lower than their neighbours, they will be pulled upwards. And where they are higher than their neighbours, they will be pulled downwards.

But unlike heat, those water molecules have mass and are subject to inertia. Thus, when pulled upwards or downwards, they accelerate in that direction. And there you have it; our physical intuition for the movements of water does match up with the Wave Equation. (Or, instead of water, you could just as well think of a guitar string or the membrane of a drum.)

Caveats

In the previous article, I mentioned that the constant of proportionality in the Heat Equation was elided to simplify the discussion. The same is true of this article’s presentation of the Wave Equation. When describing a real, physical system, a constant of proportionality is needed.

Like the Heat Equation, the Wave Equation also has versions with one, two, three, or more spacial dimensions. This article presented the one-dimensional version.

The next article in this series will connect the Heat and Wave Equations to a third differential equation, equally fundamental and worthy of attention: Laplace’s Equation.

[1] Don’t be distracted because the function is called $F$ here, rather than $T$ (as in the previous article). I did that to make the comparison between the Heat Equation and Wave Equation clearer; anyways, the name is arbitrary. ⏎