The Wave Equation

Welcome to the seconds part of WatCompPhys' Fundamental Equations series! Here, we will introduce you to, and give you hands on experience in dealing with some of the most fundamental equations. This chapter is all about modelling the movement of waves. 

Waves are the most fundamental way of transporting energy, and thats why they are all around us. Think about the waves on the ocean, or the sound waves in the air, or the waves on a vibrating violin string. These sound like distinctly different events, right? But you'd be surprised to know that they all follow the same fundamental set of rules. 

Lets begin by observing a vibrating string, where its ensds are fixed. This could be the motion of a guitar string after the guitarist holds down the string at some point along the fretboard and plucks the string. Because one end of the string is held down by the finger and the other by the bridge at the end of the guitar, the ends of this string are fixed. 

Before we dive into the finer details, lets first point out the obvious features of this type of motion. Notice that the wave is fastest when it forms a straight line as it passes through the part where y=0. Another interesting feature is that the string always slows to a stop at around the same height, and begins to fly back towards the centre after it reaches that height. 

From these observations, we can conclude that there must be something constantly pulling it back to the centre at all times. Furthermore, since the string is straight at the centre, we can extend our idea to say that there is something always pulling the string so that it returns to being straight. We can verify that the string wants to be straight by observing the strings around you. 

Notice how the strings of the violin are all perfectly straight when they are at rest. When you pluck the strings, you displace it from this perfectly straight configuration. Since the string "wants" to be straight, it will pull itself back towards its original position, and the observed motion occurs (this wave motion disturbs the air in the room and the air also undergoes this wave movement to form the sound that you hear! but thats for later). Now the key is to see that the string dosn't just stop once it becomes a straight line, and instead keeps going in that direction. This tells us that when the string is perfectly straight, nothing stops it from continuing to move. 

We can say now, that the more "unstraight" the string is, the more it will be pulled back towards the centre, and when the string is perfectly straight, there will be nothing pulling on it. That wasn't a very scientific sentence, but we're getting there. We can quantify the idea of "unstraightness" by thinking about what it means. Lets think of a string as a chain of particles connected together by some type of force (the specifics of the force is not important here, just think of it as the "glue" that holds the particles together). 

If the string is perfectly straight, then all the particles would have the same displacement from y=0 as all its neighbours. In our case, the edge particles all have a displacement of 0, this means every particle must also have a displacement of 0. If the string is not straight, that means that each particles' displacement will differ from its neighbours (think about pulling just one particle up - that "bends" the string since it makes one particle higher than its neighbours). 

We arrive at the notion that the "straightness" of a string is governed by each particle's displacement relative to its neighbours. Does this sound familiar? If you have read our article on the heat equation, you might recognise that we have dealt with this exact scenario (if you haven't or forgot, I highly recommend reading it over)! Lets use that result to treat our string. Suppose u Is the displacement of the string. Using the same finite difference method and recognising that are trillions of trillions of particles inside any given string (the same argument and process as last time in the heat equation), we can say that 

Which is the second derivative, ie. the "curvature" of the string. Now that we have the idea of "straightness" and "unstraightness" figured out via the notion of curvature, lets move on to the other half of the puzzle. From the start, we knew that the more curved the string is, the harder it will be pulled back. But what does it really mean to "pull" on something? For us, pulling means exerting a force. So, to put things into science terms, we can now say that the force on the string is proportional to its curvature, or 

So how do we relate this to the motion of the string? Newton's second law tells us that acceleration is proportional to force. Since we already know that force is proportional to curvature,  we can say that acceleration is proportional to curvature. Lets now figure out what acceleration means for our string. We know that velocity is the displacement in some period of time time, or

If we want the velocity at any given point in time, we must have 𝚫t become tiny (A tiny interval of time is approximately an instant of time). We repeat the process of making 𝚫t tend to zero and replacing 𝚫 with ∂, which gives us

What is acceleration? Acceleration is the change in velocity in some period of time (think of how we express acceleration of sports cars - "0 to 60 in 2 seconds", thats referring to a change in velocity of 60 mph in the forward direction in a period of 2 seconds), so we can say

Where we again found the acceleration at a given point in time by reducing the time interval to a tiny number (tends to 0). We can now combine the two definitions together and say that 

Which means that the acceleration is the change of the change in position. Since the change in position is what we call velocity, acceleration is the change in velocity. 

Lets get back to our string. We know that acceleration is proportional to curvature. This means that we can put our expressions for acceleration and curvature together and get 

Now to make this into an equation, all we need is a constant of proportionality, which we call k. Now, the proportionality statement becomes the following equation

Which is the wave equation. For those astute readers out there, you will notice that the equation only has the right units if k has units of speed squared (m^2/s^2), since t has units of time and u & x has units of space . That observation would be correct, and turns out k is exactly the wave speed squared, or

Where v is the speed of how fast a wave (from a macroscopic perspective) is moving horizontally, not the speed at which the individual particles (thats microscopic!) that make up the string are moving vertically. 

The wave equation is perhaps the most fundamental equation in physics. It governs all waves, from gravitational waves to sound waves to waves on a string. This is why it is worthwhile to spend a lot of time studying the wave equation in order to understand the natural world.

Written by Lachlan on Dec 3rd, 2025