The Wave Equation — Part 1
Introduction
Regardless of whether you study Physics or not, you’ve probably heard the term “wave”. Waves in Physics are not really different from things we associate waves with such as ocean waves, sound waves, even waves from our phones!
With this in mind, I wanted to do a small two to three part series on waves. This first article is essentially going to cover how we can model a wave mathematically.
This article will be more mathematically intensive than my previous articles but fear not, I’m only going to assume you have a working knowledge of trigonometry and functions.
With that being said, let’s look at how we can model a wave!
Properties of a wave
Before we get into modelling a wave, let’s look at what a wave is and the various characteristics of a wave.
A wave is essentially a disturbance that propagates through a medium. If this is too abstract for you, a wave is a transfer of energy without a transfer of matter.
Hold on, how do waves transfer energy without transferring matter?
Notice how as the wave “moves”, the brown dots are actually only moving in a vertical line at a fixed position?
Here is a clearer visualisation:
This common representation of a wave is known as a transverse wave. The reason for this is because the wave oscillates transverse (another word for perpendicular) to the direction of motion or energy propagation.
Here are the following characteristics of a wave:
The amplitude is the length of oscillation from rest. Essentially, how far the crest or trough is from the dotted line. The wavelength is the length between two consecutive crests or troughs.
The period of a wave is essentially the time to make one complete cycle on the wave. What does this mean?
It’s the time taken for the red dot, from the top, to go all the way down and back up again.
We can mathematically describe the period of a wave as:
f represents the frequency of the wave. This basically tells us how many waves passes a fixed point in a second. It has units of Hz (Hertz), which is also s^-1 (per second). Hence, the unit for period is 1/s^-1 = s.
These are basically all the properties of a wave.
Now, how do we actually represent it mathematically?
The Trigonometry
Now this is where we start using more Maths. Check out my post on trigonometry if you would like a quick refresher:
Let us assume we have some random wave that looks like this:
What I am essentially doing is plotting the displacement of the oscillation of a wave u against the position of the wave. What is the displacement of the oscillation of a wave?
Essentially, the displacement of the red dot at some particular position
Notice how the wave looks sinusoidal? Sinusoidal refers to anything that looks like a sine or cosine wave.
Why do sine and cos graphs look like waves? This is because they are considered periodic functions wherein for different values of x, they give the same y value. For example, sin(0) = sin(π) = sin(2π) = 0. If you’re familiar with the unit circle, this gif might help establish this:
The reason cosine waves are also considered sinusoidal is because a cosine graph is actually just a sine graph shifted
Note, that if I shift the blue graph (a sine graph) to the left by a certain amount, it essentially just becomes a cosine graph.
In fact, the amount I shift it to the left by is π/2 radians. Hence, I can define cos(x) as
I want to digress for a second.
If I have some function f(x), then f(x+a) is equal to shifting every point on f(x) to the left by the vector
while f(x-a) is equal to shifting every point on f(x) to the right by the vector
Hence, sin(x + π/2) is equal to shifting every point on sin(x) to the left by π/2 to give cos(x).
Great, so we’ve established that waves are sinusoidal in nature.
There’s another cool thing, the period of sin(x) and cos(x) is 2π. When we talk about the period of a sinusoidal function by itself, its simply the length of one cycle (The length from a peak to a peak or a trough to a trough, sound familiar)?
If we had sin(ax), then the period would be 2π/a
Armed with this knowledge, we can now start to build our wave equation.
Our wave equation is essentially a function of the position of the wave. Hence, we should essentially get the displacement of the oscillation u at a given position.
Therefore, if I knew what my wave equation was, I could essentially find u at any value of x.
Let’s assume this is our wave. The wave has an amplitude of A and a wavelength of λ.
If you’re familiar, the general form of a sine function is y=asin(bx) + c where a is the amplitude, b is the period and c is some constant that shifts out graph up and down.
Now, our function is not dependent on time, it is dependant on position. Hence, the b in our equation represents the distance from a peak to a peak, which is also our wavelength. Therefore,
Rearranging for b gives us
Hence, we can write our wave equation as:
Here is a wave with A=1
And here is a wave with amplitude 3, notice how our wave equation is starting to take form?
Now, notice how this wave is stationary. The wave does not always look like this.
For example
At t=5.0s, our wave looks like a cos graph (At x=0, we’re not at 0 because cos(0) is not equal to 0)
While at t=15.0s, our wave looks like a sin graph (At x=0, we’re at 0 because sin(0) = 0)
Now, this means that our wave not only depends on position but also on time. This involves multiple variables which makes our maths so much harder, however let’s stick with it.
The way we represent a function like so: f(x), x is the only input or parameter, essentially our function only depends on x. However, multivariable functions are represented as f(x, y) where now x, and y are parameters.
Hence, our wave equation would be of the form u(x, t). However, notice that
Does not have t anywhere?
Let’s back up for a second. At some particular point, we can represent our function as
However, at a different point, that function looks more like a cos wave than a sine wave. So, how do we get to a cos wave from a sine wave? Remember that we can essentially translate by adding π/2? However, our wave need not always shift by π/2, it could ideally shift by any amount. This is known as the phase of a wave and is given by φ
Hence, we can make our wave equation more accurate by adding this phase.
Therefore,
Now, waves are not static.
They do not magically go from
to
Instead, they transition between the two like so:
In order to represent this, we can model our phase based on time
To do this, we must recall the period of a wave (The one that now depends on time)
Using the similar notion of:
We get
So, when one period of the wave is done, we’re essentially covering 2π which works out perfect given that that is the period of a sinusoidal function.
Adding this term onto our equation, we now get a modified:
There you have it, every wave can be modelled by this equation.
Now, we can go a step forward if we want.
The 2π/λ term is essentially telling us how many radians we’re covering per unit distance. We call this the angular number and abbreviate this with k
We also know that
Hence,
Using the definition
Where ω is angular speed.
Hence, our wave function or wave equation can be rewritten as
As always, I hope you enjoyed reading!
