Can a wave be represented by infinite discrete oscillators?

In this article I will find the equation of motion for an arbitrary number of discrete coupled oscillators (I will call “beads”) on a string. Then we will take the limit as the number of oscillators tends towards infinity, and see if the wave equation can be reproduced. If so it would be a good representation to say that a wave can be interpreted as infinite discrete oscillators.

The wave equation, in various forms, is essential in many areas of physics and mathematics. The wave equation for a one dimensional scalar wave is given by the following:

    $$\frac{\partial^2 \psi}{\partial t^2}=v^{2}\frac{\partial^2 \psi}{\partial x^2}$$

Let us consider the following, a bounded massless string with a number n identical beads of mass m. These beads are separated by a distance \Delta x, and there is a uniform tension in the string so that T>>mg. The upward vertical displacement of each bead is given by \begin{Bmatrix}\psi_{p}\end{Bmatrix}^{n}_{p=1}:

*note here the bead labeled p=0, actually represents p=1.

Let us consider the bead p within it’s oscillating “neighbourhood”:

In the diagram \theta_{p,p+1} represent the angle to the horizontal the string makes at that point, again \psi represents the vertical displacement from equilibrium. From this we can work out the equation of motion for a bead on the string in general (for the p^{th} bead). Let’s resolve the forces vertically for bead p:

m\ddot{\psi_{p}}= Tsin\theta_{p+1} - Tsin\theta_{p}

We assume that displacements are small, thus both angle are small. Therefore small angle approximations are valid:

sin\theta_{p} \approx\theta_{p} \approx \frac{\psi_{p}- \psi_{p-1}}{\Delta x}

sin\theta_{p+1} \approx\theta_{p+1} \approx \frac{\psi_{p+1}- \psi_{p}}{\Delta x}

Substituting back into the equation of motion for bead p:

m\ddot{\psi_{p}}= T\left ( \frac{\psi_{p+1}- \psi_{p}}{\Delta x}\right ) -T\left ( \frac{\psi_{p}- \psi_{p-1}}{\Delta x} \right )

We know that we have a number n coupled equations of motion. From this we can find the normal modes of oscillation for the system, however in this case we are not interested in the normal modes, we want to find the wave equation. In essence the wave equation relates \frac{\partial^2 \psi}{\partial t^2} to \frac{\partial^2 \psi}{\partial x^2}. We recognise that we previously found a result for \frac{\partial^2 \psi}{\partial t^2}:

m\frac{\partial^2 \psi}{\partial t^2}= T\left ( \frac{\psi_{p+1}- \psi_{p}}{\Delta x}\right ) -T\left ( \frac{\psi_{p}- \psi_{p-1}}{\Delta x} \right )

Re-writing this equations allows for us to reach a very interesting result:

    $$\frac{\partial^2 \psi}{\partial t^2}= \frac{T}{\frac{m}{\Delta x}}\left (\frac{\frac{\psi_{p+1}- \psi_{p}}{\Delta x}-\frac{\psi_{p}- \psi_{p-1}}{\Delta x}}{\Delta x} \right )$$

Before the next step we must define a few things. We say that a disturbance \psi is a function of space and time such that \psi(x_{p}, t)= \psi_{p}(t) \ \forall p. With this definition, and recalling that the distance between two oscillators is \Delta x such that \psi(x_{p+1},t)= \psi(x_{p}+\Delta x, t), the equation above can be re-written: 

    $$\frac{\partial^2 \psi}{\partial t^2}(x_{p},t)= \frac{T}{\frac{m}{\Delta x}}\left (\frac{\frac{\psi({x_{p}+\Delta x},t)- \psi(x_{p},t)}{\Delta x}-\frac{\psi({x_{p}},t)- \psi(x_{p}- \Delta x,t)}{\Delta x}} {\Delta x}\right )$$

Remember that this comes from the equation of motion of n beads on a string. We can now see what happens when we take the limit n \rightarrow \infty, clearly this is equivalent to taking the limit \Delta x \rightarrow 0 so:

    $$\frac{\partial^2 \psi}{\partial t^2}(x_{p},t)= \lim_{\Delta x \rightarrow 0} \frac{T}{\frac{m}{\Delta x}}\left (\frac{\frac{\psi({x_{p}+\Delta x},t)- \psi(x_{p},t)}{\Delta x}-\frac{\psi({x_{p}},t)- \psi(x_{p}- \Delta x,t)}{\Delta x}} {\Delta x}\right )$$

For a uniform string the fraction \frac{m}{\Delta x} will be constant and will be equal to the mass per unit length (denoted by \mu). The expressions within the brackets are the first principle forms of partial derivatives with respect to x at a constant time. Thus we can re-write the equation:

    $$\frac{\partial^2 \psi}{\partial t^2}(x_{p},t)=\frac{T}{\mu}\lim_{\Delta x \rightarrow 0}\left ( \frac{\frac{\partial \psi }{\partial x}(x_{p},t)-\frac{\partial \psi }{\partial x}(x_{p}-\Delta x, t)}{\Delta x} \right )$$

Again the limit is in the form of a partial derivative in x at a constant time. This time it is the partial derivative of a partial derivative so the limit evaluates to the second spatial derivative and we have the following:

    $$\frac{\partial^2 \psi}{\partial t^2}(x,t)=\frac{T}{\mu}\frac{\partial^2 \psi}{\partial x^2}(x,t)$$

We know that the speed of propagation of a wave on a string is equal to \sqrt{\frac{T}{\mu}} therefore we get:

    $$\frac{\partial^2 \psi}{\partial t^2}=v^{2}\frac{\partial^2 \psi}{\partial x^2}$$

Which is the wave equation for a scalar wave in one dimension!

Therefore considering the coupled oscillations of n beads along a finite string, and then letting n \rightarrow \infty, allows us to manipulate the equation of motion and reproduce the wave equation. This therefore means that we can picture a wave as discrete oscillators in the infinitesimal limit, allowing for an interesting way of interpreting what a wave really is!

Can a wave be represented by infinite discrete oscillators?- We have shown that for a scalar wave in one dimension we can intuit a wave as such. 




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