Wave Equation derivation

Question

I'm curious about part of the derivation of the wave equation as is done in all references that I've seen so far (I'm gonna reproduce only the part that's puzzling me).

We apply Newton's second law to the motion of a piece of vibrating string in the vertical direction. Let's call the vertical direction $y$, and the horizontal direction $x$. Then we limit ourselves to an infinitesimal part of the string, and we write down the mass of this piece as $\rho \Delta x $. The fact that we could replace the length of this portion of the string $l$ with $\Delta x$ is because we assumed small oscillations: $l\approx \sqrt{(\Delta x)^2+(\Delta y)^2}\approx \Delta x$, as the displacement of string from the equilibrium position $\Delta y$ was assumed to be very small.

Now the equation $\partial^2y/\partial t^2=c (\partial^2y/\partial x^2)$ is derived after some more calculations ($c$ is just some constant). This equation is derived based on the assumption that the oscillation is small ($\Delta y\approx 0$) but it's obviously satisfied for non small oscillations (e.g. sine and cosine type of waves) as well.

So how does the proof generalize to the case of non-small oscillations? In other words, why does a string oscillation which is not necessarily small, satisfies the wave equation as well?

Answer 1

The wave equation for a string is indeed only true for small heights and is, as a result, only an approximation. There perhaps exists a more accurate model with a slightly altered wave equation for large heights but this is the simplest case to show how the wave equation can manifest itself in even everyday application.

There is also an important concept that must be understood here. The wave equation is a mathematical equation i.e. it represents the equation that equates the Laplacian of some quantity to its second time derivative:

\begin{equation} \nabla^2 \Psi =\frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} \end{equation}

where $c$ is some characteristic speed. For your case, the 1D string, $\Psi$ represents string height, and the laplacian becomes $\frac{\partial^2 \Psi}{\partial x^2}$. Many other physical systems can be said to similarly follow this equation or variations of it. As a result, you don't derive it but show that a system follows it.

Answer 2

The wave amplitude does have to be small for the simple wave equation to hold for a wave on a taut string. Sinusoidal waves also have to be of small amplitude (specifically, $A\ll\lambda$ in which $A$ is amplitude and $\lambda$ is wavelength).

When we derive the wave equation for a taut string we assume that the tension in the string is constant (as implied by your analysis leading to $\ell=\Delta x$). We also make various small angle approximations. These only hold for small amplitude waves.

Answer 3

Students are usually introduced to the wave equation by analyzing a vibrating string, because this can be done using only Newtonian mechanics. The resulting equation applies only to small-amplitude vibrations of the string.

However, that equation, generalized to three dimensions, applies to electromagnetic waves of any amplitude in vacuum, at least in classical electromagnetism. That wave equation, derivable from Maxwell’s equations, is not an approximation. And, unless you are a musician, EM waves are more interesting and important than vibrating violin strings.

In fact, the propagation of all non-interacting elementary particles, not just photons, in vacuum is described by the wave equation, without it being a small-amplitude approximation.

Answer 4

why does a string oscillation which is not necessarily small, satisfies the wave equation as well?

The wave equation as written in the Question doesn’t work for all physical waves. Some waves have propagation speeds that vary with frequency or amplitude, for example.

Simplifications and approximations went into the model-creation that underlies that wave equation. When those are good, the equation models the physical situation well. But when not, not.

Answer 5

Here are derivations of the 1D wave equation that do not depend on the physics of a string (or even physics at all). So there are no approximations.

If the solutions of the wave equation are specified to be of the form f(x-ct), f(x+ct), then the wave equation can be derived from that alone:

https://physics.stackexchange.com/a/403761/45664

Or just by examining the geometry alone without using physics the wave equation can be derived:

https://physics.stackexchange.com/a/110842/45664

So then its a matter of seeing whether the physics of a particular situation fits the wave equation.