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I'm teaching a geometry course covering spectral problems, using eigenvalues of the Laplace operator for shape analysis ("Can you hear the shape of a drum?"). I thought I'd cover where the wave equation $u_{tt}=c^2u_{xx}$ actually comes from in physics before discussing eigenvalues in spectral geometry. When I returned to my old undergrad PDE/physics books, however, I got myself confused!

The Strauss textbook (page 12) takes a piece of string with vertical displacement $u(x,t)$ with endpoints $x_0,x_1\in\mathbb R$. They argue in presence of constant tension $T$ the transverse forces yield the relationship $$ \left.\frac{T u_x}{\sqrt{1+u_x^2}}\right|_{x_0}^{x_1}=\int_{x_0}^{x_1}\rho u_{tt}\,dx $$ for Newton's equations. The PDE appears when taking $x_1\to x_0$ and approximating $\sqrt{1+u_x^2}=1+\frac{1}{2}u_x^2+\cdots.$ It's the approximate "$\cdots$" that I'm worried about! Why can it be ignored?

Similarly, the Wikipedia page has a strange derivation from Hooke's law where the square root doesn't appear. It's not clear in their argument if the particles linked by springs are moving vertically---in which case a $\sqrt{\cdot}$ should appear when measuring spring forces---or horizontally---in which case evaluating $u$ at positions like $x+2h$ doesn't make much sense since the particles are moving horizontally.

Is there a way to motivate the wave equation that doesn't involve a heuristic or Taylor series handwave? If not, why is it OK to solve this equation for large $t$ values rather than just differentially?

Justin Solomon
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1 Answers1

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The derivation based on Hooke's law given on Wikipedia is for the 1D case, so it's assumed that the particles can only move horizontally - hence the absence of the square root. In that derivation, $u(x+2h)$ simply means the horizontal displacement of the mass that was initially at $x+2h$. The wave equation then comes from the "usual" trick of considering these "masses" to be both very numerous ($N \to \infty$) and very close to each other ($h \to 0$). That's essentially the standard continuous approximation of solid mechanics.

The same idea applies in the 2D and 3D cases, and even though a square root shows up in the derivation (as you correctly mentioned), it can always be expanded in powers of $h$ and the higher order terms disregarded (due to the $h \to 0$ "clause").

As a side note: if you really want a derivation of the wave equation that doesn't resort to the continuous approximation, maybe the electromagnetic case is a more appropriate choice, as it only relies on the validity of Maxwell's equations.

The Theoretical Astronaut
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