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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?

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    $\begingroup$ How about this one: math.ubc.ca/~feldman/m256/wave.pdf $\endgroup$
    – Gert
    Feb 12 '19 at 22:20
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    $\begingroup$ This one at least admits there's some approximation happening ("horrendous mess"), which I appreciate! But still I'm confused why we can use a local approximation (eg Taylor series) but then analyze the solution for large ranges of t. $\endgroup$ Feb 13 '19 at 1:30
  • $\begingroup$ Taylor series handwave. This is not a handwave, it's a result of using a small interval and in such circumstances your can usually ignore higher order terms quite legitimately. As $x_1\to x_0$ it is quite fair to say that the interval is modeled by a straight line (i.e. higher order terms are irrelevant). This is the basis of calculus, not a cheat. $\endgroup$
    – StephenG
    Feb 13 '19 at 5:02
  • $\begingroup$ Hmm. The math justifying Taylor series doesn't say anything about the global behavior of solving the linearized PDE---it just stays in a small neighborhood of a point you can make this approximation. It's typical to plot/analyze/compute/describe solutions of $u_{tt}=c^2u_{xx}$ over large $t,x$ ranges, which seems unjustified as an approximation to the "true" nonlinear wave. That is, there are two hyperbolic PDE at play (linear wave equation and its nonlinear counterpart), and I don't think this Taylor series shows that one is a good approximation of the other over large time/length scales. $\endgroup$ Feb 13 '19 at 12:32
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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.

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  • $\begingroup$ Yes, but Maxwell equations do not relate directly to some visible phenomenon like a moving string. Maybe it's harder to figure out how a wave can arise from those field quantities. $\endgroup$
    – BowPark
    Feb 13 '19 at 0:40
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    $\begingroup$ Yes, I agree: it's harder to get an intuitive physical picture of what an electromagnetic wave is when compared to something like a vibrating string. I mentioned it simply because it's a case where the wave equation can be derived without resorting to a Taylor expansion. $\endgroup$ Feb 13 '19 at 11:27
  • $\begingroup$ I got your point and definitely understand. It's a straightforward procedure, without approximations. $\endgroup$
    – BowPark
    Feb 13 '19 at 12:24
  • $\begingroup$ Interesting. Are there people who work out these sorts of continuum limits carefully? I guess it's sort of close to checking if the discretization of a PDE relates to its true structure. $\endgroup$ Feb 13 '19 at 12:27
  • $\begingroup$ Upon thinking about it a little more, I think the linear wave equation is justified by the Wikipedia argument as long as it's for compression waves that only move horizontally in one dimension. Things get fishy when there's out-of-1d motion---this is where these strange Taylor arguments start to appear. We just need to stop drawing pictures of vibrating strings and acting like they're somehow justified by the linear wave equation; this is where the math is imprecise. But this is coming from a picky math person trying to do $\varepsilon$ amount of physics :-) $\endgroup$ Feb 13 '19 at 12:41

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