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I’m interested in simulating a wavetable of arbitrary shape.

Formally, suppose $R ⊂ \mathbb{R}^2$ is a region whose boundary is a simple closed curve $γ$. Let $u = 0$ on $γ$, and let $u$ satisfy the wave equation in $R$. For an initial value problem, let $u = f(x, y)$ initially.

Idea:

Perhaps one could find a complete basis of standing wave functions $u_i(x, y)\cos(ω_it)$, so that a solution could be written as a series $u = Σ_i α_i u_i \cos(ω_it)$ by finding the coefficients $α_i$?

To motivate this, consider the simple example $R = [0, π]×[0, π]$. The 2D Fourier basis $u_{nm} = \sin(nx)\sin(my) \ ∀_{(n,m)∈\mathbb{N}^2}$ is orthonormal in $\mathcal{C}(R)$, and any $u_{nm}$ is a standing wave solution with frequency $ω_{nm} = c^2(n^2 + m^2)$, so that $u_{nm}\cos(ω_{nm}t)$ solves the wave equation. Finally, we can decompose $f$ in this basis with $α_{nm} = ⟨f, u_{nm}⟩$ so that $u = Σ_{nm}α_{nm}u_{nm}\cos(ω_{nm}t)$ is the solution to our IVP.

Does this approach generalise? For general regions, I’m not sure whether standing wave solutions exist, or whether they always form a basis. Do they? If so, how might one find them?


With respect to an inner product like $⟨f, g⟩ ≡ \frac{4}{π^2}∫_0^π∫_0^π fg\,\mathrm{d}x\mathrm{d}y$.

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    $\begingroup$ In a general sense, as long as you have periodic boundary conditions and your solution is well behaved in-between, yes you can expand in plane waves. $\endgroup$ – InertialObserver Dec 21 '18 at 0:20
  • $\begingroup$ May I ask why this has the "hilbert-space" tag? $\endgroup$ – probably_someone Dec 21 '18 at 0:25
  • $\begingroup$ @probably_someone Because I describe the standing waves as elements in an inner product space. I suppose it’s not relevant enough. $\endgroup$ – Jollywatt Dec 21 '18 at 0:34
  • $\begingroup$ @InertialObserver What if the boundary conditions cannot be interpreted as periodic? (If the region does not tesselate, for example.) $\endgroup$ – Jollywatt Dec 21 '18 at 0:36
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    $\begingroup$ If I understand your approach it does generalize in theory. But you need to be able to apply the boundary conditions and get a meaningful solution. Are you planning in doing that numerically, over a finite descretization? $\endgroup$ – ggcg Dec 21 '18 at 0:47

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