# Wave equation IVP on arbitrarily shaped 2D domain

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 $$⟨f, g⟩ ≡ \frac{4}{π^2}∫_0^π∫_0^π fg\,\mathrm{d}x\mathrm{d}y$$.

• 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. – InertialObserver Dec 21 '18 at 0:20
• May I ask why this has the "hilbert-space" tag? – probably_someone Dec 21 '18 at 0:25
• @probably_someone Because I describe the standing waves as elements in an inner product space. I suppose it’s not relevant enough. – Jollywatt Dec 21 '18 at 0:34
• @InertialObserver What if the boundary conditions cannot be interpreted as periodic? (If the region does not tesselate, for example.) – Jollywatt Dec 21 '18 at 0:36
• 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? – ggcg Dec 21 '18 at 0:47