Editing Wave equation (section)

===Several space dimensions===
[[Image:Drum vibration mode12.gif|right|thumb|220px|A solution of the wave equation in two dimensions with a zero-displacement boundary condition along the entire outer edge]]
The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain {{mvar|D}} in {{mvar|m}}-dimensional {{mvar|x}} space, with boundary {{mvar|B}}. Then the wave equation is to be satisfied if {{mvar|x}} is in {{mvar|D}}, and {{math|''t'' > 0}}. On the boundary of {{mvar|D}}, the solution {{mvar|u}} shall satisfy

<math display="block"> \frac{\partial u}{\partial n} + a u = 0, </math>

where {{mvar|n}} is the unit outward normal to {{mvar|B}}, and {{mvar|a}} is a non-negative function defined on {{mvar|B}}. The case where {{mvar|u}} vanishes on {{mvar|B}} is a limiting case for {{mvar|a}} approaching infinity. The initial conditions are

<math display="block"> u(0, x) = f(x), \quad u_t(0, x) = g(x), </math>

where {{mvar|f}} and {{mvar|g}} are defined in {{mvar|D}}. This problem may be solved by expanding {{mvar|f}} and {{mvar|g}} in the eigenfunctions of the Laplacian in {{mvar|D}}, which satisfy the boundary conditions. Thus the eigenfunction {{mvar|v}} satisfies

<math display="block"> \nabla \cdot \nabla v + \lambda v = 0 </math>

in {{mvar|D}}, and

<math display="block"> \frac{\partial v}{\partial n} + a v = 0 </math>

on {{mvar|B}}.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary {{mvar|B}}. If {{mvar|B}} is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle {{mvar|θ}}, multiplied by a [[Bessel function]] (of integer order) of the radial component. Further details are in [[Helmholtz equation]].

If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are [[spherical harmonics]], and the radial components are [[Bessel function]]s of half-integer order.