Editing Wave equation (section)

==Problems with boundaries==

===One space dimension===

==== Reflection and transmission at the boundary of two media ====
For an incident wave traveling from one medium (where the wave speed is {{math|''c''<sub>1</sub>}}) to another medium (where the wave speed is {{math|''c''<sub>2</sub>}}), one part of the wave will transmit into the second medium, while another part reflects back into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be calculated by using the continuity condition at the boundary.

Consider the component of the incident wave with an [[angular frequency]] of {{mvar|ω}}, which has the waveform
<math display="block">u^\text{inc}(x, t) = Ae^{i(k_1 x - \omega t)},\quad A \in \C.</math>
At {{math|1=''t'' = 0}}, the incident reaches the boundary between the two media at {{math|1=''x'' = 0}}. Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms
<math display="block">u^\text{refl}(x, t) = Be^{i(-k_1 x - \omega t)}, \quad
 u^\text{trans}(x, t) = Ce^{i(k_2 x - \omega t)}, \quad
 B, C \in \C.</math>
The continuity condition at the boundary is
<math display="block">u^\text{inc}(0, t) + u^\text{refl}(0, t) = u^\text{trans}(0, t), \quad
 u_x^\text{inc}(0, t) + u_x^\text{ref}(0, t) = u_x^\text{trans}(0, t).</math>
This gives the equations
<math display="block">A + B = C, \quad
 A - B = \frac{k_2}{k_1} C = \frac{c_1}{c_2} C,</math>
and we have the reflectivity and transmissivity
<math display="block">\frac{B}{A} = \frac{c_2 - c_1}{c_2 + c_1}, \quad
 \frac{C}{A} = \frac{2c_2}{c_2 + c_1}.</math>
When {{math|''c''<sub>2</sub> < ''c''<sub>1</sub>}}, the reflected wave has a [[reflection phase change]] of 180°, since {{math|''B''/''A'' < 0}}. The energy conservation can be verified by
<math display="block">\frac{B^2}{c_1} + \frac{C^2}{c_2} = \frac{A^2}{c_1}.</math>
The above discussion holds true for any component, regardless of its angular frequency of {{mvar|ω}}.

The limiting case of {{math|1=''c''<sub>2</sub> = 0}} corresponds to a "fixed end" that does not move, whereas the limiting case of {{math|1=''c''<sub>2</sub> → ∞}} corresponds to a "free end".

==== The Sturm–Liouville formulation ====
A flexible string that is stretched between two points {{math|1= ''x'' = 0}} and {{math|1= ''x'' = ''L''}} satisfies the wave equation for {{math|''t'' > 0}} and {{math|0 < ''x'' < ''L''}}. On the boundary points, {{mvar|u}} may satisfy a variety of boundary conditions. A general form that is appropriate for applications is

<math display="block">\begin{align}
 -u_x(t, 0) + a u(t, 0) &= 0, \\
  u_x(t, L) + b u(t, L) &= 0,
\end{align}</math>

where {{mvar|a}} and {{mvar|b}} are non-negative. The case where {{mvar|u}} is required to vanish at an endpoint (i.e. "fixed end") is the limit of this condition when the respective {{mvar|a}} or {{mvar|b}} approaches infinity. The method of [[separation of variables]] consists in looking for solutions of this problem in the special form
<math display="block">u(t, x) = T(t) v(x).</math>

A consequence is that
<math display="block">\frac{T''}{c^2 T} = \frac{v''}{v} = -\lambda.</math>

The [[eigenvalue]] {{mvar|λ}} must be determined so that there is a non-trivial solution of the boundary-value problem
<math display="block">\begin{align}
 v'' + \lambda v = 0,& \\
 -v'(0) + a v(0) &= 0, \\
  v'(L) + b v(L) &= 0.
\end{align}</math>

This is a special case of the general problem of [[Sturm–Liouville theory]]. If {{mvar|a}} and {{mvar|b}} are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for {{mvar|u}} and {{math|''u<sub>t</sub>''}} can be obtained from expansion of these functions in the appropriate trigonometric series.

===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.