Editing Wave equation (section)

==== 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".