Editing Wave equation (section)

==Inhomogeneous wave equation in one dimension==
{{see also|Inhomogeneous electromagnetic wave equation}}

The inhomogeneous wave equation in one dimension is
<math display="block">u_{t t}(x, t) - c^2 u_{xx}(x, t) = s(x, t)</math>
with initial conditions
<math display="block">u(x, 0) = f(x),</math>
<math display="block">u_t(x, 0) = g(x).</math>

The function {{math|''s''(''x'', ''t'')}} is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the [[Lorenz gauge]] of [[electromagnetism]].

One method to solve the initial-value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point {{math|(''x<sub>i</sub>'', ''t<sub>i</sub>'')}}, the value of {{math|''u''(''x<sub>i</sub>'', ''t<sub>i</sub>'')}} depends only on the values of {{math|''f''(''x<sub>i</sub>'' + ''ct<sub>i</sub>'')}} and {{math|''f''(''x<sub>i</sub>'' − ''ct<sub>i</sub>'')}} and the values of the function {{math|''g''(''x'')}} between {{math|(''x<sub>i</sub>'' − ''ct<sub>i</sub>'')}} and {{math|(''x<sub>i</sub>'' + ''ct<sub>i</sub>'')}}. This can be seen in [[d'Alembert's formula]], stated above, where these quantities are the only ones that show up in it. Physically, if the maximum propagation speed is {{mvar|c}}, then no part of the wave that cannot propagate to a given point by a given time can affect the amplitude at the same point and time.

In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that causally affects point {{math|(''x<sub>i</sub>'', ''t<sub>i</sub>'')}} as {{math|''R<sub>C</sub>''}}. Suppose we integrate the inhomogeneous wave equation over this region:
<math display="block">
 \iint_{R_C} \big(c^2 u_{xx}(x, t) - u_{tt}(x, t)\big) \, dx \, dt = \iint_{R_C} s(x, t) \, dx \, dt.
</math>

To simplify this greatly, we can use [[Green's theorem]] to simplify the left side to get the following:
<math display="block">
 \int_{L_0 + L_1 + L_2} \big({-}c^2 u_x(x, t) \, dt - u_t(x, t) \, dx\big) = \iint_{R_C} s(x, t) \, dx \, dt.
</math>

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute:
<math display="block">
 \int^{x_i + c t_i}_{x_i - c t_i} -u_t(x, 0) \, dx = -\int^{x_i + c t_i}_{x_i - c t_i} g(x) \, dx.
</math>

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus {{math|1=''dt'' = 0}}.

For the other two sides of the region, it is worth noting that {{math|''x'' ± ''ct''}} is a constant, namely {{math|''x<sub>i</sub>'' ± ''ct<sub>i</sub>''}}, where the sign is chosen appropriately. Using this, we can get the relation {{math|1=d''x'' ± ''c''d''t'' = 0}}, again choosing the right sign:
<math display="block">\begin{align}
 \int_{L_1} \big({-}c^2 u_x(x, t) \, dt - u_t(x, t) \, dx\big) &= \int_{L_1} \big(c u_x(x, t) \, dx + c u_t(x, t) \, dt \big) \\
 &= c \int_{L_1} \, du(x, t) \\
 &= c u(x_i, t_i) - c f(x_i + c t_i).
\end{align}</math>

And similarly for the final boundary segment:
<math display="block">\begin{align}
 \int_{L_2} \big({-}c^2 u_x(x, t) \, dt - u_t(x, t) \, dx\big) &= -\int_{L_2} \big(c u_x(x, t) \, dx + c u_t(x, t) \, dt \big) \\
 &= -c \int_{L_2} \, du(x, t) \\
 &= c u(x_i, t_i) - c f(x_i - c t_i).
\end{align}</math>

Adding the three results together and putting them back in the original integral gives
<math display="block">\begin{align}
 \iint_{R_C} s(x, t) \, dx \, dt &= - \int^{x_i + c t_i}_{x_i - c t_i} g(x) \, dx + c u(x_i, t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) \\
 &= 2 c u(x_i, t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \int^{x_i + c t_i}_{x_i - c t_i} g(x) \, dx.
\end{align}</math>

Solving for {{math|''u''(''x<sub>i</sub>'', ''t<sub>i</sub>'')}}, we arrive at
<math display="block">
 u(x_i, t_i) = \frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} +
               \frac{1}{2 c} \int^{x_i + c t_i}_{x_i - c t_i} g(x) \, dx +
               \frac{1}{2 c} \int^{t_i}_0 \int^{x_i + c(t_i - t)}_{x_i - c(t_i - t)} s(x, t) \, dx \, dt.
</math>

In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices {{math|(''x<sub>i</sub>'', ''t<sub>i</sub>'')}} compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension. The difference is in the third term, the integral over the source.