Editing Wave equation (section)

== Green's function ==
Consider the inhomogeneous wave equation in <math>
  1+D
</math> dimensions<math display="block">
(\partial_{tt} - c^2\nabla^2) u = s(t, x) 
</math>By rescaling time, we can set wave speed <math> c = 1</math>.

Since the wave equation <math>
(\partial_{tt} - \nabla^2) u = s(t, x)
</math> has order 2 in time, there are two [[Impulse response|impulse responses]]: an acceleration impulse and a velocity impulse. The effect of inflicting an acceleration impulse is to suddenly change the wave velocity <math>\partial_t u</math>. The effect of inflicting a velocity impulse is to suddenly change the wave displacement <math>u</math>.

For acceleration impulse, <math>s(t,x) = \delta^{D+1}(t,x)</math> where <math>\delta</math> is the [[Dirac delta function]]. The solution to this case is called the [[Green's function]] <math>G</math> for the wave equation.  

For velocity impulse, <math>s(t, x) = \partial_t \delta^{D+1}(t,x)</math>, so if we solve the Green function <math>G</math>, the solution for this case is just <math>\partial_t G</math>.{{citation needed|reason=See Talk page#Green's function - Lack of references|date=August 2024}}

=== Duhamel's principle ===
The main use of Green's functions is to solve [[Initial value problem|initial value problems]] by [[Duhamel's principle]], both for the homogeneous and the inhomogeneous case.

Given the Green function <math>G</math>, and initial conditions <math>u(0,x), \partial_t u(0,x)</math>, the solution to the homogeneous wave equation is<ref name=":2" /><math display="block">
  u = (\partial_t G) \ast u + G \ast \partial_t u
</math>where the asterisk is [[convolution]] in space. More explicitly, <math display="block">
  u(t, x) = \int  (\partial_t G)(t, x-x') u(0, x') dx' +  \int  G(t, x-x') (\partial_t u)(0, x') dx'.
</math>For the inhomogeneous case, the solution has one additional term by convolution over spacetime:<math display="block">
  \iint_{t' < t} G(t-t', x-x') s(t', x')dt' dx'.
</math>

=== Solution by Fourier transform ===
By a [[Fourier transform]],<math display="block">
  \hat G (\omega)=  \frac{1}{-\omega_0^2 + \omega_1^2 + \cdots + \omega_D^2}, 
\quad G(t, x) = \frac{1}{(2\pi)^{D+1}} \int \hat G(\omega) e^{+i \omega_0 t + i \vec \omega \cdot \vec x}d\omega_0 d\vec\omega.  
</math>The <math>\omega_0</math> term can be integrated by the [[residue theorem]]. It would require us to perturb the integral slightly either by <math>+i\epsilon</math> or by <math>-i\epsilon</math>, because it is an improper integral. One perturbation gives the forward solution, and the other the backward solution.<ref>{{Cite web | url=http://julian.tau.ac.il/bqs/em/green.pdf | title=The green function of the wave equation | website=julian.tau.ac.il | access-date=2024-09-03}}</ref> The forward solution gives<math display="block">
  
G(t,x) = \frac{1}{(2\pi)^D} \int  \frac{\sin (\|\vec \omega\| t)}{\|\vec \omega\|} e^{i \vec \omega \cdot \vec x}d\vec \omega,
\quad
\partial_t G(t, x) = \frac{1}{(2\pi)^D} \int  \cos(\|\vec \omega\| t) e^{i \vec \omega \cdot \vec x}d\vec \omega.
 
</math>The integral can be solved by analytically continuing the [[Poisson kernel]], giving<ref name=":2">{{Cite web |last=Barnett |first=Alex H. |date=December 28, 2006 |title=Greens Functions for the Wave Equation |url=https://users.flatironinstitute.org/~ahb/notes/waveequation.pdf |access-date=August 25, 2024 |website=users.flatironinstitute.org}}</ref><ref name=":0" /><math display="block">
  G(t, x) = \lim _{\epsilon \rightarrow 0^{+}} \frac{C_D}{D-1} 
\operatorname{Im}\left[\|x\|^2-(t-i \epsilon)^2\right]^{-(D-1) / 2}
</math>where  <math display="block">
  C_D=\pi^{-(D+1) / 2} \Gamma((D+1) / 2)
</math> is half the surface area of a <math>(D + 1)</math>-dimensional [[N-sphere|hypersphere]].<ref name=":0">{{Citation |last=Taylor |first=Michael E. |title=The Laplace Equation and Wave Equation |date=2023 |work=Partial Differential Equations I: Basic Theory |series=Applied Mathematical Sciences |volume=115 |pages=137–205 |editor-last=Taylor |editor-first=Michael E. |url=https://link.springer.com/chapter/10.1007/978-3-031-33859-5_2 |access-date=2024-08-20 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-031-33859-5_2 |isbn=978-3-031-33859-5}}</ref>

=== Solutions in particular dimensions ===

We can relate the Green's function in <math>D</math> dimensions to the Green's function in <math>D+n</math> dimensions.<ref name=":1">{{Cite journal |last1=Soodak |first1=Harry |last2=Tiersten |first2=Martin S. |date=1993-05-01 |title=Wakes and waves in N dimensions |url=https://pubs.aip.org/ajp/article/61/5/395/1054318/Wakes-and-waves-in-N-dimensions |journal=American Journal of Physics |language=en |volume=61 |issue=5 |pages=395–401 |doi=10.1119/1.17230 |bibcode=1993AmJPh..61..395S |issn=0002-9505}}</ref>

==== Lowering dimensions ====
Given a function <math>s(t, x)</math> and a solution <math>u(t, x)</math> of a differential equation in <math>(1+D)</math> dimensions, we can trivially extend it to <math>(1+D+n)</math> dimensions by setting the additional <math>n</math> dimensions to be constant:
<math display="block">
	  s(t, x_{1:D}, x_{D+1:D+n}) = s(t, x_{1:D}), \quad u(t, x_{1:D}, x_{D+1:D+n}) = u(t, x_{1:D}).
</math>Since the Green's function is constructed from <math>f</math> and <math>u</math>, the Green's function in <math>(1+D+n)</math> dimensions integrates to the Green's function in <math>(1+D)</math> dimensions:
<math display="block">
	  G_D(t, x_{1:D}) = \int_{\R^n} G_{D+n}(t, x_{1:D}, x_{D+1:D+n}) d^n x_{D+1:D+n}.
</math>

==== Raising dimensions ====
The Green's function in <math>D</math> dimensions can be related to the Green's function in <math>D+2</math> dimensions. By spherical symmetry,
<math display="block">
	  G_D(t, r) = \int_{\R^2} G_{D+2}(t, \sqrt{r^2 + y^2 + z^2}) dydz.
</math>
Integrating in polar coordinates,
<math display="block">
	  G_D(t, r) = 2\pi \int_0^\infty G_{D+2}(t, \sqrt{r^2 + q^2}) qdq = 2\pi \int_r^\infty G_{D+2}(t, q') q'dq',
</math>
where in the last equality we made the change of variables <math>q' = \sqrt{r^2 + q^2}</math>.  Thus, we obtain the recurrence relation<math display="block">
	  G_{D+2}(t, r) = -\frac{1}{2\pi r} \partial_r G_D(t, r).
</math>

=== Solutions in ''D = 1, 2, 3'' ===
When <math>D=1</math>, the integrand in the Fourier transform is the [[sinc function]]<math display="block">\begin{aligned}
G_1(t, x) &= \frac{1}{2\pi} \int_\R \frac{\sin(|\omega| t)}{|\omega|} e^{i\omega x}d\omega \\
&= \frac{1}{2\pi} \int \operatorname{sinc}(\omega) e^{i \omega \frac xt} d\omega \\
&= \frac{\sgn(t-x) + \sgn(t+x)}{4} \\
&= \begin{cases}
\frac 12 \theta(t-|x|) \quad t > 0 \\
-\frac 12 \theta(-t-|x|) \quad t < 0
\end{cases}
\end{aligned}</math>
where <math>\sgn</math> is the [[sign function]] and <math>\theta</math> is the [[Heaviside step function|unit step function]]. One solution is the forward solution, the other is the backward solution.

The dimension can be raised to give the <math>D=3</math> case<math display="block">G_3(t, r) = \frac{\delta(t-r)}{4\pi r}</math>and similarly for the backward solution. This can be integrated down by one dimension to give the <math>D=2</math> case<math display="block">G_2(t, r) = \int_\R \frac{\delta(t - \sqrt{r^2 + z^2})}{4\pi \sqrt{r^2 + z^2}} dz 
= \frac{\theta(t - r)}{2\pi \sqrt{t^2 - r^2}}
</math>

=== Wavefronts and wakes ===
In <math>D=1</math> case, the Green's function solution is the sum of two wavefronts <math>\frac{\sgn(t-x)}{4} + 
\frac{\sgn(t+x)}{4}</math> moving in opposite directions.

In odd dimensions, the forward solution is nonzero only at <math> t = r</math>. As the dimensions increase, the shape of wavefront becomes increasingly complex, involving higher derivatives of the Dirac delta function. For example,<ref name=":1" /><math display="block">\begin{aligned}
& G_1=\frac{1}{2 c} \theta(\tau) \\
& G_3=\frac{1}{4 \pi c^2} \frac{\delta(\tau)}{r} \\
& G_5=\frac{1}{8 \pi^2 c^2}\left(\frac{\delta(\tau)}{r^3}+\frac{\delta^{\prime}(\tau)}{c r^2}\right) \\
& G_7=\frac{1}{16 \pi^3 c^2}\left(3 \frac{\delta(\tau)}{r^4}+3 \frac{\delta^{\prime}(\tau)}{c r^3}+\frac{\delta^{\prime \prime}(\tau)}{c^2 r^2}\right)
\end{aligned}</math>where <math>\tau = t- r</math>, and the wave speed <math>c</math> is restored.

In even dimensions, the forward solution is nonzero in <math>r \leq t</math>, the entire region behind the wavefront becomes nonzero, called a [[Wake (physics)|wake]]. The wake has equation:<ref name=":1" /><math display="block">G_{D} (t, x ) = (-1)^{1+D / 2} \frac{1}{(2 \pi)^{D / 2}} \frac{1}{c^D} \frac{\theta(t-r / c)}{\left(t^2-r^2 / c^2\right)^{(D-1) / 2}}</math>The wavefront itself also involves increasingly higher derivatives of the Dirac delta function.

This means that a general [[Huygens–Fresnel principle|Huygens' principle]] – the wave displacement at a point <math>(t, x)</math> in spacetime depends only on the state at points on [[Method of characteristics|characteristic rays]] passing <math>(t, x)</math> – only holds in odd dimensions. A physical interpretation is that signals transmitted by waves remain undistorted in odd dimensions, but distorted in even dimensions.<ref name=":3">{{Cite book |last1=Courant |first1=Richard |title=Methods of mathematical physics. 2: Partial differential equations / by R. Courant |last2=Hilbert |first2=David |date=2009 |publisher=Wiley-VCH |isbn=978-0-471-50439-9 |edition=2. repr |location=Weinheim}}</ref>{{Pg|page=698}} 

'''Hadamard's conjecture''' states that this generalized Huygens' principle still holds in all odd dimensions even when the coefficients in the wave equation are no longer constant. It is not strictly correct, but it is correct for certain families of coefficients<ref name=":3" />{{Pg|page=765}}