Editing Wave equation (section)

==Scalar wave equation in three space dimensions==
[[File:Leonhard Euler 2.jpg| thumb|right|Swiss mathematician and physicist [[Leonhard Euler]] (b. 1707) discovered the wave equation in three space dimensions.<ref name=Speiser />]]A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.

===Spherical waves===
To obtain a solution with constant frequencies, apply the [[Fourier transform]]
<math display="block">\Psi(\mathbf{r}, t) = \int_{-\infty}^\infty \Psi(\mathbf{r}, \omega) e^{-i\omega t} \, d\omega,</math>
which transforms the wave equation into an [[elliptic partial differential equation]] of the form:
<math display="block">\left(\nabla^2 + \frac{\omega^2}{c^2}\right) \Psi(\mathbf{r}, \omega) = 0.</math>

This is the [[Helmholtz equation]] and can be solved using [[separation of variables]]. In [[spherical coordinates]] this leads to a separation of the radial and angular variables, writing the solution as:<ref>{{cite book |first=John David |last=Jackson |title=Classical Electrodynamics |date=14 August 1998 | edition=3rd |publisher=Wiley |page=425 |isbn=978-0-471-30932-1 }}</ref>
<math display="block">\Psi(\mathbf{r}, \omega) = \sum_{l,m} f_{lm}(r) Y_{lm}(\theta, \phi).</math>
The angular part of the solution take the form of [[spherical harmonics]] and the radial function satisfies:
<math display="block"> \left[\frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + k^2 - \frac{l(l + 1)}{r^2}\right] f_l(r) = 0.</math>
independent of <math>m</math>, with <math>k^2=\omega^2 / c^2</math>. Substituting
<math display="block">f_{l}(r)=\frac{1}{\sqrt{r}}u_{l}(r),</math>
transforms the equation into
<math display="block"> \left[\frac{d^2}{dr^2} + \frac{1}{r} \frac{d}{dr} + k^2 - \frac{(l + \frac{1}{2})^2}{r^2}\right] u_l(r) = 0,</math>
which is the [[Bessel equation]].

====Example====
Consider the case {{math|1= ''l'' = 0}}. Then there is no angular dependence and the amplitude depends only on the radial distance, i.e., {{math|Ψ('''r''', ''t'') → ''u''(''r'', ''t'')}}. In this case, the wave equation reduces to{{clarify|reason=What "reduction" has happened? This is exactly equivalent to the wave equation described in the "Introduction" section. Since the wave "Scalar wave equation in three space dimensions" was not defined at the beginning of this section, we can only assume that the version given in "Introduction" is what was meant. There's no way for the reader to know what generalization of the wave equation including l was intended.|date=March 2024}}<math display="block">
 \left(\nabla^2 - \frac{1}{c^2} \frac{\partial^2 }{\partial t^2}\right) \Psi(\mathbf{r}, t) = 0,
</math>
or
<math display="block">
 \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}\right) u(r, t) = 0.
</math>

This equation can be rewritten as
<math display="block">\frac{\partial^2(ru)}{\partial t^2} - c^2 \frac{\partial^2(ru)}{\partial r^2} = 0,</math>
where the quantity {{math|''ru''}} satisfies the one-dimensional wave equation. Therefore, there are solutions in the form<math display="block">u(r, t) = \frac{1}{r} F(r - ct) + \frac{1}{r} G(r + ct),</math>
where {{mvar|F}} and {{mvar|G}} are general solutions to the one-dimensional wave equation and can be interpreted as respectively an outgoing and incoming spherical waves. The outgoing wave can be generated by a [[point source]], and they make possible sharp signals whose form is altered only by a decrease in amplitude as {{mvar|r}} increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.{{citation needed|date=February 2014}}

For physical examples of solutions to the 3D wave equation that possess angular dependence, see [[dipole radiation]].

====Monochromatic spherical wave====
[[File:Spherical Wave.gif|thumb|Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source]]
Although the word "monochromatic" is not exactly accurate, since it refers to light or [[electromagnetic radiation]] with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on [[#Plane-wave eigenmodes|plane-wave eigenmodes]], if we again restrict our solutions to spherical waves that oscillate in time with well-defined ''constant'' angular frequency {{mvar|ω}}, then the transformed function {{math|''ru''(''r'', ''t'')}} has simply plane-wave solutions:<math display="block">r u(r, t) = Ae^{i(\omega t \pm kr)},</math>
or
<math display="block">u(r, t) = \frac{A}{r} e^{i(\omega t \pm kr)}.</math>

From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude
<math display="block">I = |u(r, t)|^2 = \frac{|A|^2}{r^2},</math>
drops at the rate proportional to {{math|1/''r''<sup>2</sup>}}, an example of the [[inverse-square law]].

===Solution of a general initial-value problem===
The wave equation is linear in {{mvar|u}} and is left unaltered by translations in space and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let {{math|''φ''(''ξ'', ''η'', ''ζ'')}} be an arbitrary function of three independent variables, and let the spherical wave form {{mvar|F}} be a [[delta function]]. Let a family of spherical waves have center at {{math|(''ξ'', ''η'', ''ζ'')}}, and let {{mvar|r}} be the radial distance from that point. Thus

<math display="block">r^2 = (x - \xi)^2 + (y - \eta)^2 + (z - \zeta)^2.</math>

If {{mvar|u}} is a superposition of such waves with weighting function {{mvar|φ}}, then
<math display="block">u(t, x, y, z) = \frac{1}{4\pi c} \iiint \varphi(\xi, \eta, \zeta) \frac{\delta(r - ct)}{r} \, d\xi \, d\eta \, d\zeta;</math>
the denominator {{math|4''πc''}} is a convenience.

From the definition of the delta function, {{mvar|u}} may also be written as
<math display="block">u(t, x, y, z) = \frac{t}{4\pi} \iint_S \varphi(x + ct\alpha, y + ct\beta, z + ct\gamma) \, d\omega,</math>
where {{mvar|α}}, {{mvar|β}}, and {{mvar|γ}} are coordinates on the unit sphere {{mvar|S}}, and {{mvar|ω}} is the area element on {{mvar|S}}. This result has the interpretation that {{math|''u''(''t'', ''x'')}} is {{mvar|t}} times the mean value of {{mvar|φ}} on a sphere of radius {{math|''ct''}} centered at {{mvar|x}}:
<math display="block">u(t, x, y, z) = t M_{ct}[\varphi].</math>

It follows that
<math display="block">u(0, x, y, z) = 0, \quad u_t(0, x, y, z) = \varphi(x, y, z).</math>

The mean value is an even function of {{mvar|t}}, and hence if
<math display="block">v(t, x, y, z) = \frac{\partial}{\partial t} \big(t M_{ct}[\varphi]\big),</math>
then
<math display="block">v(0, x, y, z) = \varphi(x, y, z), \quad v_t(0, x, y, z) = 0.</math>

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point {{mvar|P}}, given {{math|(''t'', ''x'', ''y'', ''z'')}} depends only on the data on the sphere of radius {{math|''ct''}} that is intersected by the '''[[light cone]]''' drawn backwards from {{mvar|P}}. It does ''not'' depend upon data on the interior of this sphere. Thus the interior of the sphere is a [[Petrovsky lacuna|lacuna]] for the solution. This phenomenon is called '''[[Huygens' principle]]'''. It is only true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure.{{sfn | Atiyah | Bott | Gårding | 1970 | pp=109–189}}{{sfn | Atiyah | Bott | Gårding | 1973 | pp=145–206}}