space of an even number of dimensions, but may or may not occur in space of an odd number of dimensions. This … space with an even number of dimensions. But when the number of spatial dimensions is odd, Hadamard’s … MATHEMATICAL THEORY OF HUYGENS’ PRINCIPLE THE MATHEMATICAL THEORY OF HUYGENS’ PRINCIPLE BY BEVAN B. BAKER

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 The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. This article focuses on waves in classical physics. Quantum physics uses an These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t, x, y, z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a 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. The function 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 (xi, ti), the value of u(xi, ti) depends only on the values of f(xi + cti) and… ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 . {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}.} This equation is typically described as having only one spatial dimension x {\displaystyle x} , because the only other independent variable is the time t {\displaystyle t} . with wave number k = ω/c. The total wave function for this eigenmode is then the linear combination where complex numbers A, B depend in general on any initial and boundary conditions of the problem. Eigenmodes are useful in constructing a full solution to the wave equation, where F and 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 r increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions. For physical examples of solutions to the 3D wave equation that possess angular dependence, see dipole radiation. These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point P, given (t, x, y, z) depends only on the data on the sphere of radius ct that is intersected by the light cone drawn backwards from P. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a 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. We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain D in m-dimensional x space, with boundary B. Then the wave equation is to be satisfied if x is in D, and t > 0. On the boundary of D, the solution u shall satisfy The function 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 (xi, ti), the value of u(xi, ti) depends only on the values of f(xi + cti) and f(xi − cti) and the values of the function g(x) between (xi − cti) and (xi + cti). 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 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 (xi, ti) as RC. Suppose we integrate the inhomogeneous wave equation over this region: 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 (xi, ti) 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.