View source for Huygens principle

The statement that the propagation of oscillations, described by the [[Wave equation|wave equation]], in a space of odd dimension involves the re-appearance of a sharply-localized initial state at a later moment of time at another point as a phenomenon just as sharply localized. If the number of space variables is even, Huygens' principle is absent, and the signal from the localized initial perturbation received at the observation point will be washed away. The principle was first formulated by Chr. Huygens in 1678 [[#References|[1]]] and was subsequently developed by A. Fresnel in 1818 in his studies on the problems of diffraction (cf. also [[Diffraction, mathematical theory of|Diffraction, mathematical theory of]]).

Huygens' principle results from the mathematical fact that the solution of the wave equation at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481701.png" /> of the three-dimensional space at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481702.png" /> is expressed in terms of the values of the solution and its derivatives on an arbitrary closed surface inside which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481703.png" /> is contained, at preceding moments of time. In particular, the solution of the [[Cauchy problem|Cauchy problem]] at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481704.png" /> for the wave equation is determined only by the initial data at the intersection of the initial manifold with the characteristic cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481705.png" /> and does not depend on the initial data inside and outside the characteristic cone (cf. also [[Characteristic surface|Characteristic surface]]). A rigorous mathematical formulation of Huygens' principle was first given by H. Helmholtz (1859) and by G. Kirchhoff (1882) for the stationary and non-stationary cases, respectively.

The results of J. Hadamard [[#References|[2]]], according to which the solution of the Cauchy problem for the second-order linear hyperbolic equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481706.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048170/h0481707.png" /> depends only on the initial data at the intersection of the initial manifold with the characteristic conoid if and only if the [[Fundamental solution|fundamental solution]] of (*) has no logarithmic terms, is a generalization of Huygens' principle to include the linear hyperbolic equation (*). For an account of the entire class of equations of the form (*) for which Huygens' principle is valid, see [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Huygens, "Tractise on light" , Moscow-Leningrad (1935) (In Russian; translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N.Kh. Ibragimov, "Huygens' principle" , ''Some problems in mathematics and mechanics'' , Leningrad (1970) pp. 159–170 (In Russian) {{MR|0299961}} {{MR|0269988}} {{ZBL|}} </TD></TR></table>



====Comments====
In a weaker form, Huygens' principle refers to the propagation of singularities of the fundamental solution. Writing the latter as an oscillatory integral (or [[Fourier integral operator|Fourier integral operator]]), it can be explained in terms of the principle of stationary phase, see [[#References|[a1]]], [[#References|[a2]]] and [[Stationary phase, method of the|Stationary phase, method of the]].

An intermediate form is to define a sharp front as a piece of the [[Wave front|wave front]] such that the fundamental solution at one side has a smooth extension over the boundary. For the wave-type operators this only happens when the space dimension is odd, cf. .

The most restricted form, in which the fundamental solution is equal to zero at one side of the wave front, is called the lacunary principle. For operators with constant coefficients this has been discussed in .

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "Fourier integral operators I" ''Acta Math.'' , '''127''' (1971) pp. 79–183 {{MR|0388463}} {{ZBL|0212.46601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" ''Acta Math.'' , '''128''' (1972) pp. 183–269 {{MR|0388464}} {{ZBL|0232.47055}} </TD></TR><TR><TD valign="top">[a3a]</TD> <TD valign="top"> L. Gårding, "Sharp fronts of paired oscillatory integrals" ''Publ. R.I.M.S.'' , '''12. Suppl.''' (1967/77) pp. 53–68 {{MR|0728724}} {{MR|0470502}} {{MR|0470501}} {{ZBL|0542.35086}} {{ZBL|0373.35061}} {{ZBL|0369.35062}} </TD></TR><TR><TD valign="top">[a3b]</TD> <TD valign="top"> L. Gårding, "Sharp fronts of paired oscillatory integrals" ''Publ. R.I.M.S.'' , '''13''' (1977/78) pp. 821 {{MR|0728724}} {{MR|0470502}} {{MR|0470501}} {{ZBL|0542.35086}} {{ZBL|0373.35061}} {{ZBL|0369.35062}} </TD></TR><TR><TD valign="top">[a4a]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients I" ''Acta Math.'' , '''124''' (1980) pp. 109–189 {{MR|1290364}} {{MR|0606062}} {{MR|0470499}} {{ZBL|0208.13201}} {{ZBL|0191.11203}} </TD></TR><TR><TD valign="top">[a4b]</TD> <TD valign="top"> M.F. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients II" ''Acta Math.'' , '''131''' (1973) pp. 145–206 {{MR|0470499}} {{MR|0470500}} {{ZBL|0266.35045}} </TD></TR></table>