Huygens principle

The statement that the propagation of oscillations, described by the 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 [1] and was subsequently developed by A. Fresnel in 1818 in his studies on the problems of diffraction (cf. also Diffraction, mathematical theory of).

Huygens' principle results from the mathematical fact that the solution of the wave equation at a point $ M $ of the three-dimensional space at the moment of time $ t $ is expressed in terms of the values of the solution and its derivatives on an arbitrary closed surface inside which $ M $ is contained, at preceding moments of time. In particular, the solution of the Cauchy problem at the point $ ( M, t) $ for the wave equation is determined only by the initial data at the intersection of the initial manifold with the characteristic cone of $ ( M, t) $ and does not depend on the initial data inside and outside the characteristic cone (cf. also 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 [2], according to which the solution of the Cauchy problem for the second-order linear hyperbolic equation

$$ \tag{* } \sum _ {i, j = 1 } ^ { {n } - 1 } g ^ {ij} ( x) u _ {ij} + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) u _ {i} + c ( x) u = 0 $$

for even $ n \geq 4 $ depends only on the initial data at the intersection of the initial manifold with the characteristic conoid if and only if the 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 [4].

References

[1]	C. Huygens, "Tractise on light" , Moscow-Leningrad (1935) (In Russian; translated from French)
[2]	J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) (Translated from French)
[3]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[4]	N.Kh. Ibragimov, "Huygens' principle" , Some problems in mathematics and mechanics , Leningrad (1970) pp. 159–170 (In Russian) MR0299961 MR0269988

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), it can be explained in terms of the principle of stationary phase, see [a1], [a2] and Stationary phase, method of the.

An intermediate form is to define a sharp front as a piece of the 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

[a1]	L. Hörmander, "Fourier integral operators I" Acta Math. , 127 (1971) pp. 79–183 MR0388463 Zbl 0212.46601
[a2]	J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" Acta Math. , 128 (1972) pp. 183–269 MR0388464 Zbl 0232.47055
[a3a]	L. Gårding, "Sharp fronts of paired oscillatory integrals" Publ. R.I.M.S. , 12. Suppl. (1967/77) pp. 53–68 MR0728724 MR0470502 MR0470501 Zbl 0542.35086 Zbl 0373.35061 Zbl 0369.35062
[a3b]	L. Gårding, "Sharp fronts of paired oscillatory integrals" Publ. R.I.M.S. , 13 (1977/78) pp. 821 MR0728724 MR0470502 MR0470501 Zbl 0542.35086 Zbl 0373.35061 Zbl 0369.35062
[a4a]	M.F. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients I" Acta Math. , 124 (1980) pp. 109–189 MR1290364 MR0606062 MR0470499 Zbl 0208.13201 Zbl 0191.11203
[a4b]	M.F. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients II" Acta Math. , 131 (1973) pp. 145–206 MR0470499 MR0470500 Zbl 0266.35045