By Jean-Claude Nedelec

This ebook is dedicated to the learn of the acoustic wave equation and of the Maxwell method, the 2 most typical wave equations encountered in physics or in engineering. the most aim is to offer a close research in their mathematical and actual homes. Wave equations are time established. despite the fact that, use of the Fourier trans shape reduces their learn to that of harmonic structures: the harmonic Helmholtz equation, in relation to the acoustic equation, or the har monic Maxwell process. This publication concentrates at the examine of those harmonic difficulties, that are a primary step towards the examine of extra normal time-dependent difficulties. In each one case, we supply a mathematical atmosphere that permits us to turn out life and forte theorems. we have now systematically selected using variational formulations relating to concerns of actual strength. We learn the crucial representations of the strategies. those representa tions yield a number of fundamental equations. We examine their crucial homes. We introduce variational formulations for those critical equations, that are the root of such a lot numerical approximations. varied elements of this booklet have been taught for a minimum of ten years by way of the writer on the post-graduate point at Ecole Poly method and the collage of Paris 6, to scholars in utilized arithmetic. the particular presentation has been demonstrated on them. I desire to thank them for his or her energetic and optimistic participation, which has been tremendous necessary, and that i express regret for forcing them to profit a few geometry of surfaces.

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**Additional resources for Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems**

**Example text**

7) 42 2. 5) are equivalent. We denote by Bi the domain interior to the sphere S and by Be the domain exterior to S. We recall that for any positive integer k, Hk(Bi) is the space of functions such that all the derivatives up to the order k are in L2(Bi). We denote by Dku the operator of derivation of order k, which associates with the function u all its k-order partial derivatives. When k = 1, it is also the operator \i'u. We can now give explicit solutions to these Dirichlet problems. 9) ulYim(B,cp)rl, 1=0 m=-!

We denote by Bi the domain interior to the sphere S and by Be the domain exterior to S. We recall that for any positive integer k, Hk(Bi) is the space of functions such that all the derivatives up to the order k are in L2(Bi). We denote by Dku the operator of derivation of order k, which associates with the function u all its k-order partial derivatives. When k = 1, it is also the operator \i'u. We can now give explicit solutions to these Dirichlet problems. 9) ulYim(B,cp)rl, 1=0 m=-! and this series is absolutely convergent for r < 1.

2ll! 4. 115) For m > 0, and r = 1, the derivatives of HI are given by aHI aX3 _ 2 =i + 1/2 1- 1/2 l Jp _ m { aX2 =~ 2 l-I' l + 1/2 [ . 118) I)Yi~11]. 112) follows from the homogeneity of the polynomial HI' which must satisfy HI(x) = rlytm (~). 31). 80). 121 ) 30 2. 124) = -2/(l + m)(l + m d - 1) ( dx )1+m-2 (1 - X2)1-1. 125) I L (-l)P p=E( 1+'~±1) I = L p=E( 1+'~±2) (-l)P l! (2p)! (2p-I-m)! 126) l- m ' l! (2p)! x2p-l-m. (1 - p)! (2p -I - m - I)! 4. 121) is obtained by subtraction. The two relations ( 1 d )l+m+l (1 - X 2 )l dx l-1 (2 2)' "(-l)PCP+l p+ .