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22 Physics Formulary by ir. J.C.A. Wevers 5.4 Green functions for the initial-value problem This method is preferable if the solutions deviate much from the stationary solutions, like point-like excitations. Starting with the wave equation in one dimension, with 2 = 2 / x 2 holds: if Q ( x,x ± ,t ) is the solution with initial values Q ( ± , 0) = δ ( x - x ± ) and Q ( ± , 0) t =0 , and P ( ± ) the solution with initial values P ( ± , 0) = 0 and P ( ± , 0) t = δ ( x - x ± ) , then the solution of the wave equation with arbitrary initial conditions f ( x )= u ( x, 0) and g ( x u ( x, 0) t is given by: u ( x,t ± -∞ f ( x ± ) Q ( ± ) dx ± + ± -∞ g ( x ± ) P ( ± ) dx ± P and Q are called the propagators . They are deFned by: Q ( ± 1 2 [ δ ( x - x ± - vt )+ δ ( x - x ± + )] P ( ± ² 1 2 v if | x - x ± | <vt 0 if | x - x ± | >vt ±urther holds the relation: Q ( ± P ( ± ) t 5.5 Waveguides and resonating cavities The boundary conditions for a perfect conductor can be derived from the Maxwell equations. If ± n is a unit vector
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

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