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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 ( x, x , 0) = δ ( x - x ) and Q ( x, x , 0) t = 0 , and P ( x, x , t ) the solution with initial values P ( x, x , 0) = 0 and P ( x, x , 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 ( x, x , t ) dx + -∞ g ( x ) P ( x, x , t ) dx P and Q are called the propagators . They are defined by: Q ( x, x , t ) = 1 2 [ δ ( x - x - vt ) + δ ( x - x + vt )] P ( x, x , t ) = 1 2 v if | x - x | < vt 0 if | x - x | > vt Further holds the relation: Q ( x, x , t ) = P ( x, x , t ) 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 the surface, pointed from 1 to 2, and K
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