22Physics Formulary by ir. J.C.A. Wevers5.4Green functions for the initial-value problemThis 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/∂x2holds: ifQ(x, x , t)is the solution withinitial valuesQ(x, x ,0) =δ(x-x)and∂Q(x, x ,0)∂t= 0, andP(x, x , t)the solution with initial valuesP(x, x ,0) = 0and∂P(x, x ,0)∂t=δ(x-x), then the solution of the wave equation with arbitrary initialconditionsf(x) =u(x,0)andg(x) =∂u(x,0)∂tis given by:u(x, t) =∞-∞f(x)Q(x, x , t)dx+∞-∞g(x)P(x, x , t)dxPandQare called thepropagators. They are defined by:Q(x, x , t)=12[δ(x-x-vt) +δ(x-x+vt)]P(x, x , t)=12vif|x-x|< vt0if|x-x|> vtFurther holds the relation:Q(x, x , t) =∂P(x, x , t)∂t5.5Waveguides and resonating cavitiesThe boundary conditions for a perfect conductor can be derived from the Maxwell equations. Ifnis a unitvector⊥the surface, pointed from 1 to 2, andK
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ez, Transversal Magnetic modes, Transversal Electric modes, modes TE m,n