General Fourier solution and causality in attenuating media

It has been previously reported that a general electric field solution and its initial condition, E/sub g/(z,t) and E/sub g/(z,0), respectively, are not causal when formed by a superposition of time-harmonic waves in an attenuating medium. However, this is not the case. Further, the relationship between attenuation and phase velocity as well as their dependence on frequency arise simply from the form chosen for the time harmonic particular solutions. Even though causality is not introduced during the solution to the wave equation, the general solution can subsequently be shown to be a time convolution of a causal boundary condition (time history of the electric field as it crosses the z=0 plane, E/sub g/(0,t)), and the medium's impulse response g(z,t), which can be shown to be causal. Hence, the general solution is also causal. The initial condition occurs at the instant, t=0, when the electric field arrives at the z=0 plane, and it has been previously reported that the initial condition depends on the boundary condition for times after the initial time thereby violating causality. A re-examination shows that the initial condition does not depend on times after the initial time. Hence, the initial condition obeys causality, and it can also be shown to be properly determined (E/sub g/(z,0)=0 for z>0) even when the boundary condition is not zero. It has also been reported that limiting expressions for the boundary and initial conditions, E/sub g/(0,t/spl rarr/0) and E/sub g/(z/spl rarr/0,0), respectively, are not equal. However, a re-examination reveals that E/sub g/(0,t/spl rarr/0)=E/sub g/(z/spl rarr/0,0).<
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