I am a bit confused by the question 9.4 (and similarly 10.2) in the book. It asks to solve the dispersion equation for complex k and hence find the decay length of the wave. I solved for k, using the exponential expression for sin, and got k equal to some fairly nasty imaginary thing with a natural log. But I'm not sure where to go from there to find the decay length (which I'm interpreting as the distance for the wave to decay to 1/e of it's initial amplitude).
I tried subbing this k into the ansatz wave expression, but this didn't make much sense, because there is no distance to solve for - the ansatz wave has the fixed distance n*a in it, not a general x. So I'm lost as to how to find the decay length...