Posted by Steve Simon on April 18, 2014, 9:05 am, in reply to "Decaying waves in 1D chains"
I'm glad you are trying some of these extra problems!
Yes, you are supposed to get a complex k (and solutions should come in complex conjugate pairs). If you write k = Real + i Imag or k=Real - I imag, then the waveform is
e^{i w t - i k n a}
as usual, but k is complex. Rewrite this exponential as
e^{i somethingreal} e^{+n a Imag}
or
e^{i something real} e^{-n a Imag}
which shows this particular wave has a decaying envelope with the decay length set by the imaginary part of k.
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