Mathematical Descriptions of Opacity - Complex Wavenumber, Propagation Constant

Complex Wavenumber, Propagation Constant

Another way to incorporate attenuation is to use essentially the original expression:

but with a complex wavenumber (as indicated by writing it as instead of k). Then the intensity of the wave satisfies:

i.e.,

Therefore, comparing this to the absorption coefficient approach,

,

(k is the standard (real) angular wavenumber, as used in any of the previous formulations.) In accordance with the ambiguity noted above, some authors use the complex conjugate definition,

A closely related approach, especially common in the theory of transmission lines, uses the propagation constant:

where is the propagation constant.

Comparing the two equations, the propagation constant and complex wavenumber are related by:

(where the * denotes complex conjugation), or more specifically:

(This quantity is also called the attenuation constant, sometimes denoted .)

(This quantity is also called the phase constant, sometimes denoted .)

Unfortunately, the notation is not always consistent. For example, is sometimes called "propagation constant" instead of, which swaps the real and imaginary parts.