Thermal wave propagation in finite velocity

Hello All,

For the Cattaneo's hyperbolic heat equation with relaxation time,

usually a simple plane thermal wave Τ=T0 * exp[ ¡ (kx - ωt) ], with definition k is the (complex) wave number and ω is the the (real) frequency, is adopted to obtain the dispersion relations of the heat equation.

And consequently, the velocity is defined as: [ω / Re(k)].

My confusion is that is it necessary or obligatory to let k be complex and w be real?

If the answer is yes, then why?

If no, then it's possible to let both k and w are real?

Thanks in advance and welcome for all comments.

Weijie