How to identify a travelling wave?

Question

I have the following equation: $u(x,t)= e^{-t} f(x-ct)$ . I have to figure out whether it is a traveling wave or not? The amplitude of the wave in this case is time-dependent i.e. $e^{-t}$. Does it still qualify as a traveling wave even though it is not in the form $g(x-ct)$? Please suggest.

Answer 1

The wave you write respects the identity:

$u(x,t)=u(x-vt,0)e^{-t}=u_0(x-vt)e^{-t}$

where I called $u_0(x)=u(x,0)$ the wave at time $0$.

The wave at time $t$ can be obtained by the wave at time $t=0$ first translating $u_0$ by a distance $vt$, an operation that does not change its shape, and than multiplying by a uniform decaying factor $e^{-t}$ that does not change the shape but brings the amplitude to zero.

If we made a video, this wave would vanish over time. I would leave to other users/formal books to define whether this is a travelling wave or not, even if in my opinion it is not.