How to determine whether a wave is travelling or standing?
I have been told that wave function of the form $f(k_1x \pm k_2t)$ is a travelling wave.
But then even a standing wave equation $2A \cos \omega t \sin kx$ can be written as superposition of two waves
$$A\sin\frac{kx-\omega t}{2} + A\sin\frac{kx+\omega t}{2}$$
This is of the form $f(k_1x \pm k_2t)$ but still not a travelling wave.
In particular is $y=\cos x \sin t \ + \cos 2x \sin 2t$ a travelling wave?
Best Answer
The "rule" you have given is a little simplistic. To use it you have to be able to write the wave solely as a function of $(kx-\omega t)$ or of $(kx + \omega t)$. That is because the thing in the brackets, the phase of the wave, has to be kept constant to apply a meaning to a direction of travel.
e.g. take $f(kx-\omega t)$. If $t$ increases, then you can only keep the phase constant by increasing $x$. So this wave travel towards positive $x$ as $t$ increases. Try experimenting with this simulation. A standing wave cannot be written solely as $f(kx-\omega t)$ or $f(kx+\omega t)$, so is not a wave travelling in a single direction. It is the superposition of two waves of equal frequency and amplitude travelling in opposite directions.
In general, waves can always be written as the superposition of multiple waves travelling in different directions.
Your final example can be decomposed into 4 travelling waves of the same speed, but different wavelengths, 2 travelling towards positive $x$ and two towards negative $x$. $$ y = \frac{1}{2}(\sin [x-t] + \sin[x+t] +\sin[2x-2t] + \sin[2x+2t])$$