Rituraj Tiwari
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−ωt)or of(kx+ωt). 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. takef(kx−ωt). Ift increases, then you can only keep the phase constant by increasingxx. So this wave travel towards positivex asttincreases. Try experimenting withthis simulation. A standing wave cannot be written solely asf(kx−ωt) orf(kx+ωt), 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 positivexxand two towards negativex.y=12(sin[x−t]+sin[x+t]+sin[2x−2t]+sin[2x+2t])
