Simple Harmonic Progressive Wave

[SDewan]

Given a Simple Harmonic Progressive Wave with the equation y=A*sin(ωt-kx+φ) where A is amplitude, k is wave number, ω is frequency of wave and φ is the initial phase.
How to determine in what direction is the wave propagating?

 

[andrewkirk]

Consider a wave crest. If you sit on the wave crest then your phase (the value inside the parenthesis above) always remains the same. So as time goes by, the change in the distance term $-kx$ must exactly offset the increase in the time term $\omega t$.
So the crest must be traveling in the positive x direction, as long as k is positive, and vice versa if k is negative.
A more realistic scenario might be sitting on the point of the wave that has greatest downward slope - half way between crest and trough. That's where a surfer would be!