Why Is Phase Velocity Defined as v=w/k?

[asdf1]

why is phase velocity v=w/k ?

 

[arildno]

Let us look at a typical harmonic wave, $$f(x,t)=a\sin(kx-\omega{t})$$
Here, a,k,$$\omega$$ are constants, where a is the amplitude, k the wave number and $$\omega$$ the frequency.

Rewrite this in the following form:
$$f(x,t)=a\sin(k(x-\frac{\omega}{k}t))$$
Do you see that if $$x-\frac{\omega}{k}t=s$$ where s is a CONSTANT, makes the value of f constant as well (equal to $$a\sin(ks)$$)?
But that means, that the signal value $$a\sin(ks)$$ can be regarded as MOVING ALONG THE POSITIVE X-DIRECTION WITH VELOCITY $$\frac{\omega}{k}$$!
For, (remembering that s is constant) we have the trajectory for our signal:
$$x=s+\frac{\omega}{k}t$$
and this simply shows what the propagation velocity of our signal is..

 

[asdf1]

@@a
"But that means, that the signal value can be regarded as MOVING ALONG THE POSITIVE X-DIRECTION WITH VELOCITYw/k !"
can you explain that a little clearer?