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?