The equation for a progressive wave is given by y=Asin(kx-wt). y is the displacement from equilibrium and A is the amplitude. k is called the wave number and is 2*pi/wavelength. w (omega) is the angular velocity. It is 2*pi/T, where T is the time period. A wave in periodic so w is how fast things go round in a circle, measured in the number of radians covered in a second. So what is k? Well, it must be measured in radians per metre so I'd say that it is a kind of angular gradient. It is about turning a distance into an angle in radians. It's an odd gradient because it is periodic and resets every wavelength. I was thinking about what happens if you choose a fixed position. Take x = 0. Then displacement y = ASin(-wt). The particle just goes down and up on the spot. Now freeze the time. Take time t=0. y=Asinkx. You get the familiar wave shape.
