There are 3 important variables for a wave, namely, i) the height y of the wave, ii) the distance x that the wave travels and iii) the time t that the wave travels. When visualizing a wave we usually think of a y - x plot or a y - t plot. The y - x plot represents an instant of time t and is similar to a photograph or snapshot of a water wave that we would take at the beach. The distance between wave crests (that we could measure from our snapshot) is called the wavelength λ. The y - t plot represents a single location x and is similar to a movie of a buoy bobbing up and down in the water as a wave passes through. The buoy is anchored to the ocean floor at a fixed distance x. The time it takes the buoy to bob up and down once is called the period T of the wave. Thus, to summarize, λ is determined from the y - x graph (instant of time t) whereas T is determined from the y - t graph (fixed distance x). Thus y is a function of both x and t, written as y(x, t). Now the y - x graph can be written

where we have taken the instant of time to be t = 0. The reason we have written sin kx and not just sin x is because the domain of the sine function is an angle. We can only ever have sin θ where θ is an angle. Thus we cannot write sin x because x is not an angle. Actually x is a distance with units of m. However we want to use x as a plotting variable. To do this we have to multiply it by something called k, so that the quantity kx is an angle, i.e. θ ≡ kx. Now what is k? Well if kx is an angle then after one complete wave cycle, the angle kx must be 2π. Now after one complete cycle the distance the wave moves is x = λ. Thus we must have

or

giving

which is called the wave number. Similarly, the y - t graph can be written

where we have taken the fixed distance to be x = 0. We did not write sin t because t is not an angle, whereas t is an angle.  is the angular speed that we have discussed before. Again after one complete wave cycle t must be 2π and after one cycle the time t will just be one period T. Thus we must have

or

giving

which is often called the angular frequency . We previously defined  wave can be written

Does this agree with what we had before? Yes. We can see that y(x, 0) = y_m sin kx and y(0, t) = y_m sin t.