Consider a single-frequency transverse wave.

Each particle in the medium experiences simple harmonic motion in the y-direction. The motion of an individual particle is described by:

y(t) = A sin(ωt + φ)

In general, different particles in the medium oscillate out of phase with one other. For a wave traveling right, the particle to the right lags behind the particle to the left.

The phase difference is proportional to the distance between the particles. If we say the motion of a particle at x=0 is given by:

y(0,t) = A sin(ωt)

The motion of a particle at another x-value is:

y(x,t) = A sin(ωt - kx)

where k is a constant known as the wave number. Note: this k is not the same as the k we used for spring constant.

For a wave traveling left the motion of the particle to the right is ahead of that of the particle to the left, and we'd have a +kx for the phase instead.

What is this k thing anyway? A particle a distance x = 1 wavelength away from another particle would have a phase difference of 2π.

kx = 2π when x = λ, so:

the wave number

k

=

2π λ

The wave number is related to wavelength the same way the angular frequency is related to the period.

Our final equation describing a wave moving right is:

y(x,t) = A sin(ωt - kx)

This is completely equivalent to the book's general equation for a wave:

y(x,t) = A sin(kx - ωt)

Any function where the x and t dependence is of the form (kx - ωt) represents a traveling wave of some shape.