We'll now turn our attention to transverse waves, and look at a single-frequency sine wave.

Focus on a single particle in the medium - if the wave is traveling in the x-direction it experiences simple harmonic motion in the y-direction. The motion of an individual particle in the wave can therefore be described by an equation of the form:

y(t) = A sin(ωt + δ)

Now compare two different particles in the medium. They both experience the same simple harmonic motion but, in general, they're out of phase with each other. For a wave traveling left-to-right, the particle to the right is behind the particle to the left.

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

y(t) = A sin(ωt)

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

y(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 right-to-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

Our final equation describing a wave moving to the right is:

y(x,t) = A sin(ωt - kx)

A is the amplitude of the wave
The angular frequency ω = 2π/T
The wave number k = 2π/λ

One final note. The book's general equation for the wave is:

y(x,t) = A sin(kx - ωt)

For our purposes the two equations are equivalent.

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