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 the simulation, the medium is represented by 81 particles, each one experiencing simple harmonic motion. We could write out 81 separate equations, one for each particle, and our collection of 81 equations would fully describe the wave.

Which parameters would be the same in all 81 equations and which would change?

1. The amplitude is the only one that would stay the same.
2. The angular frequency is the only one that would stay the same.
3. The phase is the only one that would stay the same.
4. The amplitude and angular frequency would stay the same but the phase would vary.
5. The amplitude and phase would stay the same but the angular frequency would vary.
6. The angular frequency and phase would stay the same but the amplitude would vary.
7. All three parameters would change.

Every particle oscillates with the same amplitude and with the same frequency, but the phase angle would be different.

How is it different?

For a wave traveling right, a particle to the right lags behind a 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.

Instead of 81 equations, one for each particle, this one works for the entire wave.

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.

What if our particle at x = 0 has some non-zero displacement at t = 0? Then we'd have to use:

y(0,t) = A sin(ωt + φ)

The phase angles for all the other particles would also be shifted by the same amount, so our final equation describing a wave moving right is:

y(x,t) = A sin(ωt - kx + φ)

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

y(x,t) = A sin(kx - ωt + φ)

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