Consider a single-frequency transverse wave.

Each particle experiences simple harmonic motion in the y-direction. The motion of any particle is given by:

y(t) = A sin(wt + f)

For the simulation 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.

Each particle oscillates with the same amplitude and frequency, but with its own phase angle.

For a wave traveling right, particles to the right lag behind particles to the left. The phase difference is proportional to the distance between the particles. If we say the motion of the particle at x=0 is given by:

y(0,t) = A sin(wt)

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

y(x,t) = A sin(wt - 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 equation works for the entire wave.

For a wave traveling left we'd have +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 2p.

kx = 2p when x = l, so:

the wave number

k

=

2p l

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

the angular frequency

w

=

2p T

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