Waves/Waves in One Dimension

The Mathematics Of Waves
We start our discussion of waves by taking the equation for a very simple wave and describing its characteristics. The basic equation for such a wave is


 * $$y=a\ \sin \left(\frac{2\pi x}{\lambda} - 2\pi ft +\alpha \right)$$

where $$y$$ is the height of the wave at position $$x$$ and time $$t$$. This equation describes a fairly simple wave, but most complex waves are just sums of simpler ones. If we freeze this equation in time at $$t=0$$, we get


 * $$y=a\ \sin \left(\frac{2\pi x}{\lambda} +\alpha \right)$$

which looks like this: [TODO - Add a Graph]

From the graph we can see that each of the three parameters has a meaning. $$a$$ is the amplitude of the wave, how high it is. $$\lambda$$ is the wavelength, the distance from a part of the wave in one cycle to the same part of the wave in the next cycle. $$\alpha$$ is the phase of the wave, which shifts the wave to the left or right. The wavelength is a distance, and is usually measured in meters, millimeters or even nanometers depending on the wave. Phase is an angle, measured in radians.

Now that we have mapped out the wave in space, let's instead set $$x=0$$ and see how the wave changes over time


 * $$y=a\ \sin(- 2\pi ft +\alpha)$$

Amplitude $$a$$ and phase $$\alpha$$ remain, but the wavelength is gone and a new quantity has appeared: $$f$$, which is the frequency, or how rapidly the wave moves up and down. Frequency is measured in units of inverse time: in a fixed period of time, how many times does the wave move up and down? The unit usually used for this is the hertz, or inverse second.

Now let's combine these two pictures and see how the wave moves. Figure 3 is a diagram of how the wave looks when you plot it in both space and time. The straight lines are the places where the simple wave reaches a maximum, minimum, or zero (where it crosses the x axis).

We can look at the zeros to determine the phase velocity of the wave. The phase velocity is how fast a part of the wave moves. We can think of it as the speed of the wave, but for more complicated waves it is only one type of speed - more on that in later sections.

We can get an equation for the zeros by setting our equation to zero.


 * $$0=a\ \sin \left(\frac{2\pi x}{\lambda} - 2\pi ft +\alpha \right)$$


 * $$0=\frac{2\pi x}{\lambda} - 2\pi ft +\alpha$$


 * $$x = f \lambda t - \frac{\alpha \lambda}{2 \pi}$$

You see here that we have the equation for a straight line, describing a point that is moving at velocity $$f\lambda$$. This gives us the equation for the phase velocity of the wave, which is


 * $$\mbox{velocity}=\mbox{frequency} \times \mbox{wavelength} \quad v=f\lambda$$