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 that we will look at is

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

Here $y$ describes the height of the wave at position $x$ and time $t$.

Although this is only a specific type of wave, looking at this equation is particularly valuable since as we shall see later, much more complex waves can be treated as a sum of simple waves. Now if we freeze this equation in time, $t=0$, we get

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

Graphing this out we get: [TODO - Add a Graph]

From figure one 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, which is how wide the wave is in one cycle. $\alpha$ is the phase of the wave, essentially what the offset is. Wavelength is measured in length. Phase is an angle which you can measure in degrees or radians.

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

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

We see that we still have our amplitudes $a$ and phase $\alpha$, but we have a new quantity, $f$, which is the frequency, or how rapidly the wave moves up and down. Frequency is measured in units of inverse time. In other words how many times does the wave move up and down in one unit of time. Typically, we can talk about this unit as the hertz, which is the number of times per second the wave moves up and down.

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

We can look at the zero crossings to get a value for the phase velocity of the wave. The phase velocity is how fast a part of the wave moves across. We can think of it as the speed of the wave, although for more complicated waves it is only one type of speed. We'll talk more about that later.

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

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

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

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

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

$\text{velocity}=\text{frequency}\times \text{wavelength}v=f\lambda$

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