Traveling Waves

I. Plane Waves.

We can describe a traveling wave with the following function. We use x for the spatial coordinates and t for the time.

. (1)

Where k is called the wave number and ω is the angular speed. If we measure the spatial length of the wave we call that the wavelength, λ. Also, T is the period of the wave. So:

, and . (2)

Let’s do some algebra and simplify:

(3)

It turns out that is the speed of our traveling wave! So

, (4)

describes a wave traveling with a speed . More accurately this is called a plane wave: it is a wave that moves over a flat area in a straight line. Its wavefronts are all flat.

II. Spherical Waves.

When you clap your hands, throw a rock in a pond, or turn on a light bulb, waves go out circularly from the center. We can find the shape of the wave a radial distance r from the center:

. (5)

From this equation we see the wave size gets smaller the farther we get from the center. Can you easily plot this?