The Building Blocks of a B-Spline

Properties of Waves

So far, I've talked generally about sine waves, but I haven't really shown an equation for them. The following shows an equation for a sine wave including all the properties that shape the wave. Figure 2.5 shows a plot of the wave, labeling those properties.

Figure 2.5: Properties of waves.

(2.2) Sine wave equation. 

The sine function gives the overall shape, as shown in Figure 2.2. The amplitude parameter defines the maximum and minimum values for the values on the plot. If you refer back to Figure 2.1, the amplitude is equal to the radius of the circle.

The wavelength is the length of one cycle of the wave. If you refer back to Figure 2.2, the wavelength of the basic sine function is obviously 2pi (360 degrees). After that, the wave repeats itself. The inverse of the wavelength is the frequency, or how many times the wave repeats itself in a certain interval. A shorter wavelength equals a higher frequency because more waves can fit in a given interval. In Equation 2.2, the frequency scales the angle value. If you plot this, you will see that this scaling factor causes more or less waves to be drawn in the same interval.

Finally, the phase is an offset. It shifts the wave along the x-axis by offsetting the angle parameter. As I pointed out before, the cosine wave is just an offset version of a sine wave. The following equation describes cosine as a function of sine.

(2.3) Cosine as a function of sine. 

Once you understand these properties, you'll be able to use sine waves in many different ways. In the next section, I will talk about a few of them, but I encourage you to experiment and become familiar with these concepts.