Simple Harmonic Oscillations and Hook's Law

As I mentioned in my first post, the sine wave is the basic building block of sound; this idea is developed through an understanding of Fourier's Theorem. A sine wave models simple harmonic motion, and because of this, understanding simple harmonic motion is critical to understanding sound. In this post, I am temporarily moving away from the direct physics of sound in order to focus on this foundational concept in another context.

The most traditional way of modeling the sine wave, through a simple harmonic oscillator, requires a simple setup that involves a spring and a mass. The spring hangs from a surface, and the mass is attached to the bottom. One then applies a force to the spring, by pushing it up or pulling it down. An interesting concept that occurs in this system is Hook's Law, which states that the displacement of the object on the spring is directly proportional to the force applied to the string. This is represented by the equation F = kX, where k is the spring constant, X is the displacement, and F is the force. The integral of this equation, or an equation derived with geometry, shows that the elastic energy of the spring is equal to x*x*k/2. The elastic energy stored in the spring causes the mass to bounce up and down in a sinusoidal manner. I was able to record the displacement over time of a mass on a spring:

The shape of the simple harmonic motion is clear in the spring system, and it is interesting to compare this graph to a musical wave. The following image is taken from my post about the sound of an ocarina and depicts the waveform of an ocarina through the relative air pressure over time: 

I would argue that based on this visual comparison, an ocarina acts as a better and more interesting simple harmonic oscillator based on the purity of the shape graph that it generated, but both systems model simple harmonic motion very well.

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.