The Equation for Motion of a Simple Resonant System

So far, we've spent our time making observations of the mechanical disturbance corresponding to sound. This disturbance, we found, can generally be described as motion that oscillates. We have now to consider the causes of this oscillatory motion -- why oscillations, and why different oscillations for different instruments? The answer lies in an expression we call thewave equation.

E-WORD DETECTED, ALL SYSTEMS SHUTTING DOWN!!

Hold on - there's actually very little math involved. I'll show you. First, let's take a trip back in time to your high school physics class....dootly-doot, dootly-doot... OK we're back. So why oscillations? Let's consider what happens when I pluck the string of guitar. I pull on the string and let it go. The string is elastic so there is a force (the elastic force) that acts upon the string to return it to its original position. Back it goes.... But wait! Does it stop? No! It goes right through its original position and continues going. Why? - because of Newton's First Law. The string has gained inertial force that carries it through its original position. There's more. If the elastic force (Fe = kx) and the inertial force (Fi = ma) are the only two forces acting upon the string, then according to Newton's Third Law they must always counterbalance one another, ma = -kx.

OK, SO WHAT?

Well this happens to be the equation for motion of the string. If we study it a little closer we notice that only acceleration (a) and displacement (x) of the string are changing over time, both string mass (m) and stiffness (k) are constant. This means that displacement is always opposite in direction to acceleration.

Question: For what kind of motion is displacement always opposite in direction to acceleration?

THATS IT!! Simple harmonic motion! Sinusoidal motion! The string oscillates sinusoidally because sinusoidal motion is a type of motion that satifies the equation for motion.

Keywords to watch for: resonance, frequency, equilibrium