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Harmonic Oscillators

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Harmonic oscillator

From Wikipedia, the free encyclopedia.

Introduction

A harmonic oscillator is any physical system that varies above and below its mean value with a characteristic frequency, f. Common examples of harmonic oscillators include pendulums, masses on springs, and RLC circuits.

The following article discusses the harmonic oscillator in terms of classical mechanics. See the article quantum harmonic oscillator for a discussion of the harmonic oscillator in quantum mechanics.

Full Mathematical Definition

Most harmonic oscillators, at least approximately, solve the differential equation:

$\frac{d^2x}{dt^2} - b \frac{dx}{dt} + {\omega_0}^2x = A_0 \cos(\omega t)$

where t is time, b is the damping constant, ω_o is the characteristic angular frequency, and A_ocos(ωt) represents something driving the system with amplitude A_o and angular frequency ω. x is the measurement that is oscillating; it can be position, current, or nearly anything else. The angular frequency is related to the frequency, f, by:

$f = \frac{\omega}{2 \pi}$

Simple Harmonic Oscillator

A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:

$\frac{d^2x}{dt^2} + {\omega_0}^2x = 0$

Physically, the above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a spring and an LC circuit.

In the case of a mass hanging on a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:

- ky = ma

where k is the spring constant, m is the mass, y is the position of the mass, and a is its acceleration. Rewriting the equation, we obtain:

$\frac{d^2y}{dt^2} = -\frac{k}{m}y$

The easiest way to solve the above equation is to recognize that when d²z/dt² ∝ -z, z is some form of sine. So we try the solution:

y = Acos(ωt + δ)

$\frac{d^2y}{dt^2} = -A \omega^2 \cos(\omega t + \delta)$

where A is the amplitude, δ is the phase shift, and ω is the angular frequency. Substituting, we have:

$-A \omega^2 \cos(\omega t +\delta) = -\frac{k}{m} A \cos(\omega t + \delta)$

and thus (dividing both sides by -A cos(ωt + δ)):

$\omega = \sqrt{\frac{k}{m}}$

The above formula reveals that the angular frequency of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions (those are represented by A and δ). That means that what was labelled ω is in fact ω_o. This will become important later.