Harmonic oscillator

Remark — Application to physics: the harmonic oscillator

The equation $y'' + \omega^2 y = 0$ has characteristic equation $\lambda^2+\omega^2=0$, with roots $\lambda=\pm i\omega$ ($\alpha=0$, $\beta=\omega$). The solution is: $y = C_1\cos(\omega x)+C_2\sin(\omega x) = A\cos(\omega x + \varphi),$ that is, a harmonic oscillation of frequency $\omega$.

With damping — $y''+2\gamma y'+\omega^2 y=0$ — the roots have real part $\alpha=-\gamma<0$: the oscillation is damped exponentially. It is the model of the spring with friction.

The graph compares the undamped oscillation $\cos(\omega x)$ with the damped one $e^{-\gamma x}\cos(\omega x)$, enclosed between the two envelope curves $\pm e^{-\gamma x}$.

Undamped (blue) and damped (red) harmonic oscillation: the amplitude of the latter decays within the envelope $\pm e^{-\gamma x}$.

Links

Topics: Differential equations
Concepts: Second-order linear ODE · Characteristic equation · Harmonic oscillator · Damping
Functions: Cosine
Skills: Interpreting a graph · Solving equations