# Undamped Resonance of a Classical Harmonic Oscillator

Consider an undamped harmonic oscillator. It may be driven at it's natural frequency, $\omega_0^2 = \frac{k}{m}$.

According to Feynman, and other sources, were this to happen, the amplitude of the oscillations would be infinite. I have tried solving the differential equation myself, and the result I got seemed slightly different.

In my solution, the amplitude of the oscillations is directly proportional to time, meaning that the oscillations will get very big quite quickly, and will only become infinite as $t \to \infty$. This page seems to confirm my result:

In contrast, the second case, $\omega = \omega_0$, will have some serious issues at t increases. The addition of the t in the particular solution will mean that we are going to see an oscillation that grows in amplitude as t increases. This case is called resonance and we would generally like to avoid this at all costs.

Am I correct in this? I haven't found any other material which supports this idea, but perhaps I haven't looked hard enough.

• It's $\omega_0^2=k/m$ or $\omega_0=\sqrt{k/m}$. I've proposed an edit to that affect. – eepperly16 Jun 11 '15 at 17:06

Absolutely. What Feynman means when he says the amplitude is infinite is indeed that the amplitude grows limitless as $t \to \infty$.