# Undamped oscillations of sound wave

I read in google that When we hit some metal or object, then a sound is generated by that object. If we hit that object with more force, then we can hear a sound of more amplitude than previous one but with same frequency. that means frequency won't change in either of these cases. Frequency means vibrations of atoms inside that metal,my question is why those atoms always vibrate with same frequency even when we apply large force, why only amplitude changes?

What you say is generally true if you consider the linear nature of resonant materials. But no materials are perfectly linear. Furthermore the frequency is not the frequency at which individual atoms resonate, but rather the system of atoms that form a resonant structure.

The shape, size, etc also has to do with what frequency you get. In real structures you can have many resonant frequencies also called modes. Modes can get filled with energy and spill into other modes. Therfore you don't always get the same frequency if you continue to bang harder

As docscience has already commented, it's actually not true in general. Usually the linear models in which the frequency is amplitude independent are accurate enough for many purposes but not for all of them. I would recommend you to study some weak nonlinear systems where the hybrid solution of slightly generalised linear solution are good enough.

Example: The linear mathematical pendulum of length $\ell$ have (angular) eigenfrequency:

$$\omega = \sqrt{\frac{g}{\ell}}$$

and the deflection angle $\phi$:

$$\phi=A\cos \omega t$$

so the frequency and amplitude are independent but the models works only for small amplitudes. It could be generalised:

$$\phi = A\cos \omega t - \frac{1}{192}A^3\cos 3\omega t$$

with the angular eigenfrequency:

$$\omega=\sqrt{\frac{g}{\ell}}(1-\frac{1}{16}A^2)$$

and that's amplitude dependent! I am not going to put the whole derivation here but basically you need to get one more term of Taylor's polynome of sine and the neglect all terms of $O(A^4)$ or higher.

Generally the large amplitudes can even excite different vibrational modes. Example: if you pluck the real 3D string with small force, you can excite only one (or almost only one) linear polarization of vibration. In large amplitudes, you practically always excite both of them, because their eigenfrequencies are close enough and resulting oscillation in eliptically polarised.

• Sound waves can be elliptically polarized? Dec 8, 2015 at 18:16
• @Viktor Are you refering to the last example? That is string, i.e. mechanical wave of an elastic object. Dec 8, 2015 at 18:30