Determine in what way and how many times will the fundamental tone frequency, of a stretched wire, change, if its length is shortened by $35\%$ and the tension is increased by $70\%$.

To determine the sound propagation velocity in air by acoustic resonance technique, one can use a pipe with a piston and a sonic membrane, closing one of its ends. Find the velocity of sound, if the distance between the adjacent positions of the piston, at which resonance is observed, at a frequency $v=2000$ $\mathrm{Hz}$, is equal to $l=8.5$ $\mathrm{cm}$.

Find the number of possible natural oscillations of air column in a pipe, whose frequencies lie below $v_0=1250$ $\mathrm{Hz}$. The length of the pipe is $l=85$ $\mathrm{cm}$. The velocity of sound is $v=340 \mathrm{m} {\mathrm{s}}^{-1}$. Consider the two cases:

(a) the pipe is closed from one end;

(b) the pipe is opened from both ends. The open ends of the pipe are assumed to be the anti nodes of displacement.

A string of mass $m$ is fixed at both ends. The fundamental tone oscillations are excited with circular frequency $\omega$ and maximum displacement amplitude $a_{\max }$. Find:

(a) the maximum kinetic energy of the string;

(b) the mean kinetic energy of the string averaged over one oscillation period.

A stationary observer receives sonic oscillations from two tuning forks, one of which approaches, and the other recedes with the same velocity. As this takes place, the observer hears the beatings with frequency $f_{\mathrm{B}}=2.0$ $\mathrm{Hz}$. Find the velocity of each tuning fork if their oscillation frequency is $f_0=680 \mathrm{Hz}$ and the velocity of sound in air is $v=340$ $\mathrm{m} {\mathrm{s}}^{-1}$.