Question $3(17$ marks$)$

Two identical piano wires have a $7^{th}$ harmonic frequency of $1400$ Hz when kept under the same tension.

$($a$)$ What is their fundamental resonant frequency $($or $1^{st}$ harmonic frequency$)?$

$(2$ marks$)$

$($b$)$ What fractional increase in the tension of one wire will lead to the occurrence of $1$ beats$/$s in their fundamental resonance? $($Modified from tutorial exercise$)$

$(9$ marks$)$

$($c$)$ A tube has its $3^{rd}$ harmonic resonant frequency matching the $5^{th}$ harmonic resonant frequency of the piano wires under the original tension. If the tube is open at one end only, what is the length of this tube? Given sound speed in air $=343\frac{m}{s}.$

$(6$ marks$)$