Question

How long must a flute be in order to have a fundamental frequency of262 Hz(this frequency corresponds to middle C on the evenly tempered chromatic scale) on a day when air temperature is 20°C. It is open at both ends.

Short answer

The length of the tube is 65.4 cm.

Step-by-step solution

Given Data

The fundamental frequency is 262 Hz.

The temperature is \({\rm{T = }}\left( {{\rm{20 + 273}}} \right)\;{\rm{K = 293}}\;{\rm{K}}\).

Calculation of the length of the tube

The resonance frequencies of a tube open at both the ends are,

\(\begin{aligned}{{\rm{f}}_{\rm{n}}}{\rm{ = n}}\frac{{\rm{v}}}{{{\rm{2l}}}}{\rm{,}}\;{\rm{n = 1,2,}}\;{\rm{3,}}\;...\\\end{aligned}\)

The speed of the sound at that temperature is,

\(\begin{aligned}{\rm{v = 331}}\sqrt {\frac{{{\rm{293}}}}{{{\rm{273}}}}} \\{\rm{ = 342}}{\rm{.9}}\;{\rm{m/s}}\end{aligned}\)

The fundamental frequency is

\(\begin{aligned}{\rm{262 = }}\frac{{{\rm{342}}{\rm{.9}}}}{{{\rm{2l}}}}\\{\rm{l = }}\frac{{{\rm{342}}{\rm{.9}}}}{{{\rm{2 \times 262}}}}\\{\rm{ = 0}}{\rm{.6543}}\;{\rm{m}}\\{\rm{ = 65}}{\rm{.4}}\;{\rm{cm}}\end{aligned}\)

Hence, the length of the tube is 65.4 cm.