Question

A uniform narrow tube \({\bf{1}}{\bf{.70}}\;{\bf{m}}\) long is open at both ends. It resonates at two successive harmonics of frequencies \({\bf{275}}\;{\bf{Hz}}\) and \({\bf{330}}\;{\bf{Hz}}\) . What is (a) the fundamental frequency, and (b) the speed of sound in the gas in the tube?

Short answer

(a) The fundamental frequency is \(55\,{\rm{Hz}}\)

(b) The velocity of the sound wave in tube is \(187\;{\rm{m/s}}\).

Step-by-step solution

Formula of the fundamental frequency

The fundamental frequency is referred to simply as the fundamental, and it is described as the least frequency of a periodic waveform.

The formula for the fundamental frequency for standing waves is,

\(f = \frac{v}{{2l}}\).

Given information

Given data:

The two successive harmonics are\({f_1} = 275\;{\rm{Hz}}\;{\rm{and }}{f_2} = 330\;{\rm{Hz}}\).

The length of the tube is \(l = 1.70\;{\rm{m}}\).

Calculation of fundamental frequency

The expression for the fundamental frequency is given as,

\(\begin{array}{c}f = {f_2} - {f_1}\\f = 330\;{\rm{Hz}} - 275\;{\rm{Hz}}\\f = 55\;{\rm{Hz}}\end{array}\)

Thus, the fundamental frequency is \(55\;{\rm{Hz}}\).

Calculation of velocity of sound wave

The expression for the fundamental frequency is given as,

\(\begin{array}{c}f = \frac{v}{{2l}}\\v = 2fl\end{array}\)

Substitute the values in the above equation,

\(\begin{array}{c}v = 2\left( {1.70\;{\rm{m}}} \right)\left( {55\;{\rm{Hz}}} \right)\\ = 187\;{\rm{m/s}}\end{array}\)

Thus, the velocity of the sound wave in tube is \(187\;{\rm{m/s}}\).