Question

Question :(I) If you were to build a pipe organ with open-tube pipes spanning the range of human hearing (20 Hz to 20 kHz), what would be the range of the lengths of pipes required?

Short answer

The pipe length varies from $8.65\mathrm{m}$ to $8.65\times {10}^{-3}\mathrm{m}$ with the frequencies $20\mathrm{H}\mathrm{z}$ to $20000\mathrm{H}\mathrm{z}$.

Step-by-step solution

Step-1:- Understanding the frequency of an open pipe

The frequency of an open pipe gets influenced by the difference in speed of the wave and the wavelength.

Step-2:-Given data

The lowest frequency range of human hearing is $f_{\mathrm{m}\mathrm{i}\mathrm{n}}=20\mathrm{H}\mathrm{z}$.

The highest frequency range of human hearing is $f_{\mathrm{m}\mathrm{a}\mathrm{x}}=20000\mathrm{H}\mathrm{z}$.

Step-3:-Calculation of the pipe length

The length of the pipe for the minimum frequency is calculated as,

$\begin{array}{c}{l}_1=\frac{\nu }{2f_{\mathrm{m}\mathrm{i}\mathrm{n}}}\\ =\frac{346\mathrm{m}/\mathrm{s}}{2\times \left(20\mathrm{H}\mathrm{z}\right)}\\ =8.65\mathrm{m}\end{array}$

The length of the pipe for the maximum frequency is calculated as,

$\begin{array}{l}{l}_2=\frac{\nu }{2f_{\mathrm{m}\mathrm{a}\mathrm{x}}}\\ =\frac{346\mathrm{m}/\mathrm{s}}{2\times \left(20000\mathrm{H}\mathrm{z}\right)}\\ =8.65\times {10}^{-3}\mathrm{m}\end{array}$

The range of the lengths of pipes is $8.65\mathrm{m}$ to $8.65\times {10}^{-3}\mathrm{m}$.