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College Physics (Urone)
Found in: Page 595

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Short Answer

The ear canal resonates like a tube closed at one end. (See Figure \(17.39\).) If ear canals range in length from \(1.80\)to\(2.60\;{\rm{cm}}\)in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be\(37.0{\rm{^\circ C}}\), which is the same as body temperature. How does this result correlate with the intensity versus frequency graph (Figure \(17.37\) of the human ear?

The range of the fundamental frequencies is\({\rm{3}}{\rm{.39}}\;{\rm{kHz}}\)to\({\rm{4}}{\rm{.9}}\;{\rm{kHz}}\)

See the step by step solution

Step by Step Solution

Given Data

The temperature is\({{\rm{t}}_{\rm{1}}}{\rm{ = 37}}{\rm{.0^\circ C = }}\left( {{\rm{37 + 273}}} \right)\;{\rm{K = 310}}\;{\rm{K}}\)

The length of the canal ranges from \({{\rm{l}}_{\rm{1}}}{\rm{ = 1}}{\rm{.80}}\;{\rm{cm = 0}}{\rm{.018}}\;{\rm{m}}\) to \({{\rm{l}}_{\rm{2}}}{\rm{ = 2}}{\rm{.60}}\;{\rm{cm = 0}}{\rm{.026}}\;{\rm{m}}\)

The diagram of Human Ear is shown below:

The human ear

The diagram of intensity Vs frequency is shown below:

The intensity-frequency graph

Calculation of the ratio of the frequencies  

For a closed tube, the resonance frequencies are,

\({{\rm{f}}_{\rm{n}}}{\rm{ = n}}\frac{{\rm{v}}}{{{\rm{4l}}}}{\rm{,}}\;{\rm{n = 1,3,}}\;{\rm{5,}}\;....\)

For an open tube, the resonance frequencies are,

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

The speed of sound at\({t_1}\) temperature is

\(\begin{array}{c}{\rm{v = 331}}\sqrt {\frac{{{{\rm{t}}_{\rm{1}}}}}{{{\rm{273}}}}} \\{\rm{ = 331}}\sqrt {\frac{{{\rm{310}}}}{{{\rm{273}}}}} \\{\rm{ = 352}}{\rm{.7}}\;{\rm{m/s}}\end{array}\)

The frequencies are,

\(\begin{array}{c}{{\rm{f}}_{\rm{1}}}{\rm{ = }}\frac{{{\rm{352}}{\rm{.72}}}}{{{\rm{4 \times 0}}{\rm{.018}}}}\\{\rm{ = 4898}}{\rm{.89}}\;{\rm{Hz}}\\{\rm{ = 4}}{\rm{.9}}\;{\rm{kHz}}\end{array}\)

\(\begin{array}{c}{{\rm{f}}_{\rm{2}}}{\rm{ = }}\frac{{{\rm{352}}{\rm{.72}}}}{{{\rm{4 \times 0}}{\rm{.026}}}}\\{\rm{ = 3391}}{\rm{.54}}\;{\rm{Hz}}\\{\rm{ = 3}}{\rm{.39}}\;{\rm{kHz}}\end{array}\)

Our ear can hear\({\rm{ - 20}}\;{\rm{dB}}\) to\({\rm{10}}\;{\rm{dB}}\) with\({\rm{0}}\;{\rm{phon}}\), \({\rm{10}}\;{\rm{dB}}\)to\({\rm{40}}\;{\rm{dB}}\)with maximum of\({\rm{40}}\;{\rm{phon}}\) and\({\rm{40}}\;{\rm{dB}}\)to\({\rm{60}}\;{\rm{dB}}\)with maximum of\({\rm{60}}\;{\rm{phon}}\).

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