91Ó°ÊÓ

To find the ratio of the intensities, use the concept of amplitudes of the successive overtones and their frequencies which is given as \(I = 2{\pi ^2}v\rho {f^2}{A^2}\) where v is the speed of sound in air,\(\rho \)is the density of the medium, f is the fundamental frequency, and A is the amplitude of the wave.

Given data:

The sound spectra for violin is given.

For the first overtone, the equation for the intensity can be given as:

\({I_1} = 2{\pi ^2}v\rho f_1^2A_1^2\) … (1)

Here, \({f_1}\) is the frequency of the first overtone and \({A_1}\) is the amplitude of the first overtone.

For the second overtone, the equation for the intensity can be given as:

\({I_2} = 2{\pi ^2}v\rho f_2^2A_2^2\) … (2)

Here, \({f_2}\) is the frequency of the second overtone and \({A_2}\) is the amplitude of the second overtone.

After dividing the equation (2) by (1), you get:

\(\begin{array}{c}\frac{{{I_2}}}{{{I_1}}} = \frac{{2{\pi ^2}v\rho f_2^2A_2^2}}{{2{\pi ^2}v\rho f_1^2A_1^2}}\\\frac{{{I_2}}}{{{I_1}}} = {\left( {\frac{{{f_2}}}{{{f_1}}}} \right)^2}{\left( {\frac{{{A_2}}}{{{A_1}}}} \right)^2}\end{array}\) … (3)

From figure 12-15, the value of the ratio of amplitude \(\left( {\frac{{{A_2}}}{{{A_1}}}} \right)\) is \(0.4\) , and the value of the ratio of frequency \(\left( {\frac{{{f_2}}}{{{f_1}}}} \right)\) is \(2\). Hence, on substituting the given values in equation (3), you get:

\(\begin{array}{c}\frac{{{I_2}}}{{{I_1}}} = {\left( 2 \right)^2}{\left( {0.4} \right)^2}\\\frac{{{I_2}}}{{{I_1}}} = 0.64\end{array}\)

Similarly, the ratio of the third harmonic to the first harmonic can be given as:

\(\frac{{{I_3}}}{{{I_1}}} = {\left( {\frac{{{f_3}}}{{{f_1}}}} \right)^2}{\left( {\frac{{{A_3}}}{{{A_1}}}} \right)^2}\) … (4)

From figure 12-15, the value of the ratio of amplitude \(\left( {\frac{{{A_3}}}{{{A_1}}}} \right)\) is \(0.15\) , and the value of the ratio of frequency \(\left( {\frac{{{f_3}}}{{{f_1}}}} \right)\) is \(3\). Hence, on substituting the given values in equation (4), you get:

\(\begin{array}{c}\frac{{{I_3}}}{{{I_1}}} = {\left( 3 \right)^2}{\left( {0.15} \right)^2}\\\frac{{{I_3}}}{{{I_1}}} = 0.20\end{array}\)

The sound level of the first two overtones can be calculated as:

\(\begin{array}{c}{\beta _{2 - 1}} = 10\log \left( {\frac{{{I_2}}}{{{I_1}}}} \right)\\{\beta _{2 - 1}} = 10\log \left( {0.64} \right)\\{\beta _{2 - 1}} = - 1.93\;{\rm{dB}} \approx - {\rm{2}}\;{\rm{dB}}\\{\beta _{2 - 1}} = - {\rm{2}}\;{\rm{dB}}\end{array}\)

Similarly, the sound level of the third overtone to the first overtone can be calculated as:

\(\begin{array}{c}{\beta _{3 - 1}} = 10\log \left( {\frac{{{I_3}}}{{{I_1}}}} \right)\\{\beta _{3 - 1}} = 10\log \left( {0.20} \right)\\{\beta _{3 - 1}} = - 6.98\;{\rm{dB}} \approx - 7\;{\rm{dB}}\\{\beta _{3 - 1}} = - 7\;{\rm{dB}}\end{array}\)

Hence, the intensities of the first two overtones of a violin are \(\frac{{{I_2}}}{{{I_1}}} = 0.64\) and \(\frac{{{I_3}}}{{{I_1}}} = 0.20\) and the sound level of the first two overtones are \({\beta _{2 - 1}} = - {\rm{2}}\;{\rm{dB}}\) and \({\beta _{3 - 1}} = - 7\;{\rm{dB}}\).