Question: (II) Approximately what are the intensities of the first two overtones of a violin compared to the fundamental? How many decibels softer than the fundamental are the first and second overtones? (See Fig. 12–15.)

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 \({\beta _{3 - 1}} = - 7\;{\rm{dB}}\).