QUESTION SIX 114 MARKSI (a) Define the terms:beats and intensity of sound. (b) A closed pipe has a length of 30cm and end correction of 1cm cal the fundamental frequency , the next two frequencies after the funda frequency and the wavelength of second overtone. (c) At a cocktail party ,38 people were speaking equally loudly, if on person was speaking, the sound level would be 72dB. Find the soun when all the 38 people were talking.

(a) Beats: Beats are the periodic variations in sound intensity that occur when two waves of slightly different frequencies interfere. They are heard as a pulsing or throbbing effect in the sound.<br /><br />Intensity of sound: Intensity of sound refers to the power per unit area carried by a sound wave. It is measured in watts per square meter (W/m²) and is a measure of the loudness or strength of the sound.<br /><br />(b) For a closed pipe, the fundamental frequency (first overtone) is given by the formula:<br /><br />\[ f_1 = \frac{v}{4L} \]<br /><br />where \( v \) is the speed of sound and \( L \) is the length of the pipe. Given \( L = 30 \) cm and end correction = 1 cm, the effective length \( L_{\text{eff}} = 30 + 1 = 31 \) cm.<br /><br />The next two frequencies after the fundamental frequency are the second and third overtones. For a closed pipe, the second overtone is given by:<br /><br />\[ f_2 = \frac{3v}{4L} \]<br /><br />\[ f_3 = \frac{5v}{4L} \]<br /><br />The wavelength of the second overtone is given by:<br /><br />\[ \lambda_2 = \frac{2v}{3f_2} \]<br /><br />(c) The sound level in decibels (dB) is given by:<br /><br />\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]<br /><br />where \( I \) is the intensity and \( I_0 \) is the reference intensity. If one person is speaking at 72 dB, the intensity of one person's speech is \( I_1 \). When 38 people are speaking, the total intensity \( I_{\text{total}} \) is:<br /><br />\[ I_{\text{total}} = 38 \times I_1 \]<br /><br />The sound level when all 38 people are talking is:<br /><br />\[ L_{\text{total}} = 10 \log_{10} \left( \frac{I_{\text{total}}}{I_0} \right) \]<br /><br />\[ L_{\text{total}} = 10 \log_{10} \left( \frac{38 \times I_1}{I_0} \right) \]<br /><br />\[ L_{\text{total}} = 10 \log_{10} (38) + 10 \log_{10} \left( \frac{I_1}{I_0} \right) \]<br /><br />\[ L_{\text{total}} = 10 \log_{10} (38) + 72 \]<br /><br />\[ L_{\text{total}} \approx 10 \times 1.59 + 72 \]<br /><br />\[ L_{\text{total}} \approx 15.9 + 72 \]<br /><br />\[ L_{\text{total}} \approx 87.9 \text{ dB} \]<br /><br />So, the sound level when all 38 people are talking is approximately 87.9 dB.