Guitar String. One of the 63.5-cm-long strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is in-creased by 1.0%, what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 m/s, find the frequency and wavelength of the sound wave produced in the air by the vibration of the B3 string. How do these compare to the frequency and wavelength of the standing wave on the string?
Solution 49E Introduction The length of the string, frequency of the fundamental note is given, we have to calculate the speed of sound in the string. The we have to calculate the change in frequency for the 1% change in tension. Then we have to calculate the frequency and the wavelength of the same sound in the air. Step 1 (a) Standing wave is produced in the guitar string. Now the wave length for the standing wave in the string is given by = 2L = 2(63.5 cm) = 127 cm = 1.27 m Now the speed of the sound is given by v = f = (245 Hz)(1.27 m) = 3.11 m/s Step 2 (b) The speed of the wave is given by v = Where is the tension in the string and is the linear mass density of the string. Now after increasing the tension by 1%, the new tension is 1.01 Hene the speed of the wave is v = 1.01= 1.01 = 1.005v Now the new frequency is given by v 1.005v f= = = 1.005f = 1.005(245 Hz) 246 Hz So the new fundamental frequency of the string will be 246 Hz.