Three resonant frequencies of a string are 90, 150 and 210 Hz. (a) Find the highest possible fundamental frequency of vibration of this string. (b) Which harmonics of the fundamental are the given frequencies? (c) Which overtones are these frequencies? (d) If the length of the string is 80 cm, what would be the speed of a transverse wave on this string?

Sol. The resonant frequencies of a string are F base 1 = 90 Hz, f base 2 = 150 Hz, f base 3 = 120 Hz a) The highest possible fundamental frequency of the string is f = 30 Hz [because f base 1, f base 2 and f base 3 are integral multiple of 30 Hz] b) The frequencies are f base 1 = 3f, f base 2 = 5f, f base 3 = 7f So, f base 1, f base 2 and f base 3 are 3^rd harmonic, 5^th harmonic and 7th harmonic respectively. c) The frequencies in the string are f, 2f, 3f, 4f, 5f, ………. So, 3f = 2^nd overtone and 3^rd harmonic 5f = 4^th overtone and 5^th harmonic 7f = 6^th overtone and 7^th harmonic d) length of the string is l = 80 cm ⇒ f base 1 = (3/2l)v (v = velocity of the wave) ⇒ 90 = {3/(2*80)} * K ⇒ K = (90 * 2 * 80) / 3 = 4800 cm/s = 48 m