Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (

Question

Two strings on a musical instrument are tuned to play at 392 Hz (G) and 494 Hz (B). (a) What are the frequencies of the first two overtones for each string? (b) If the two strings have the same length and are under the same tension, what must be the ratio of their masses (mG/mB)? (c) If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths (lG/lB)? (d) If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?

Explanation / Answer

a) f1 = 2f0 = 2*392 = 784 Hz..........f2 = 3f0 = 3*392=1176 Hz

f1' = 2f0' = 2*494 = 988 Hz..........f2 '= 3f0' = 3*494=1482 Hz

b) f = (1/2L)*sqrt(TL/m)

mG/mB = fo'/fo^2 = (494/392)^2 = 1.588

c) lG/lB = fo'/fo = 1.26

d) TG/TB = fo/fo')^2 = 0.63

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