The frequency of two continuous overtones of a vibrating string are 290 Hz and 340 Hz.
(a) What will be the frequency of the fundamental note formed ?
(b) By how many decibels will the intensity level change if amplitude of the sound wave is made 2 times ?

Explanation

(a) v is the fundamental frequency
Let frequency of `(n - 1)^(th)` overtone = nv = 290 Hz
Let frequency of `n^(th)` overtone = (n + 1) v = 340 Hz
(n + 1)n - nv = 340 - 290
v = 70 Hz
`a_(2) = 2a_(1)`,
`(a_(2))/(a_(1)) = 2`
`(I_(2))/(I_(1)) = ((a_(2))/(a_(1)))^(2) = (4)/(1)`
Intensity in decibels = `10 "log"_(10) (I_(2))/(I_(1))`
`"log"_(10) (I_(2))/(I_(1)) = 10 log_(10) (4)`
`"log"_(10) (I_(2))/(I_(1)) = 10 log_(10) 2^(2)`
`"log"_(10) (I_(2))/(I_(1)) = 20 log_(10) 2 rArr I_(2) = 2^(20) I_(1)`