A vibrating standing wave on a string radiates a sound wave with intensity proportional to the square of the standing-wave amplitude. When a piano key is struck and held down, so that the string continues to vibrate, the sound level decreases by 8.0 dB in 1.0 s.
What is the string's damping time constant τ ?
The Attempt at a Solution
From 8dB, I got change in sound intensity is 6.31*10^-12 w/m2
Since the intensity is proportional to the the square of amplitude, the amplitude will decrease by (6.31*10^-12)^1/2
So I got
(6.31*10^-12)^1/2 = 2a - 2a•e^-t/τ =
2a (1- e^-t/τ)
I assumed sin(kx)=1 because we are looking at the greatest amplitude.
I don’t know where to go from here as I don’t know what a is.
Since I don’t know what the original sound intensity is, I won’t be able to know the percentage the sound intensity has decreased in order to calculate the percentage the amplitude has decreased.
Any help is appreciated:)