An observer `A` located at a distance `r_(A)=5.0 m ` from a ringing tuning fork notes the sound to fade away `tau =19 s ` later than an observer `B` who is located at a distance `r_(B)=50m` from the tuning fork. Find the damping coefficient `beta` of oscillations of the tuning fork. The sound velocity `v=340 m//s`.

Explanation

We treat the fork as a point source. In the absenece of damping the oscillation has the form
`(Const. )/(r) cos (omegat-kr)`
Because of the fork the amplitude of oscillation decreases exponentially with the retarted tme `(i.e.` the time at which the wave started from the source.`)` Thus we write for the wave amplitude.
`xi=(Const. ) /(r)e^(-beta(t-(r)/(v)))`
Thus means that `(e^(-beta(t + tau-(r_(A))/(v))))/( r_(A))=(e^(-beta(t+tau-(r_(B))/( v))))/( r_(B))`
Thus `e^(-beta(tau+(r_(B)-r_(A))/( v)))=(r_(A))/( r_(B))` or `beta=(1n(r_(B))/(r_(A)))/(tau+(r_(B)-r_(A))/(v))=0.12.s^(-1)`