“A model for vibrations of a wine glass is x’’ + λx’ + (w^2)x = 0. Suppose that when struck the glass vibrates at 660 Hz. Show that (4w^2 - λ^2)^(1/2) = 2640
“If it takes about 3 seconds for the sound to die away, and this happens when the original vibrations have reduced to 1/100 of their initial level, show that λ = (2log100)/3 and hence that λ = 3.07 and w = 4.15*10^3"
So, I figured out the first part easily. frequency=f=660 Hz. w = 2*pi*f. I plugged it into the given expression, and assumed no damping, λ = 0, because the wine glass has just been struck. But I can't figure out how to get the second part. I was thinking about equating the equation of an underdamped system as x(0) = x(3), but that yields nothing.
w^2=k/m
λ = c/m (c is damping coefficient)
damped equation: x(t) = e^(-λt/2) * (Acos[.5*(4w^2 - λ^2)^1/2] + Bsin[.5*(4w^2 - λ^2)^1/2] )