phy53 lect dr. brown 23 november 2010
recall damped harmonic oscillator
now treat energy -- potential; total energy
defined "quality factor" = \frac{2 \pi}{\frac{\Delta E}{E}} = \frac{m \omega_o}{b}= \omega_o \tau
\tau is the exponential damping constant [units of time]
low Q => poor oscillator ; high Q = good oscillator, small damping losses
driven harmonic oscillators -- and resonance
driving force = F_o cos{\omega t}
2nd order linear inhomogeneous ordinary differential equation
transient and steady state solutions -- we'll only deal with the steady state (particular) solution
x(t) = A cos(\omega t - \phi) steady state solution
work done by driving force = energy lost to damping force
consider power -- via energy arguments [not F \dot velocity]
plot average power vs. \omega !
Q = \frac{\omega_o}{\Delta \omega}
\Delta \omega is the full width at half peak average power
Next -- most objects in some stable equilib -- oscillate and damp back down ...now consider coupled oscillators and waves in continuous medium ; sound waves, light waves, ... derive wave equation as solution to N2
begin with stretched string...
form of the WAVE EQUATION
\frac{d^2 y}{dx^2} -\frac{1}{v^2} \frac{d^2 y}{dt^2} = 0
for the stretched string v^2 = \frac{T}{\mu}