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}