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}