Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory " Lab Sections! (Tentative) Sections have been defined! Monday 6 PM and Tuesday 4 PM, starting next week! See Course Web " Sections! (Tentative) Office Hours! Masahiro: Wed :30-4:00 PM & Thu 3:30-5:00 PM! Antonio: Wed 4:00-6:00 PM
What We Did Last Time! Analyzed a simple harmonic oscillator! The equation of motion:! The general solution: x() t = acosωt+ bsinωt! Studied the solution! Frequency and period! Energy tossing m! Completeness of the solution d x t () = kx() t ω = k m Goals for Today! Just how common are harmonic oscillators?! Very Physics is filled with them! But why?! Introduce complex exponentials:! It replaces sine and cosine e ±ix! It makes the math easier once you get used to it! Attack a damped oscillator! Exercise of complex math! Tool of the day: Taylor expansion
Taylor Expansion! Any (smooth) function f(x) can be approximated around a given point x = a as: 1 1 n n f( x) f( a) + f ( a)( x a) + f ( a)( x a) +! + f ( a)( x a) +! n!! You know this already, right?! The approximation is better when x a is small.! Because the higher-order terms (x a) n shrinks faster. Ubiquity of Harmonic Oscillators! Harmonic oscillator s equation of motion: d x m = kx Hooke s Law force! The restoring force kx is linear with x! This is not exactly true in most cases! Springs do not follow Hooke s law beyond elastic limits! Still, the physical world is full of almost-harmonic oscillators! And for a good reason
Example: Pendulum! A pendulum swings because of the combined force of the gravity mg and the string tension T! Combined force is mgsinθ! Displacement from the equilibrium is Lθ Force is not linear with displacement Pendulum is not a harmonic oscillator T mg θ mg sinθ Small-Angle Approximation! Taylor-expand F = mgsinθ around θ = 0 1 sinθ = sin 0 + (sin θ) θ= 0θ + (sin θ) θ= 0θ +! 1 3 1 5 = θ θ + θ +! 6 10! For small angle θ,! Equation of motion after approximation: d ( Lθ ) m = ml "" θ = mgθ This is linear! Solving this is pretty easy θ = acosωt+ bsinωt F = mgsinθ mgθ g ω = L
Potential Energy! Look at the same problem using the potential energy! At angle θ, the mass m is higher than the lowest position by h = L(1 cosθ)! The potential energy is E P = mgh = mgl(1 cosθ)! You get the force by differentiating E P with the distance x = Lθ de 1 h P dep F = = = mgsinθ dx L dθ Remember this? L x θ Small-Angle Approximation! We could Taylor-expand the potential energy E = mgl(1 cos θ ) mglθ P! The force is dep 1 d 1 F = = mglθ = mgθ dx L dθ! OK, we got the linear force again 1 cosθ = 1 θ + θ! 1 1 4 4 Why are we doing this?
Linearizing Equation of Motion! We can often linearize the equation of motion for small oscillation E around a stable point (equilibrium)! Why?! Anything that is stable is at a minimum of the potential energy E! Let s call it x = 0 0! Taylor expansion of E near x = 0 is 1 1 3 Ex ( ) = E(0) + E (0) x+ E (0) x + E (0) x+... 6 = 0 > 0 x Linearizing Equation of Motion 1 1 Ex = E + E x + E x+ 6 3 ( ) (0) (0) (0)...! If the oscillation is small enough, the terms x 3, x 4, can be ignored.! E(x) looks like a parabola.! This gives a force de F = = E (0) x dx! This works for just about any stable physical system Linear E A harmonic oscillator! 0 x
Ubiquity of Harmonic Oscillators! Every physically stable object can make harmonic oscillation! Stable object sits where the potential energy is minimum! The potential near the minimum looks like a parabola! Its derivative gives a linear restoring force! This is true for small oscillation! How small depend on how the potential looks like! We observe oscillation only when small is large enough! Invisibly small oscillation will become important with quantum mechanics Complex Numbers! I assume you are familiar with complex numbers! A few reminders to make sure you got the key concepts i = 1 Complex plane ib z = a+ ib Real part Imaginary part Re( z) = a Im( z) = b a ib z* = a ib Complex conjugate Im( z*) = Im( z) z+ z* z z* Re( z) = Im( z) = i
Argument! For a complex number z,! The distance z from 0 is the absolute value: z = a + b! The angle θ is the argument, or phase: θ = arg( z)! z may be expressed as: z = z (cosθ + isin θ) = z i e θ What s this? θ z z = a + ib Euler s Identity e iθ = cosθ + isinθ Euler s Identity! This is a natural extension of the real exponential! Check this with Taylor expansion 1 1 3 1 4 1 5 e x = 1+ x + x + x + x + x +... 6 4 10 e ix 1 i 3 1 4 i = 1+ ix x x + x + x 6 4 10 5 +... 1 1 cos( x) = 1 x + x 4 4... 1 3 1 sin( x) = x x + x 6 10 5...
Complex Plane! e iθ goes around the unit circle on the complex plane. Im e iπ / = i e iθ = cosθ + isinθ iπ e = 1 θ 0 e =1 Re e i3 π / = i Complex Solutions! Let s try a complex exponential on a simple harmonic oscillator: d x = ω x Xt! Try x = e Xt de Solutions Xt = X e = ω e Xt xt () = i t e ± ω! We got complex solutions to a harmonic oscillator! But we need a real solution for a physical system! We need an interpretation of the complex solution X = ω X = ± iω
Complex " Real Solutions i t! The two complex solutions x() t = e ± ω are complex conjugates of each other.! Generally, if you have a complex solution z(t), the complex conjugate z*(t) must also be a solution d z( t) d z* ( t) = ω z( t) = ω z*( t)! Any linear combination of z(t) and z * (t) is a solution.! In particular: zt () + z*() t Re( zt ( )) = is a solution Linearity! Generality! Because of linearity, we can multiply the solution by any complex number iωt ( a ib) e = ( a ib)(cosωt+ isin ωt) = acosωt+ bsin ωt+ i( asinωt bcos ωt) Real part gives the general solution
Recipe! Instead of cosωt and sinωt, use e Xt where X is a complex number! This simplifies the differential math: d d ( cos t) ω = ω sin ωt ( sin t) ω = ω cosωt! When you get the final answer, take the real part! Let s look at an example d e Xt = X e Xt Damped Oscillator! An ideal LC oscillator is a capacitor C and an inductor L connected in parallel! It s a simple harmonic oscillator! Real-world LC oscillators have resistance R, or damping! Let s take such an oscillator, give charge q 0 to the capacitor, and throw the switch on at t = 0! What happens to q = q(t)? q C q R I I = dq Charge conservation L
Damped Oscillator! The voltages across C, R, and L add up to zero VC + VR + VL = 0! Define voltages clockwise q VC = Capacitance C = RI Ohm s Law VR V L di = L Inductance V C q q I V R I = dq V L Equation of Motion q C + dq R d q + L = 0 Note if R = 0, d q q L = C Damped Oscillator q C + dq d q R + L = 0! Now try q( t) = q0e 1 + RX + LX q0e C X + R L Xt = 0! Assume R is small! Oscillator is weakly-damped Xt 1 X + LC = 0! X is a complex number X = R L C q q ± R L R I 1 LC L This is small
Weakly-Damped Oscillator R X = ± L = γ ± γ R L ω! Suppose γ << ω X γ ± iω! The complex solution becomes ( i ) t t i t qt () qe γ ± ω qe γ ± = = e ω! Take the real part: γ t qt () = qe cosωt 1 LC 0 0 0 γ R, ω 1 L LC very weak damping Weakly-Damped Oscillator t q t) = q e γ cosωt ( 0! Oscillation shrinks exponentially due to the damping term.! How long it keeps ringing depends on the ratio ω = γ R L C q 0 q γt 0e q 0
Complex vs. Real! We could solve the damped oscillator problem without complex exponential. (H&L Section 1.6)! The calculation is more difficult and less elegant! One must assume that the answer is e γt cosωt! It comes out naturally with complex math! Different assumption must be made if the damping is strong! We will deal with more and more complex examples in the future! Get used to it now! Strongly-Damped Oscillator! What if the resistor R is large?! The solution is still the same e Xt where X is X = γ ± γ ω γ =, ω =! With large R, γ ω >> ( γ ) R 1 L LC ω ω X γ ± γ γ γ! The two solutions are both exponentially decreasing! No oscillation. Just slowly going toward zero, ω t γ t γ 1 qt () = qe + qe q 1 and q must be fixed by the initial conditions.
Critically-Damped Oscillator! What if the resistor R is chosen just so that γ = ω?! The two solutions for X becomes identical X = γ ± γ ω = γ! The second solution is qt () te γ t = Check this yourself.! This condition is called critical damping! Let s compare weak/strong/critical damping for a same ω and same initial condition Damping Strength q 0 Critically-damped Strongly-damped ωt Weakly-damped q 0
Critical Damping! Critically damped oscillator stops most quickly! The energy is dissipated in R as fast as possible! This is useful when you are trying to control the movement! Shock absorbers on automobiles! Feedback control systems (e.g. thermostats)! Underdamped oscillation can cause disasters! Overloaded tracks. Broken shock absorbers Summary! Studied how the equation of motion can be linearized for small oscillations! Taylor expansion of the potential near the minimum! This makes harmonic oscillators very common! Learned to deal with complex exponentials! Makes it easy to solve linear differential equations! Analyzed a damped oscillator! Tried using complex exponential! Behavior depends on the damping strength! Next target: forced oscillation