Physics 5B Lecture 8 January 27 2012 Chapter 15 Waves Damped Harmonic Motion m d 2x dt 2 dx kx b dt The solution depends on how the damping constant compares with the spring constant and mass Let x t Ae t cos t x t B C t e t 2 2 b 2m underdamped critically damped x t A1e t A2 e t e t where k m overdamped 2 2 Damped p Oscillator Underdamped Critically damped Overdamped Example plots for two different initial conditions Forced Damped p Oscillator Mechanical example move the wall to which the spring is attached back and forth sinusoidally sinusoidally Small damping h h Q high 0 Q Eventually the mass will oscillate with the driving frequency not the natural oscillator frequency 0 Large damping low Q Drive force cos t A0 0 f i ti friction http www walter fendt de ph14e resonance htm k m Forced Damped p Oscillator Important points After start up transients have died out the response is at the same frequency f as the h drive di The response amplitude is small far from resonance and large at resonance For high Q small damping the response can get huge at resonance when the drive operates at the natural frequency of the oscillator The phase difference between drive and response is zero far below resonance drive and oscillator move together 90 degrees at resonance e g the oscillator is at its maximum amplitude when the drive passes through its equilibrium position 180 degrees d far f above b resonance drive d and d oscillator ll move opposite each other The width of the resonant curve amplitude vs frequency is wide d for f large l d damping low l Q Q narrow for small damping high Q A Few Resonance Examples Car suspension springs and shock absorbers b b A Low Q B High Q FM radio receiver A Low Q B High Q Optical cavity in a laser A Low ow Q B High Q Wave Motion Contrast The wave transports energy and momentum from left to right but the individual parts of the rope are just moving up and down down Simple Wave Demos Speed of transverse waves p Restoring force v FT Caution This is the speed of the wave pulse NOT the speed of the particles making up the rope http phet colorado edu simulations sims php sim Wave on a String Wave Equation for a Stringg q Apply Newton s 2nd law to the vertical motion of an infinitesimal segment of the string x x x The magnitude of the tension FT is the same everywhere in the string But due to the curvature of the string the vertical component of FT is less at x x than it is at x provides a net restoringg force to p pull this p point on the stringg This p back to equilibrium one dimension Wave Equation one 2D x 2 1 2D v 2 t 2 v FT for a string It is easy to verify by differentiation and substitution that any function of x v t is a solution to this partial d ff differential l equation D x t D x v t A harmonic wave is one very important example of this D x t A sin k x t A sin k x v t Consider the following Gaussian shaped wave pulse it where Which of the formulas below best represents it 0 2 m and v 0 1 m s A B C 1 y x t 2 1 y x t 2 1 y x t 2 x vt 2 2 2 e x vt 2 2 2 e e x 2 2 vt 2 Moves to the right Longitudinal vs Transverse Waves g Sound Waves in a Fluid Longitudinal a gas or liquid cannot support any shear force Sound waves in solid can be longitudinal and or transverse