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Welcome back to Physics 211

Welcome back to Physics 211. Today’s agenda: Waves – general properties mathematical description superposition, reflection, standing waves. Reminder. MPHW 6 due this Friday 11 pm. Waves – general features. many examples water waves, musical sounds, seismic tremors, light, gravity etc

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Welcome back to Physics 211

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  1. Welcome back to Physics 211 Today’s agenda: Waves – general properties mathematical description superposition, reflection, standing waves

  2. Reminder • MPHW 6 due this Friday 11 pm

  3. Waves – general features • many examples • water waves, musical sounds, seismic tremors, light, gravity etc • system disturbed from equilibrium can give rise to a disturbance which propagates through medium (carries energy) • periodic character both in time and space • interference

  4. waves – structure • wave propagates by making particles in medium execute simple harmonic motion • motion along direction of wave (longitudinal) eg. sound • perpendicular to direction (transverse) eg waves on string, light

  5. picture

  6. waves - properties • waves propagate at constant speed through medium • speed depends on properties of medium (density and elastic forces) • not same as velocity of particles in medium

  7. Speed of wave • wave on string • what is v ? • expect it depends on tension and mass density of string • dimensional analysis: v2=T/m i.e (kgm/s2)/(kg/m)=m2/s2

  8. periodic wave x distance between maxima=wavelength l height of crest = amplitude

  9. periodic wave t time for 1 oscillation=frequency f T=1/f

  10. wave speed • Notice: v=fl number of wave crests passing by in 1 second times distance between crests  speed of wave • Also, can define w=2pf and k=2p/l • w angular frequency, k wave number v= w/k

  11. If you double the frequency of a wave what happens to the speed of the wave ? • doubles • halves • stays same • depends on wavelength

  12. simple wave on rods demo • Notice: • mean position of rods does not change. • but energy transported! • each rod undergoes periodic motion • speed of wave does not depend on how fast or what magnitude of driving force • ex. traveling periodic wave

  13. Mathematical description • Describe wave by wave function which tells you the size of the wave at each point in space (x) and time (t) Y=Y(x,t) • Think of sinusoidal waves for simplicity • At fixed point in space have SHM: Y= acos(wt)=acos(2pft)

  14. description continued • Not full story – the amplitude of wave depends on position as well as time • Wave is collection of SHM oscillators where each oscillator has different phase f(x) • Y= acos(wt-f) • Simplest case f=kx= (2p/l)x

  15. wavefunction • Y= acos(wt-kx)=acos(2p(ft-x/l)) or Y= acos(2pf(t-x/v)) sinusoidal wave

  16. rough picture • as one part of wave wiggles – sets off another neighboring region • neighbor wiggles at same frequency but lags the first • lag just depends on how far away it is • lag is equivalent to different phase

  17. another way of thinking • consider particle at x=0 and t=0. Wiggles with SHM • wave travels distance to some point x in time x/v (v = wave speed) • So motion of wave at time t position x is same as initial point at t=0, x=0 (sinusoidal) • thus replace tt-x/v in original Y

  18. direction of wave (1D) ? • Y= acos(wt-kx)=acos(2p(ft-x/l)) wave traveling in positive x direction • Y= acos(wt+kx)=acos(2p(ft+x/l)) traveling in negative x direction

  19. Reflection ? • consider rod demo again. If hold final rod fixed – wave reflects back but with opposite sign

  20. Interference • When two waves propagate through same region – combine to give some new wave motion • interfere • Resultant wave motion is simply sum of individual wave motions (superposition)

  21. Examples constructive interference destructive interference

  22. interference demo • can produce `no motion’ if two waves interfere destructively – need equal frequencies and equal and opposite amplitudes for complete cancellation.

  23. Standing Waves • Consider wave on rubber hose (demo) • If I drive system with just right frequency • wave exhibits standing wave pattern • some parts of wave never move, others oscillate always maximally. Motion of different parts of medium in phase • no energy transport

  24. Picture of standing wave wave motion =0 at ends nodes

  25. mathematics of standing waves Y= acos(wt-kx)- acos(wt+kx) =2asin(wt)sin(kx) possible values of k ? length L, need kL=p,2p,3p,...

  26. picture – normal modes

  27. Power in a wave • Consider energy of SHM E=1/2ka2 =1/2mw2a2 what is m ? also need power=energy per unit time delivered by wave in time T total mass of excited oscillators is Tvm E/T=P=1/2vmw2a2

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