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Understanding Wave Dynamics: Properties, Mathematical Descriptions, and Interference

In this session, we explore the fundamental properties of waves, including superposition, reflection, and standing waves. We will examine various wave types—water waves, musical sounds, seismic tremors, light, and more—focusing on their general features. Key concepts include wave speed, frequency, wavelength, and the mathematical descriptions of wave functions. We’ll also delve into phenomena such as interference and standing waves, providing demonstrations and practical examples. Be sure to review the material before the assignment deadline!

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Understanding Wave Dynamics: Properties, Mathematical Descriptions, and Interference

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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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