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Vibrations and waves

Vibrations and waves. Physics 123, Spring 2007. Lecture III: Vibrations and Waves. SHO: resonance Waves Transverse and longitudinal waves Reflection Interference Standing waves. Waves. Matter. Propagating oscillation = wave .

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Vibrations and waves

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  1. Vibrations and waves Physics 123, Spring 2007 Lecture III

  2. Lecture III: Vibrations and Waves • SHO: resonance • Waves • Transverse and longitudinal waves • Reflection • Interference • Standing waves Lecture III

  3. Waves Matter • Propagating oscillation = wave. • Waves transport energy and information, but do not transport matter. • Examples: • Ocean waves • Sound • Light • Radio waves Wave Lecture III

  4. Waves • Wave velocity: • v=l/T=lf • The only equation that you need to remember about waves. • Wave velocity is NOT the same as particle velocity of the medium • Wavelength – l • Period T • Frequency f=1/T Lecture III

  5. Transverse and longitudinal waves Matter • In transverse wave the velocity of particles of the medium is perpendicular to the velocity of wave. Wave Lecture III

  6. Transverse and longitudinal waves Matter • In longitudinal wave the velocity of particles of the medium is parallel (or anti-parallel) to the wave velocity. Wave Lecture III

  7. Description of waves • w=2pf – cyclic frequency, k=2p/l –wave vector • D=D0sin(kx-wt+d), d-phase at t=0, x=0 • Riding the wave kx-wt+d=const • kx-wt=c x=c/k+(w/k)t = x0+vt • Thus, wave velocity v=w/k=2pf/ (2p/l)=fl = l/T • D=D0sin(kx-wt) – wave is moving in +x direction • D=D0sin(kx+wt) – wave is moving in -x direction Lecture III

  8. Average intensity • Displacement D follows harmonic oscillation: • Intensity (brightness for light) I is proportional to electric field squared • Average over time (one period of oscillation) I: Lecture III

  9. Energy transported by waves • Intensity of oscillation I (energy per unit area/ per sec) is proportional to amplitude squared D2 • 3D wave (from energy conservation): D12 4pr12= D22 4pr22 D1/D2=r2/r1 • Amplitude of the wave is inversely proportional to the distance to the source: Lecture III

  10. Interference of waves • When two or more waves pass through the same region of space, we say that they interfere. • Principle of superposition (fancy word for sum of waves): the resultant displacement is the algebraic sum of individual displacements created by these waves. Lecture III

  11. Constructive and destructive interference in phase out of phase not in phase Constructive Destructive Partially destructive A 2A <A 0 Lecture III

  12. Adding waves • Two waves observed at a certain location, can set it to be x=0: D1=D10sin(wt), D2=D20sin(wt+d) • Suppose for simplicity D10=D20 • (principle of superposition): • Amplitude of oscillation 2D0cos(d/2) is determined by the relative phase shift d • Intensity of the sum is proportional to cos2(d/2) Lecture III

  13. Reflection of a transverse wave pulse • Reflection from fixed end –inverted pulse • d=p •  • Reflection from loose end – the pulse is not inverted • d=0 •  Lecture III

  14. Standing waves • Interference of a wave with its reflection creates a standing wave. • Only standing waves corresponding to resonant frequencies (e.g. nodes at fixed ends) persist for long. Lecture III

  15. Standing waves • Add two waves traveling in opposite directions: • D1 =Dsin(kx-wt) • D2 =Dsin(kx+wt) D=D1+D2=2Dsin(kx)cos(wt) • Boundary condition • Sin(kL)=0  kL=pn • K=pn/L; l=2L/n Lecture III

  16. Standing waves • First harmonic or fundamental frequency: • L=l1 /2 l1=2L • f1=v/l1 f1=v /(2L) • Second harmonic: • L=l2l2=L • f2=v /l2 • f2=v /L=2f1 • N-th harmonic: • L=nln /2 ln=2L/n • fn=v /ln Lecture III

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