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Physics 123C Waves

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  1. Physics 123C Waves Lecture 10 (T&M: 15.4)Waves & Barriers April 23, 2008 (27 Slides) John G. Cramer Professor of Physics B451 PAB cramer@phys.washington.edu

  2. Sound Wave Intensity Physics 123C - Lecture 10

  3. Sound Intensities Physics 123C - Lecture 10

  4. Hearing Response of the Ear Physics 123C - Lecture 10

  5. n Waves in an Open-Open Pipe Physics 123C - Lecture 10

  6. Waves in an Open-Closed Pipe Physics 123C - Lecture 10

  7. Pipes and Modes Open-Open or Closed-Closed Open-Closed Physics 123C - Lecture 10

  8. Example:The Length of an Organ Pipe An organ pipe open at both ends sounds its 2nd harmonic at a frequency of 523 Hz (one octave above middle C). What is the length of the pipe from sounding hole to end? Physics 123C - Lecture 10

  9. Clicker Question 1 An open-open tube of air supports standing waves of frequencies of 300 Hz and 400 Hz, with no frequencies between these two. The second harmonic (m=2) of this tube has frequency: (a) 100 Hz; (b) 200 Hz; (c) 400 Hz; (d) 600 Hz; (e) 800 Hz. Physics 123C - Lecture 10

  10. Woodwinds vs. Strings Many woodwind instruments are effectively an open-closed pipe. This means they have only odd harmonics. Their fundamental frequency will be: The vibrating string of a stringed instrument is the equivalent of a closed-closed pipe. This means it will have both odd and even harmonics.Its fundamental frequency is: Note that for wind instruments, Lis the only adjustable parameter, while for stringed instruments, L, Ts and m can, in principle, be varied. However, wind instruments can be played at relatively pure harmonic frequencies, while strings cannot. Physics 123C - Lecture 10

  11. Example:The Notes of a Clarinet A clarinet (an open-closed instrument) is 66 cm long. The speed of sound in warm air is 350 m/s. What are the frequencies of the lowest note on a clarinet and of the next highest harmonic? Physics 123C - Lecture 10

  12. Reflection from a Boundary When a traveling wave encounters a “terminating” discontinuity in the medium (mR=¥), there is a complete negative reflection at the discontinuity. All of the wave energy is reflected as the negative of the incoming wave. While the wave is inverted in displacement direction, its amplitude is unchanged. At the boundary point the wave and its reflection always subtract to produce zero deflection. The situation can be simulated as an un-terminated string with positive and negative amplitude waves moving in opposite directions and meeting at the boundary. Note that the reflected wave has the same speed and wavelength (and energy) as the incident wave. Physics 123C - Lecture 10

  13. Creating Standing Waves Plucking a Standing Wave Considering the reflections at boundaries, it is easy to see how string vibration occur. When a string is plucked in the middle, waves travel in both directions to the boundaries, where they are reflected and propagate back and forth along the string. The net result is a superposition of right and left moving traveling waves that produce a standing wave. The waves so produced must have nodes at both boundaries. Physics 123C - Lecture 10

  14. Standing Wave Normal Modes Standing waves have the form: D(x,t) = (2a sin kx)cos wt The two string boundary conditions are:D(x=0, t) = 0 and D(x=L, t) = 0. Therefore, 2a sin kL = 0, which implies that kL = mp, where m is an integer. Butk = 2p/l, so: The frequency f is related to the wavelength l by: f = v/l, so the allowed waves on a string of length L will have frequencies: Physics 123C - Lecture 10

  15. About Normal Modes • The integer m is the number of antinodes of the standing wave. The number of nodes of the wave is m + 1. • The fundamental mode, with m = 1, has wavelength l1=2L (notL). Half a wavelength fits on the string, because the spacing between nodes is l/2. • The frequencies of the normal modes of a string form an arithmetic series: f, 2f, 3f, 4f, … Therefore, the fundamental frequency f1 can be found as the difference between the frequencies of any two adjacent modes, i.e.,Df = fm+1 - fm = f1. Physics 123C - Lecture 10

  16. Clicker Question 1 A standing wave on a string vibrates as shown. If the tension is quadrupled while the frequency and distance between boundaries remain the same, which diagram represents the new vibration? Physics 123C - Lecture 10

  17. Waves, Power, and Energy Physics 123C - Lecture 10

  18. Reflection and Transmission Slow to Fast Transition When a traveling wave encounters a “speed-up” discontinuity in the medium (mL>mR), there is a positive reflection at the discontinuity. Part of the wave energy is reflected and part is transmitted. Fast to Slow Transition When a traveling wave encounters a “slow-down” discontinuity in the medium (mL<mR), there is a negative reflection at the discontinuity. Again, part of the wave energy is reflected and part is transmitted. Slow® Fast Þ positive reflection Fast ® Slow Þ negative reflection Physics 123C - Lecture 10

  19. Transmission Coefficients Physics 123C - Lecture 10

  20. Example:Two Soldered Wires Two wires with different linear mass densities are soldered end-to-end and then stretched to a tension FT. The wave speed v1 on the first wire is twice the wave speed v2 on the second wire • If the incident wave amplitude is A, what are the amplitudes Ar and At of the reflected and transmitted waves? • What is linear mass density ratio m1/m2 of the wires? • What fraction of the incident average power is reflected at the junction, and what fraction is transmitted? Physics 123C - Lecture 10

  21. Transparent Optical Media Rather surprisingly, there are typesof matter, solids, liquids, and gasses,that are transparent and that transmitlight almost unimpeded. When youconsider that such matter is made ofatoms, electrically charged nucleiorbited by clouds of electrically chargedelectrons, it is quite remarkable thatelectromagnetic radiation, the carrierof electric fields that interact stronglywith these charged particles, is not immediately absorbed. Instead, within the transparent medium the bound electrons vibrate together at the frequency of the incoming electric field to “help along” the incident light without absorbing its energy. This usually reduces its speed through the material as it is transmitted. Physics 123C - Lecture 10

  22. The Index of Refraction Light travels through transparent media at a speed less than its speed c in vacuum. We define the index of refraction in a transparent medium as: Is nalways greater than 1? Almost always. There are a few media in which the phase velocity of light waves is greater than c. However, this super-luminal speed cannot be used to send signals or energy at a speed greater than c. Physics 123C - Lecture 10

  23. Waves vs. Particles If twopitchingmachinessimultane-ously throwbaseballs,they willcollide andbounce.Twoparticlescannotoccupy the same space point at the same time. On theother hand,if two loud-speakersmake soundwaves atthe sametime, theywill passthrougheach otherwithout collision. Two waves can occupy the same space point at the same time. Physics 123C - Lecture 10

  24. Particles Waves Diffraction Physics 123C - Lecture 10