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UW-Madison PHYSICS 207 - Lecture 29

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Lecture 29, Dec. 10Superposition & InterferenceSlide 3Exercise SuperpositionOrgan Pipe ExampleStanding Wave QuestionSample ProblemSlide 8Example problemSlide 10Slide 11Slide 12Sample problemSlide 14Slide 15An exampleThe Full Cyclic ProcessSlide 18Slide 19Slide 20Slide 21An exampleSlide 23Slide 24An example problemSlide 26Chapter 1Important ConceptsChapter 2Chapter 3Chapter 4Slide 32Chapter 5Chapter 5 & 6Chapter 6Chapter 7Chapter 7 (Newton’s 3rd Law) & Chapter 8Chapter 8Chapter 9Chapter 10Slide 41Slide 42Chapter 11Slide 44Chapter 12Slide 46Slide 47Slide 48Angular MomentumHooke’s Law Springs and a Restoring ForceSimple Harmonic MotionResonance and dampingFluid FlowDensity and pressureResponse to forcesIdeal gas equation of statepV diagramsWork, Pressure, Volume, HeatChapter 18Thermal EnergyRelationshipsChapter 19RefrigeratorsCarnot CyclesWork (by the system)Chapter 20Displacement versus time and positionSinusoidal Waves (Sound and Electromagnetic)Doppler effectChapter 21Standing WavesBeatsSlide 73Physics 207: Lecture 29, Pg 1Lecture 29, Dec. 10To do :To do :•Chapter 21Chapter 21 Understand beats as the superposition of two waves of unequal frequency.Prep for exam. Room 2103 Chamberlain Hall Sections: 602, 604, 605, 606, 610, 611, 612, 614 Room 5208 Sewell Social Science (1180 Observatory Dr.)Sections: 601, 603, 607, 608, 609, 613Room 5310: McBurney and by prior arrangement9:25 Evaluations•AssignmentAssignment HW12, Due Friday, Dec. 12th , 10:59 PMPhysics 207: Lecture 29, Pg 2Superposition & InterferenceConsider two harmonic waves A and B meet at t=0. They have same amplitudes and phase, but  2 = 1.15 x 1.The displacement versus time for each is shown below:A(1t)B(2t)CONSTRUCTIVEINTERFERENCEDESTRUCTIVEINTERFERENCEC(t) = A(t) + B(t) Beat SuperpositionPhysics 207: Lecture 29, Pg 3Superposition & Interference Consider A + B, DA(x,t)=A cos(k1x–t) DB(x,t)=A cos(k2x–t) And let x=0, D=DA+DB=2A cos[2(f1 – f2)t/2] cos[2(f1 + f2)t/2] and |f1 – f2| ≡ fbeat = = 1 / Tbeat A(1t)B(2t)C(t) = A(t) + B(t)TbeattPhysics 207: Lecture 29, Pg 4Exercise SuperpositionThe traces below show beats that occur when two different pairs of waves are added (the time axes are the same). For which of the two is the difference in frequency of the original waves greater?A. Pair 1B. Pair 2C. The frequency difference was the same for both pairs of waves.D. Need more information.Physics 207: Lecture 29, Pg 5Organ Pipe ExampleA 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in this pipe?L=0.9 mf = 382 Hz and f  = v with  = 2 L / m (m = 1)v = 382 x 2(0.9) m  v = 687 m/sPhysics 207: Lecture 29, Pg 6Standing Wave QuestionWhat happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)?Recall: f  = v(A) Increases (B) Same (C) DecreasesPhysics 207: Lecture 29, Pg 7Sample ProblemThe figure shows a snapshot graph D(x, t = 2 s) taken at t = 2 s of a pulse traveling to the left along a string at a speed of 2.0 m/s. Draw the history graph D(x = −2 m, t) of the wave at the position x = −2 m.Physics 207: Lecture 29, Pg 8Sample ProblemHistory Graph:23647time (sec)2-25Physics 207: Lecture 29, Pg 9Example problemTwo loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity.As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s.a. What is the frequency of the sound?b. Draw, as accurately as you can, a wave-front diagram. On your diagram, label the positions of the two speakers, the point at which the intensity is maximum, and the point at which the intensity is minimum.c. Use your wave-front diagram to explain why the intensity is a minimum at a point 3.0 m directly in front of one of the speakers.Physics 207: Lecture 29, Pg 10Example problemTwo loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity.As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s.What is the frequency of the sound? v = f   but we don’t know f or DRAW A PICTUREConstructive Interference in-phaseDestructive Interference out-of-phasePhysics 207: Lecture 29, Pg 11Example problemTwo loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity.v = 340 m/s. PUT IN GEOMETRY v = f   but we don’t know f or AC - BC = 0 (0 phase differenc) AD - BD =  /2( phase shift)AD = (3.02+1.82)1/2 BD = 3.0  = 2(AD-BD) =1.0 mConstructive Interference in-phaseDestructive Interference out-of-phase3.0 m1.8 mABCDPhysics 207: Lecture 29, Pg 12Example problemTwo loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity.b. Draw, as accurately as you can, a wave-front diagram. On your diagram, label the positions of the two speakers, the point at which the intensity is maximum, and the point at which the intensity is minimum.c. Use your wave-front diagram to explain why the intensity is a minimum at a point 3.0 m directly in front of one of the speakers.Physics 207: Lecture 29, Pg 13Sample problemA tube, open at both ends, is filled with an unknown gas. The tube is 190 cm in length and 3.0 cm in diameter. By using different tuning forks, it is found that resonances can be excited at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no frequencies in between these.a. What is the speed of sound in this gas?b. Can you determine the amplitude of the wave? If so, what is it? If not, why not?Physics 207: Lecture 29, Pg 14Sample problemA tube, open at both ends, is filled with an unknown gas.


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UW-Madison PHYSICS 207 - Lecture 29

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