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Physics 151: Lecture 35 Today’s Agenda

Physics 151: Lecture 35 Today’s Agenda. Topics Waves on a string Superposition Power.  and T are related !. Travleing 1-D wave: y(x,t):. Review: Wave Properties.

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Physics 151: Lecture 35 Today’s Agenda

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  1. Physics 151: Lecture 35 Today’s Agenda • Topics • Waves on a string • Superposition • Power

  2.  and T are related ! • Travleing 1-D wave: y(x,t): Review: Wave Properties... • The speed of a wave (v) is a constant and depends only on the medium, not on amplitude (A), wavelength () or period (T). • remember : T = 1/ f and T = 2 / 

  3. Example • Bats can detect small objects such as insects that are of a size on the order of a wavelength. If bats emit a chirp at a frequency of 60 kHz and the speed of soundwaves in air is 330 m/s, what is the smallest size insect they can detect ? • a. 1.5 cm • b. 5.5 cm • c. 1.5 mm • d. 5.5 mm • e. 1.5 um • f. 5.5 um

  4. Example • Write the equation of a wave, traveling along the +x axis with an amplitude of 0.02 m, a frequency of 440 Hz, and a speed of 330 m/sec. • A. y = 0.02 sin [880 (x/330 – t)] • b.y = 0.02 cos [880 x/330 – 440t] • c.y = 0.02 sin [880(x/330 + t)] • d.y = 0.02 sin [2(x/330 + 440t)] • e.y = 0.02 cos [2(x/330 - 440t)]

  5. Example • For the transverse wave described by y = 0.15 sin [ p(2x - 64 t)/16] (in SI units), determine the maximum transverse speed of the particles of the medium. • a. 0.192 m/s • b. 0.6 m/s • c. 9.6 m/s • d. 4 m/s • e. 2 m/s

  6. Lecture 34, Act 4Wave Motion • A heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. • As the wave travels up the rope, its speed will: v (a) increase (b) decrease (c) stay the same • Can you calcuate how long will it take for a pulse travels a rope of length L and mass m ?

  7. See text: 16.4 Superposition • Q: What happens when two waves “collide” ? • A: They ADD together! • We say the waves are “superposed”. Animation-1 Animation-2 see Figure 16.8

  8. Aside: Why superposition works • It can be shown that the equation governing waves (a.k.a. “the wave equation”) is linear. • It has no terms where variables are squared. • For linear equations, if we have two (or more) separate solutions, f1 and f2 , then Bf1+Cf2 is also a solution ! • You have already seen this in the case of simple harmonic motion: linear in x ! x = Bsin(t)+Ccos(t)

  9. will add constructively will add destructively See text: 16.4 Superposition & Interference • We have seen that when colliding waves combine (add) the result can either be bigger or smaller than the original waves. • We say the waves add “constructively” or “destructively” depending on the relative sign of each wave. • In general, we will have both happening see Figure 16.8

  10. Superposition & Interference • Consider two harmonic waves A and B meeting. • Same frequency and amplitudes, but phases differ. • The displacement versus time for each is shown below: A(t) B(t) What does C(t) =A(t)+B(t) look like ??

  11. Superposition & Interference • Add the two curves, • A = A0 cos(kx – wt) • B = A0 cos (kx – wt - f) • Easy, • C = A + B • C = A0 (cos(kx – wt) + cos (kx – wt + f)) • formula cos(a)+cos(b) = 2 cos[ 1/2(a+b)] cos[1/2(a-b)] • Doing the algebra gives, • C = 2 A0 cos(f/2) cos(kx – wt - f/2)

  12. Superposition & Interference • Consider, • C = 2 A0 cos(f/2) cos(kx – wt - f/2) A(t) B(t) Amp = 2 A0 cos(f/2) C(kx-wt) Phase shift = f/2

  13. Lecture 35, Act 1Superposition • You have two continuous harmonic waves with the same frequency and amplitude but a phase difference of 170° meet. Which of the following best represents the resultant wave? Original wave (other has different phase) B) A) D) C) E)

  14. D) Lecture 35, Act 1Superposition • The equation for adding two waves with different frequencies, C = 2 A0 cos(f/2) cos(kx – wt - f/2). • The wavelength (2p/k) does not change. • The amplitude becomes 2Aocos(f/2). Withf=170, we have cos(85°) which is very small, but not quite zero. • Our choice has same l as original, but small amplitude.

  15. See text: 16.8 Wave Power • A wave propagates because each part of the medium communicates its motion to adjacent parts. • Energy is transferred since work is done ! • How much energy is moving down the string per unit time. (i.e. how much power ?) P

  16. See text: 16.8 Wave Power... • Think about grabbing the left side of the string and pulling it up and down in the y direction. • You are clearly doing work since F.dr > 0 as your hand moves up and down. • This energy must be moving away from your hand (to the right) since the kinetic energy (motion) of the string stays the same. P

  17. x  Power P = F.v x F v See text: 16.8 How is the energy moving? • Consider any position x on the string. The string to the left of x does work on the string to the right of x, just as your hand did: see Figure 16-15

  18. y  x vy F v dy  dx Recall sin   cos  1 for small  tan   See text: 16.8 Power along the string. • Since v is along the y axis only, to evaluate Power = F.v we only need to find Fy = -Fsin -F  if  is small. • We can easily figure out both the velocity v and the angle  at any point on the string: • If

  19. But last time we showed that and See text: 16.8 Power... • So:

  20. We are often just interested in the average power movingdown the string. To find this we recall that the averagevalue of the function sin2(kx - t) is 1/2 and find that: See text: 16.8 Average Power • We just found that the power flowing past location x on the string at time t is given by: • It is generally true that wave power is proportional to thespeed of the wave v and its amplitude squared A2.

  21. Recap & Useful Formulas: y  A x • General harmonic waves • Waves on a string tension mass / length

  22. Lecture 35, Act 2Wave Power • A wave propagates on a string. If both the amplitude and the wavelength are doubled, by what factor will the average power carried by the wave change ? i.e. Pfinal/Pinit = X (a) 1/4 (b) 1/2 (c) 1 (d) 2 (e) 4 initial final

  23. 3-D Representation RAYS Wave Fronts Waves, Wavefronts, and Rays • Up to now we have only considered waves in 1-D but we live in a 3-D world. • The 1-D equations are applicable for a 3-D plane wave. • A plane wave travels in the +x direction (for example) and has no dependence on y or z,

  24. 3d representation Shading represents density wave fronts rays Waves, Wavefronts, and Rays • Sound radiates away from a source in all directions. • A small source of sound produces a spherical wave. • Note any sound source is small if you are far enough away from it.

  25. Waves, Wavefronts, and Rays • Note that a small portion of a spherical wave front is well represented as a plane wave.

  26. Waves, Wavefronts, and Rays • If the power output of a source is constant, the total power of any wave front is constant. • The Intensity at any point depends on the type of wave.

  27. Lecture 35, Act 3Spherical Waves • You are standing 10 m away from a very loud, small speaker. The noise hurts your ears. In order to reduce the intensity to 1/2 its original value, how far away do you need to stand? (a) 14 m (b) 20 m (c) 30 m (d) 40 m

  28. Lecture 35, Act 4Traveling Waves Two ropes are spliced together as shown. A short time after the incident pulse shown in the diagram reaches the splice, the ropes appearance will be that in • Can you determine the relative amplitudes of the transmitted and reflected waves ?

  29. Lecture 35, Act 3bPlane Waves • You are standing 0.5 m away from a very large wall hanging speaker. The noise hurts your ears. In order to reduce the intensity you walk back to 1 m away. What is the ratio of the new sound intensity to the original? (a) 1 (b) 1/2 (c) 1/4 (d) 1/8 speaker 1 m

  30. Recap of today’s lecture • Chapter 16 • Waves on a string • Superposition • Power

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