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Lecture 29

Lecture 29. Goals:. Chapter 20, Waves. Final test review on Wednesday. Final exam on Monday, Dec 20, at 5:00 pm. HW 11 due Wednesday. Relationship between wavelength and period. v. D(x,t=0). x. x 0. l. T= l /v. Exercise Wave Motion.

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Lecture 29

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  1. Lecture 29 Goals: • Chapter 20, Waves • Final test review on Wednesday. • Final exam on Monday, Dec 20, at 5:00 pm. • HW 11 due Wednesday.

  2. Relationship between wavelength and period v D(x,t=0) x x0 l T= l/v

  3. Exercise Wave Motion • A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, how long Dt does it take the boat to go from the top of a crest to the bottom of a trough ? (Recall T = / v ) (A) 2 sec(B) 4 sec(C) 8 sec t t + Dt

  4. Mathematical formalism D(x=0,t) D(0,t) ~ A cos (wt + f) • w: angular frequency • w=2p/T t T λ D(x,t=0) D(x,0) ~ A cos (kx+ f) • k: wave number • k=2p / l x

  5. Mathematical formalism • The two dimensional displacement function for a sinusoidal wave traveling along +x direction: D(x,t) = A cos (kx - wt + f) A : Amplitude k : wave number w : angular frequency f : phase constant

  6. Mathematical formalism • Note that there are equivalent ways of describing a wave propagating in +x direction: D(x,t) = A cos (kx - wt + f) D(x,t) = A sin (kx - wt + f+p/2) D(x,t) = A cos [k(x – vt) + f]

  7. Why the minus sign? • As time progresses, we need the disturbance to move towards +x: at t=0, D(x,t=0) = A cos [k(x-0) + f] at t=t0, D(x,t=t0) = A cos [k(x-vt0) + f] vt0 v x

  8. Which of the following equations describe a wave propagating towards -x: D(x,t) = A cos (kx – wt ) D(x,t) = A sin (kx – wt ) C)D(x,t) = A cos (-kx + wt ) D) D(x,t) = A cos (kx + wt )

  9. Speed of waves • The speed of mechanical waves depend on the elastic and inertial properties of the medium. • For a string, the speed of the wave can be shown to be: Tstring: tension in the string m=M / L : mass per unit length

  10. Waves on a string • Making the tension bigger increases the speed. • Making the string heavier decreases the speed. • The speed depends only on the nature of the medium, not on amplitude, frequency etc of the wave.

  11. Exercise Wave 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

  12. Sound, A special kind of longitudinal wave λ Individual molecules undergo harmonic motion with displacement in same direction as wave motion.

  13. Waves in two and three dimensions • Waves on the surface of water: circular waves wavefront

  14. Plane waves • Note that a small portion of a spherical wave front is well represented as a “plane wave”.

  15. Intensity (power per unit area) • A wave can be made more “intense” by focusing to a smaller area. I=P/A : J/(s m2) R

  16. Exercise Spherical 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 1/4 its original value, how far away do you need to stand? (A) 14 m (B) 20 m (C) 30 m(D)40 m

  17. Intensity of sounds • The range of intensities detectible by the human ear is very large • It is convenient to use a logarithmic scale to determine the intensity level,b I0: threshold of human hearing I0=10-12 W/m2

  18. Intensity of sounds • Some examples (1 pascal  10-5 atm) :

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