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Lecture 29: Relationships between Wavelength, Period, and Wave Propagation

This lecture covers the relationship between wavelength and period, mathematical formalism of wave displacement, speed of mechanical waves, waves on a string, sound waves, intensity of waves, spherical waves, and the Doppler effect.

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Lecture 29: Relationships between Wavelength, Period, and Wave Propagation

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  1. Lecture 29 Goals: • Chapter 20, Waves • No labs this week. • HW 11 is due tomorrow night. • HW 12 (a short one) is due Thursday night.

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

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

  4. 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 ω : angular frequency ϕ: phase constant

  5. 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+π/2) D(x,t) = A cos [k(x – vt) + f]

  6. 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

  7. 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 )

  8. 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 μ=M/L : mass per unit length

  9. 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.

  10. 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

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

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

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

  14. 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

  15. 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

  16. 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

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

  18. The Doppler effect • The frequency of the wave that is observed depends on the relative speed between the observer and the source. observer

  19. vs observer

  20. The Doppler effect • Approaching source: f=f0/(1-vs/v) • Receding source: f=f0/(1+vs/v)

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