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Announcements

Announcements. Final exam is on Monday, April 28 th 3:30-5:30 PM Comprehensive with about 20-25% from Units 19-24 All MC Must earn at least 25% to pass class Section 001 – S BEH AUD Section 006 – FMAB AUD Next Friday, April 25

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Announcements

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  1. Announcements • Final exam is on Monday, April 28th 3:30-5:30 PM • Comprehensive with about 20-25% from Units 19-24 • All MC • Must earn at least 25% to pass class • Section 001 – S BEH AUD • Section 006 – FMAB AUD • Next Friday, April 25 • Professor Gerton will be available for consultation ALL DAY in the Browning Building (Geology/Geophysics) near the 2 Creek Coffee House. Discounted drinks will be available (coffee, hot chocolate, etc.) • We are trying to set up review sessions for that day also: stay tuned.

  2. Lecture 23: Harmonic Waves and the Wave Equation Today’s Concept: Harmonic Waves

  3. What is a Wave? A wave is a traveling disturbance that transports energy but not matter. Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Largest in history??? Water waves (water moves up & down) Light waves (what moves?)

  4. Types of Waves Transverse: The medium oscillates perpendicular to the direction the wave is moving. This is what we study in Physics 2210. Longitudinal:The medium oscillates in the same direction as the wave is moving.

  5. From sinusoidal function to traveling wave Start with a drawing of θ -π π 2π -2π

  6. From sinusoidal function to traveling wave Start with a drawing of • Write angle as a function of position: • Wavelength: • Wave number:

  7. From sinusoidal function to traveling wave Start with a drawing of • Write angle as a function of position: • Wavelength: • Wave number: Shift the wave by adding a “phase angle”:

  8. From sinusoidal function to traveling wave Start with a drawing of • Write angle as a function of position: • Wavelength: • Wave number: Shift the wave by adding a “phase angle”: Let the phase angle vary in time:

  9. From sinusoidal function to traveling wave Multiply by an amplitude to give physical dimensions: y A

  10. Wave Properties Amplitude: The maximum displacement Aof a point on the wave. λ A Wavelength: The distance λbetween identical points on the wave. Period: The time Pit takes for an element of the medium to make one complete oscillation.

  11. λ Wave speed vs. Rope speed Imagine we have a sinusoidal wave on a rope: A If then gives: A) The horizontal speed of the wave. B) The vertical speed of a segment of rope. This is the speed of a segment of rope at position xand time t.

  12. λ Wave speed vs. Rope speed Imagine we have a sinusoidal wave on a rope: A To find the speed of the wave, first pick out a distinctive feature of the wave (e.g., a peak). That feature corresponds to a particular value of the cosine (or sine) function, which does not change in time: Constant in time

  13. ACT y = f (x-vt) If a function moving to the right with speed vis described by f (x - vt) then what describes the same function moving to the left with speed v? y y y v x x x 0 0 y A)y = - f (x - vt) v B)y = f (x + vt) x 0 C) y = f (-x +vt)

  14. y y y 0 v x x x y v x 0 Suppose the function has its maximum at f (0). y x – vt = 0 v y = f (x - vt) x =vt x 0 x – vt = 0 v A) y = - f (x - vt) x = vt x + vt = 0 v x =-vt B) y = f (x +vt) -x + vt = 0 v x = vt C) y = f (-x + vt) Moves to the right when the signs in front of the x and t terms are different Moves to the left when the signs in front of the x and t terms are the same

  15. CheckPoint: Wave Equation We have shown that the functional form y(x,t) = Acos(kx-ωt) represents a wave moving in the +xdirection. y A x Which of the following represents a wave moving in the –x direction? A) y(x,t)=Acos(ωt– kx) B) y(x,t)=Asin(kx– ωt) C) y(x,t)=Acos(kx+ ωt) A) y(x,t) =Acos(ωt-kx)I have no clue but this is the only answer with a “-” in front of the x. Plus, its derivative, velocity, is negative so that helps with my reasoning. C) y(x,t) = Acos(kx+ ωt)As the wave reaches its amplitude, kx+wt=0; which means if t is increasing constantly at all times then x must increase constantly in the - direction as well.

  16. How to make a Function Move Suppose we have some function y = f (x): y f(x - a) is just the same shape moveda distance a to the right: x 0 x = a y Let a =vtThen f(x -vt) will describe the same shape moving to the right with speed v. v x 0 x = vt y x 0

  17. Harmonic Wave Consider a wave that is harmonic in x and has a wavelength ofλ. y Now, if this is moving to the right with speed vit will be described by: v x λ y A x If the amplitude is maximum atx = 0 it has the functional form:

  18. Transverse Wave on a String

  19. Energy

  20. CheckPoint: Wave Frequency 2 y A x y B x Waves A and Bshown above are propagating in the same medium with the same amplitude. Which one carries the most energy per unit length? A) A B) B C) They carry the same energy per unit length

  21. CheckPoint Results: Wave Frequency 2 y A x y B x Which one carries the most energy per unit length? A) A B) BC) Same Also, vwave is the same for both: So:

  22. Homework Problem

  23. Homework Problem

  24. Homework Problem

  25. Homework Problem Average speed = distance traveled / time taken Distance traveled by a piece of rope during one period is 4A

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