Wave Motion (Cont.)

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Chapter 16

• Wave Motion (Cont.)

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Example 16.1

• A wave pulse moving to the right along the x axis
  is represented by the wave function
• where x and y are measured in cm and t is in
  seconds.
• Plot the wave function at t 0, t 1.0 s, and t
  2.0 s.
• Find the speed of the wave pulse.

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Outline

• Sinusoidal waves
• Basic variables
• Wave function y(x, t)
• Sinusoidal waves on strings
• The motion of any particle on the string
• The speed of waves on strings

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Basic variables of a sinusoidal wave

• Crest The point at which the displacement of the
  element from its normal position is highest.
• Wavelength ? The distance between adjacent
  crests, adjacent troughs, or any other comparable
  adjacent identical points.
• Period T The time interval required for two
  identical points of adjacent waves to pass by a
  point. Frequency f 1/T.
• Amplitude A The maximum displacement from
  equilibrium of an element of the medium.

What is the difference between (a) and (b)?
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Wave function y(x, t) of a sinusoidal wave

• y(x, 0) A sin(2?x/?)
• At any later time t
• If the wave is moving to the right,
• 
  
• If the wave is moving to the left,
• Relationship between v, ?, and T (or f) v ?/T
  ?f
• An alternative form of the wave function

Consider a special case suppose that at t 0,
the position of a sinusoidal wave is shown in the
figure below, where y 0 at x 0.
(Traveling to the right)
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Periodic nature of the wave function

• The periodic nature of
• At any given time t, y has the same value at the
  positions x, x?, x 2?, and so on.
• At any given position x, the value of y is the
  same at times t, tT, t2T, and so on.
• Definitions
• Angular wave number k 2?/?
• Angular frequency ? 2?/T
• v ?/T ?f ?/k
• An alternative form y A sin(kx - ?t) (assuming
  that y 0 at x 0 and t 0)
• General expression y A sin(kx -?t ?)
• ? the phase constant.
• kx -?t ? the phase of the wave at x and at t

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Example 16.2

• A sinusoidal wave traveling in the positive x
  direction has an amplitude of 15.0 cm, a
  wavelength of 40.0 cm, and a frequency of 8.00
  Hz. The vertical displacement of the medium at t
  0 and x 0 is also 15.0 cm, as shown in the
  figure.
• (a) Find the angular wave number k, period T,
  angular frequency ?, and the speed v of the wave.
• (b) Determine the phase constant ?, and write a
  general expression for the wave function.

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Sinusoidal waves on strings

• Producing a sinusoidal wave on a string The end
  of the blade vibrates in a simple harmonic motion
  (simple harmonic source).
• Each particle on the string, such as that at P,
  also oscillates with simple harmonic motion.
• Each segment oscillates in the y direction, but
  the wave travels in the x direction with a speed
  v a transverse wave.

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The motion of any particle on the string

• Example (Problem 17) A transverse wave on a
  string is described by the wave function y
  (0.120 m) sin (?x/8) 4 ?t.
• (a) Determine the transverse speed and
  acceleration at t 0.200 s for the point on the
  string located at x 1.60 m.
• (b) What are the wavelength, period, and speed of
  propagation of this wave?

• Assuming wave function is y A sin(kx - ?t)
• Transverse speed
• vy,max?A
• Transverse acceleration
• ay,max?2A

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The speed of waves on strings

• Example 16.4 A uniform cord has a mass of 0.300
  kg and a length of 6.00 m. The cord passes over a
  pulley and supports a 2.00-kg object. Find the
  speed of a pulse traveling along this cord.

• The speed of a transverse pulse traveling on a
  taut string
• T Tension in the string.
• ? Mass per unit length in the string.

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Homework

• Ch. 16, P. 506, Problems 10, 12, 13, 16, 21.
• Hints
• In 10, find the wave speed v first.
• In 12, set the phase difference at point A and B
  to be (Phase at B) (Phase at A) ? ?/3 rad
  then solve for xB.