Physics 151: Lecture 35 Today’s Agenda

1
Physics 151 Lecture 35 Todays Agenda

• Topics
• Waves on a string
• Superposition
• Power

2
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? / ???

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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)

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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

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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 ?

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Superposition
See text 16.4

• 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
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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)
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Superposition Interference
See text 16.4

• 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
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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 ??
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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(ab)
  cos1/2(a-b)
• Doing the algebra gives,
• C 2 A0 cos(f/2) cos(kx wt - f/2)

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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
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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)
A)
B)
D)
C)
E)
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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). With f170, we
  have cos(85) which is very small, but not quite
  zero.
• Our choice has same l as original, but small
  amplitude.

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Wave Power
See text 16.8

• 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
Wave Power...
See text 16.8

• 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 gt 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
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How is the energy moving?
See text 16.8

• 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
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Power along the string.
See text 16.8

• 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

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Power...
See text 16.8

• So

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Average Power
See text 16.8

• 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.

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Recap Useful Formulas
y
?
A
x

• Waves on a string

• General harmonic waves

tension
mass / length
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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
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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,

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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.

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Waves, Wavefronts, and Rays

• Note that a small portion of a spherical wave
  front is well represented as a plane wave.

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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.

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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
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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 ?

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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
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Recap of todays lecture

• Chapter 16
• Waves on a string
• Superposition
• Power