Physics 106P: Lecture 15 Notes

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Electric Christmas tree lights were first used
in what year?a. 1925b. 1897c. 1882d. 1939
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Physics 211 Lecture 27Todays Agenda

• Wave power
• Flow of energy
• Superposition Interference
• The wave equation

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Lecture 27, Act 1Wave 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?

(a) 2 sec (b) 4 sec (c) 8 sec
t
t Dt
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Lecture 27, Act 1 Solution

• We know that v l / T, hence T l / v

• In this case l 20 m and v 5 m/s, so T 4
  sec

• The time to go from a crest to a trough is T/2
  (half a period)

So Dt 2 sec
t
t Dt
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Wave Power

• 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
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Wave Power...
Slinky

• 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
Movie (pump)
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How is the energy moving?

• 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

x
?
x
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Power along the string

• Since v is along the y axis only, to evaluate
  Power F.v we only need to find Fy -F sin ? ?
  -F ? if ? is small.
• We can easily figure out both the velocity v and
  the angle ? at any point on the string
• If

y
?
x
Fy
F
v
dy
?
dx
Recall sin ? ? ? cos ? ? 1
for small ?
tan ? ? ?
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Power...

• So

• But last time we showed that and

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Average Power

• We just found that the power flowing past
  location x on the string at time t is given by

• We are often just interested in the average power
  movingdown the string. To find this we recall
  that the averagevalue of the function sin2 (kx -
  ?t) is 1/2 and find that

• It is generally true that wave power is
  proportional to thespeed of the wave v and its
  amplitude squared A2.

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Energy of the Wave

• We have shown that energy flows along the
  string.
• The source of this energy (in our picture) is the
  hand that is shaking the string up and down at
  one end.
• Each segment of string transfers energy to (does
  work on) the next segment by pulling on it, just
  like the hand.

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Power Example

• A rope with a mass of ? 0.2 kg/m lays on a
  frictionless floor. You grab one end and shake
  it from side to side twice per second with an
  amplitude of 0.15 m. You notice that the
  distance between adjacent crests on the wave you
  make is 0.75 m.
• What is the average power you are providing the
  rope?
• What is the average energy per unit length of the
  rope?
• What is the tension in the rope?

f 2 Hz
? 0.75 m
A 0.15 m
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Power Example...

• We know A, ? and ? 2?f. We need to find v!
• Recall v ?f (.75 m)(2 s-1) 1.5 m/s .
• So

Average power
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Power Example...

• So

Average energy per unit length
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Power Example...

• We also know that the tension in the rope is
  related to speed of the wave and the mass
  density

Tension in rope F 0.45 N
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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 27, 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? (The velocity of the wave is unchanged).

(a) 1 (b) 2 (c) 4
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Lecture 27, Act 2 Solution

• We have shown that the average power

So
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Lecture 27, Act 2 Solution

• But since v lf lw / 2p is constant,

i.e. doubling the wavelength halves the frequency.
Pi
Pf
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Lecture 27, Act 2 Solution
So
same power
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Superposition
Shive model

• Q What happens when two waves collide?
• A They ADD together!
• We say the waves are superposed.

Movie (super_pulse)
Movie (super_pulse2)
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Aside Why superposition works

• As we will see shortly, 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 B sin(?t) C cos(?t)
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Superposition Interference
Bead model

• 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

Movie (super)
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Superposition Interference

• Consider two harmonic waves A and B meeting at
  x0.
• Same amplitudes, but ?2 1.15 x ?1.
• The displacement versus time for each is shown
  below

A(?1t)
B(?2t)
What does C(t) A(t) B(t) look like??
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Superposition Interference

• Consider two harmonic waves A and B meeting at x
  0.
• Same amplitudes, but ?2 1.15 x ?1.
• The displacement versus time for each is shown
  below

A(?1t)
B(?2t)
C(t) A(t) B(t)
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Beats
Sound generator

• Can we predict this pattern mathematically?
• Of course!
• Just add two cosines and remember the identity

where
and
cos(?Lt)
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The Wave Equation

• Harmonic waves have the form y(x,t) A cos(kx -
  ?t).
• In general, a wave traveling to the right with
  velocity v is given by y(x,t) f(x - vt)
• How do we know a wave of this form really
  satisfies Newtons 2nd Law??
• We will now prove this is the case.

where
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The Wave Equation...

• Suppose we have the plucked string shown below
• The displacement is greatly exaggerated in the
  picture...?1 and ?2 are both close to 0.

F
dm
?2
?1
y
F
x
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The Wave Equation...
sin ? ? ? cos ? ? 1 for small ? tan ? ?
?

• ?FX F cos(?2) ? F cos(?1) 0 (no net force
  in x direction)
• But ?FY F sin(?2) ? F sin(?1) ? F(?2 - ?1) Fd?

F
y
dm
?2
?1
F
dx
x
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The Wave Equation...

• So ?FY F

?FY

• Use

and
dx
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Finally

• The is called the Wave Equation
• Any time the analysis of a physical system leads
  to an equation of this form, that system supports
  waves propagating with speed v.
• y f(x - vt) is a solution of the above, see
  Tipler chapter 15.

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Recap of todays lecture

• Wave power and intensity
• Superposition
• Interference
• The Wave Equation