Physics 2211: Lecture 40 Todays Agenda

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Physics 2211 Lecture 40Todays Agenda

• Superposition Interference
• The wave equation
• Standing waves

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Lecture 40, Act 1Wave Motion

• The wavelength of microwaves generated by a
  microwave oven is about 3 cm. At what frequency
  do these waves cause the water molecules in your
  burrito to vibrate?

(a) 1 GHz (b) 10 GHz (c) 100 GHz
1 GHz 109 cycles/sec
The speed of light is c 3x108 m/s
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Superposition

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

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Aside Why superposition works

• As we will see, 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

• 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

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Superposition Interference

• Consider two harmonic waves A and B meeting at x
  0.
• Same amplitudes, but ?2 1.15 ?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

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

where
and
cos(?Lt)
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Lecture 40, Act 2Interference

• The maximum frequency a typical human ear can
  detect is about 20 kHz. If you walk into a room
  where two speakers are emitting sound waves at
  100 kHz and 110 kHz respectively, will you hear
  anything?

(a) yes (b) no (c) huh?

• You should assume the ear superposes the sounds
  waves in the same linear fashion we have been
  discussing in lecture.

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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.
• We will show that y f(x - vt) is a solution of
  the above.

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Check
is a solution to
Left Side
Right Side
So
So
Left side Right side !!
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Check OK !

• So we found that y f(x - vt) is a solution to
• You should verify that y Acos (kx - ?t) is a
  solution to this equation!
• Waves do not have to be harmonic!
• They can be pulses, pulse trains, or anything we
  can think of having the form y f(x - vt).

Aside We should really have written this in
terms of partial derivatives (but dont worry
about this for now)
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Standing Waves

• Recall what happens when two harmonic transverse
  waves having the same amplitude and frequency,
  travelling in opposite directions, meet
• The resulting superposition
• Has the same wavelength.
• Has twice the amplitude.
• Seems to be standing still.

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Standing Waves

• Suppose the incoming waves have the forms

yR(x,t) A cos(kx - w t)
yL(x,t) A cos(kx w t)
cos(a) cos(b) 2 cos cos
Recall that
ySUM(x,t) 2A cos(w t) cos(kx)
stationary wave
oscillating amplitude
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Standing Waves
y 2A cos(w t) cos(kx)

• Lets look at

We call this a standing wave since it is not
moving to the leftor to the right Each point
moves up and down with simple harmonic motion
having the same phase.
Notice that there are points that dont move at
all.These are called Nodes
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Standing Waves

• The string could be clamped (fixed) at two of the
  stationary points and the rest of the string
  would not be affected!
• This kind of wave can exist on a string whose end
  are fixed
• Guitar string

• You can think of the wave as bouncing back and
  forth betweenthe fixed boundaries.

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Lecture 40, Act 3Make it so

• You are in the Enterprise traveling at half the
  speed of light (v c / 2), heading toward a
  Borg spaceship. You fire your lasers and you see
  the light waves you are creating leaving your
  ship at the speed of light c 3x108 m/s toward
  the Borg.
• What do the Borg measure as the speed of the
  laser pulses approaching them?

(a) .5 c (b) c (c) 1.5 c
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Recap of todays lecture

• Superposition Interference
• The wave equation
• Standing waves
• Review Chapter 16 in Tipler.