Physics 2211: Lecture 38 Todays Agenda

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Physics 2211 Lecture 38Todays Agenda

• Conceptual discussion of wave motion
• Wave properties
• Mathematical description
• Waves on a string

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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)
• Water waves (water moves up down)
• String waves (string moves up down)
• Light waves (what moves??)

3
Types of Waves

• Transverse The medium oscillates perpendicular
  to the direction the wave is moving.
• Water waves
• Wave on a string
• Light waves
• Slinky

• Longitudinal The medium oscillates in the same
  direction as the wave is moving
• Sound waves
• Slinky

• We will deal only with transverse waves in this
  course!

4
Wave Properties

• Wave Number k 2p/l

y
x
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Wave Properties...

• Period The time T for a point on the wave to
  undergo one complete oscillation.

• Frequency The number of complete oscillations
  per second, f 1/T.

• Speed A traveling harmonic wave moves one
  wavelength ? in one period T so its
  speed is v ??/ T ? f.

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t 0
t T/4
t T/2
t 3T/4
t T
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Wave Properties...

• We will show that the speed of a wave is a
  constant that depends only on the medium, not on
  amplitude, wavelength, or frequency.
  ? and f are related by v
• For a given frequency and medium, wavelength is
  determined ? v / f
• Note that v ??/k since ?? 2? f ?and k
  2? / ? ?????????????
• Units f cycles/sec or hertz (Hz)
• ? rad/sec
• k m-1

8
Wave Forms

• So far we have examined continuous waves that
  go on forever in each direction!

• We can also have pulses caused by a brief
  disturbanceof the medium

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Mathematical Description

• Suppose we have some function y f(x)

0

• Let a v t Then
• f(x - v t) will describe the same shape
  moving to the right with speed v.

v
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Math...
y
?

• Consider a wave that is harmonic in x and has a
  wavelength of ?.

A
x
In general, this wave has the function form
v
y

• Now, if this is moving tothe right with speed v
  it will be described by

x
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Math...

• So we see that a simple harmonic wave moving with
  speed v in the x direction is described by the
  equation

• By using

and
we can write this as
(what about moving in the -x direction?)
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Traveling Wave Summary
?
y

• describes a
  harmonic wave ofamplitude A moving in the x
  direction.

A
x

• Each point on the wave oscillates in the y
  direction withsimple harmonic motion of angular
  frequency ? 2pf.

• The quantity k is called the wave number.

• The speed of the wave is .

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Wave Particle Kinematics
?
y

• Vertical speed of a point on the wave

A
x

• Maximum vertical speed of a point on the wave

• Vertical acceleration of a point on the wave

• Maximum vertical acceleration of a point on the
  wave

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Waves on a string

• What determines the speed of a wave?
• Consider a pulse propagating along a string

• Snap a rope to see such a pulse
• How can you make it go faster?

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Waves on a string...
Suppose

• The tension in the string is F
• The mass per unit length of the string is ?
  (kg/m)
• The shape of the string at the pulses maximum is
  circular and has radius R

F
?
R
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Waves on a string...

• Consider moving along with the pulse
• Apply F ma to the small bit of string at the
  top of the pulse

v
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Waves on a string...

• The total force FNET is the sum of the tension F
  at each end of the string segment.
• The total force is in the -y direction.

?
?
F
F
FNET 2F ? ?
(since ? is small, sin ? ?)
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Waves on a string...

• The mass m of the segment is its length (R x 2?)
  timesits mass per unit length ?.

m R 2???
?
?
??
R
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Waves on a string...

• The acceleration a of the segment is v 2/ R
  (centripetal)in the -y direction.

v
a
R
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Waves on a string...

• So FNET ma becomes

a
FTOT
m

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Waves on a string...

• So we find

v
tension F
mass per unit length ?

• Making the tension bigger increases the speed.
• Making the string heavier decreases the speed.
• As we asserted earlier, this depends only on the
  nature of the medium, not on amplitude,
  frequency, etc. of the wave.

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Recap of todays lecture

• Conceptual discussion of wave motion
• Wave properties
• Mathematical description
• Waves on a string
• Review Chapter 15 in Tipler Read
  Chapter 16.