Waves Chapter 15

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Waves Chapter 15

• PHYS 2326-24

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Concepts to Know

• Transverse Waves
• Longitudinal Waves
• Periodic Waves
• Sinusoidal Waves
• Wavelength
• Wave Number
• Wave Velocity (Phase Velocity)
• Particle Velocity
• Group Velocity

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Transverse Wave

• A wave that vibrates the medium perpendicular to
  the direction of travel
• Examples
• Ocean waves rise and fall
• Electromagnetic waves
• A rope or a slinky shifted up and down or
  sideways
• Earthquake S waves (secondary waves)

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Longitudinal Waves

• Sound waves compressing the air
• A slinky (or spring) being compressed or expanded
• Earthquake P waves (primary waves)
• Some waves have both longitudinal and transverse
  components

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Periodic Waves

• Waves that repeat with the same waveform
• A nonperiodic wave is often called a pulse

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Sinusoidal Wave

• A sinusoidal wave is described by a sine or
  cosine function.

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Wavelength Period
y
Variation at a point x over time
T
A
t
?
y
Variation at a time over x
A
x
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Equation of a Wave
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Wave Number

• The wave number or angular wave number is defined
  as k 2p / ?
• Remember the angular frequency is
• ? 2p / T 2 p f
• Using these our wave eqn turns into 16.10
• y A sin(kx ?t)
• Also, our speed v ?/k f ?
• NOTE this k is not the spring constant

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Wave Velocity

• Wave Velocity or Phase Velocity is the speed that
  the phase such as the crest or trough of the wave
  travels through the medium

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Group Velocity

• Group velocity is the speed that the envelope or
  the variations in the shape of the wave travel
• This is valid when there is a group of waves of
  slightly different frequency
• This eqn will not be in any test

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Particle Velocity

• The velocity of a particle (real or imagined) as
  it transmits the wave
• This motion may be longitudinal as in sound or
  transverse as a a guitar string on ocean wave.
• The equation is for a velocity in the y
  direction, regardless of whether the wave is
  moving in the y direction

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Partial Differential Wave Equation

• The linear wave equation 16.27 gives a complete
  description of the wave motion including the wave
  speed
• Partials are used since the function is of both t
  and x
• This is good for traveling waves such as
  transverse displacement on a string or
  longitudinal displacement from equilibrium for
  soundwaves

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Example Problem

• Given a transverse wave
• y(x,t) 0.02 sin(3t2x)
• Find a) amplitude b) wavelength, c) frequency, d)
  period, e) velocity, f) particle displacement at
  t1sec and x2m, g) particle velocity there, h)
  acceleration there, i) the concavity of the wave
  there

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• a) amplitude A 0.02 m provided
• b) wavelength, wave eqn has (2p/?)x 2x for
  our problem, so ? p
• c) frequency, wave eqn. has ?t 3t for our
  problem, ? 3 angular frequency
• ? 2 p f, or f ? /2 p 3/6.28 0.478 Hz
• d) period, T1/f 2.09 seconds
• e)velocity, k2 p/ ?, v ?/k 3/2 -1.5 m/s or
  1.5 m/s in the x direction.

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• f) particle displacement y, at 1 second, 2
  meters, y(x,t) A sin (kx?t)
• y(2,1)(0.02) sin(2(2) 3(1)) (0.02)sin(7r)
  sin(458)(0.02) 0.0131 m
• g)particle velocity, ,v(2,1) ? A
  cos(kx?t)
• (3)(0.02)cos(2(2) 3(1)) (0.06)cos(7r)
  0.0456m/s
• h) particle accel.
  ,a-0.118 m/s2
• i)Concavity of the wave,
• rate of change of the change
• -(2)2 cos(7r) -0.05256

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Periodic Motion

• Motion that regularly returns to a given
  position. Mechanical example is an object
  attached to a spring
• Equilibrium position point where the spring is
  neither compressed nor stretched
• Restoring force force directed towards the
  equilibrium position
• When the acceleration of the object is
  proportional to its position, it is Simple
  Harmonic Motion, SHM.

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Hookes Law

• F -kx
• Restoring force always directed towards
  equilibrium
• Newtons second law Fma
• ma-kx, a -kx/m or (k/m) x
• k is the spring constant, m the mass and a is the
  acceleration

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Differential Notation

• dx/dt v velocity
• dv/dt a acceleration
• d2x/dt2 dv/dt a
• d2x/dt2 -(k/m) x
• Let k/m ?2 - renaming this ratio to something
  else
• d2x/dt2 -(?2) x
• find a satisfactory solution for a 2nd order
  differential equation that is a function of the
  position x

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• x(t) Acos(?tF) is a function that is a
  solution to the 2nd differential as well
• dx/dt - ?Asin(?tF) v
• d(dx/dt)/dt - ?2Acos (?tF) a
• note this is - ?2x
• Note the cosine is periodic with ?t
• when ?t reaches 2p radians it has completed a
  full cycle from 0 radians and
• ? sqrt(k/m) because we defined it that way
• F is a phase constant or initial phase angle at
  time t 0

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Amplitude

• x(t) Acos(?tF)
• A is the amplitude or peak displacement
• Since cos varies between /-1.0, A determines
  just how far the object moves away from the
  equilibrium point
• Note A doesnt affect how x changes in time and
  the angle doesnt change the peak amplitude

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Velocity Acceleration

• dx/dt - ?Asin(?tF) v
• d(dx/dt)/dt - ?2Acos (?tF) a
• Since sin varies between /- 1 so
• ?A becomes the peak velocity
• Note while Amplitude (position) doesnt depend
  upon ?, velocity acceleration do

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Period Frequency

• ? is the angular frequency and it is normally in
  radians / second when t is in seconds. Since a
  cycle is complete for 2p radians, then one can
  have a frequency f
• f ?/ 2p or ? 2pf
• f is normally in Hertz, Hz
• The time it takes to complete a cycle is T, the
  period. In mks units T is in seconds
• f 1/T or T 1/f

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Circle of Reference

• Given a circle of diameter A and angle ?t which
  is a function of time one can note that the
  position along the x axis becomes Acos(?t). Also
  the y position is A sin(?t)
• Velocity is proportional to y A sin(?t)
• Observe when x /-A, y0
• when x 0 y is maximum
• Acceleration is proportional to x
• so when x is /- A, a maximum
• when x is 0, a0

y
?t
x
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Energy

• Total energy is the sum of kinetic, Ek or K, and
  potential energy, Ep or U.
• Ek ½ mv2
• E K U
• From chapter 7.8 The work done within a system by
  a conservative force equals the decrease in
  potential energy of the system
• Since F-kx
• U1/2 kx2

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Total Energy