Classical I.e. not Quantum Waves

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• Classical (I.e. not Quantum) Waves

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• Find speed of wave on a string, which is flexible
  and not displaced much
• method of Tait
• the force downward, radial with respect to the
  curve at the top of the bump, is equal to the
  component of the tension on each side

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• The mass of a segment of a string with mass per
  unit length ? is
• The idea of this calculation is to equate the
  radial force with the centripetal force
• solving for v

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• Another calculation, with the kink, is
  instructive

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• There is an unbalanced force on the kink. The
  right hand end has no y component force, the left
  end has a force
• the impulse equals change in momentum
• and now we use the small angle approximation that
  sin x tan x, so

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• Reflections at the end
• open end pulse reflected with no change in sign
• fixed end pulse reflected inverted
• momentum conservation can be used to conclude
  this behavior

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• Reflections at the end
• open end pulse reflected with no change in sign
• fixed end pulse reflected inverted
• momentum conservation can be used to conclude
  this behavior

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• Traveling Waves-start with Harmonic Waves,
  exciting with a simple harmonic motion of
  frequency f, wavelength ? v/f
• We have seen that the travelling wave travels
  with speed v, so in full generality the wave
  function is (NON-DISPERSIVE)
• and the harmonic wave excited by SHM is
• Note k is called wave number and has nothing to
  do the the k in Hookes law used in the previous
  slide on SHM!!
• It is useful to write the wave function in some
  other ways

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• Sometimes we want the choice of the origin of x
  and t to be arbitrary, so we have to throw in an
  initial phase ?
• We like to define another 2? killer ?2?f
• The key here is the traveling wave form, which
  comes from the wave equation, which we will soon
  derive for the string

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• Review of Simple Harmonic Motion

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• SHM formula application to wave
• Energy transported by wave--any bit of the string
  is performing SHM, with amplitude A
• and in a time dt the oscillation moves along by a
  distance dxvdt, to the power passing by in one
  second is

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• String fixed at 0 and L Standing Waves
• the end points have to stay at zero y0
• at the point x0, this follows from the form sin
  kx. At the other end, it implies a relation
  between k and L, sin kL 0

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• Some trig what if we have two traveling waves
  going in opposite directions
• and with identity
• sin(ab)sin a cos b cos a sin b
• and we see that this is the wave function for the
  standing waves

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• What if we add (traveling) waves with different
  wave lengths?
• So we get a difference frequency or beat
  frequency -this can be written as a oscillation
  with frequency f1f2 and variable amplitude
• amplitude

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• Wave Equation on String, generalize our analysis
  of kink
• the mass of this bit of string is ? ds? dx, and
  Fma ?

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• Or, canceling the dx
• We have noted that this has any solution of the
  form
• that is, traveling, non-dispersive waves, which
  lead to standing wave on the finite string.

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• Solutions of the Wave Equation 1. Traveling
  Wave solutions

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• 2. The Normal Modes of the string with fixed
  ends
• A successful separation of variables! The
  equation has to be true for all x and t, can only
  be so if each is equal to the same constant, call
  it -?2

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• This is the familiar SHO equation, with solutions
• but the cos kx does not fit the fixed ends,
  y(0)y(L)0, so knn?/L, so?nn ?v/L
• The wave equation is linear, so sums of solutions
  are also solutions. The shape at time zero can
  be what we please, so we need both sin and cos in
  t, and then the general solutions are

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Special Problem, preparation for Test, Due
Monday 18 October, questions answered 13,15 Oct.

• Power Spectrum of String-
• A string is 4 meters long, fixed at both ends.
  It weighs 80 grams in total, and carries a
  tension of 100 Newtons.
• The middle half is lifted from the equilibrium
  position by 5mm, like this
• What is the power in the vibrations in the first
  ten active modes, or harmonics?