Spectral Analysis of Wave Motion

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Spectral Analysis of Wave Motion

• Dr. Chih-Peng Yu

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Elastic wave propagation

• Unbounded solids
• P-wave, S-wave
• Half space
• Surface (Rayleigh) wave
• Double bounded media
• Lamb waves
• Slender member

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2001 Fall

• Waves in Slender members
• longitudinal wave
• flexural wave
• torsional wave
• Waves by different approximate theories
• Elementary member
• Deep member

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2002 Spring

• General derivation of waves in solids
• P-wave, S-wave, Surface (Rayleigh) wave
• Modification due to bounded media
• Lamb waves

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General Function of Space and Time

• At a specific point in space, the spectral
  relationship can be expressed as
• In general, at arbitrary position

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• Imply (discrete) Fourier Transform pairs
• Or, in a simpler form as

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Spectral representation of time derivatives

• Assuming linear functions
• Or, for simplicity,

discrete
continuous
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• Derivatives of general order
• It is clear to see the advantage of using
  spectral approach
• time derivatives replaced by algebraic
  expressions in Fourier coefficients gt simpler

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Spectral representation of spatial derivatives

• Nothing special
• Or, for simplicity,

discrete
continuous
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• It is clear to see another advantage of using
  spectral approach
• partial differential equation becomes ordinary
  differential equation in Frequency domain gt
  solution form is solvable or at least easier to
  be solved
• This is also true for using other transform
  integrals, such as Laplace Transform,
  Bessel-Laplace Transform

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Spectral relation

• Consider a general, linear, homogeneous
  differential equation for u(r,t)
• with all coefficients independent of time
• Assume one dimensional problem

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• The spectral representation of the general
  differential equation becomes

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• ei?nt is independent for all n. Thus the spectral
  representation results in n simultaneous
  equations as
• Or, in a general form as
• Aj depend on frequency and are complex.

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• When Aj(x,?) independent of position, the
  original partial differential equation has been
  transformed into n simultaneous ordinary linear
  differential equation.
• The solution form is e?t , the transformed ODE
  becomes then
• The equation in the ( ) is called
  characteristic equation, which can be solved to
  give values for ?
• ? can be complex, so the solution form is in a
  form as

with ? ? ik
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with ? ? ik

• ? is referred to as the attenuation factor of the
  wave motion. It represents the non-propagating
  and the attenuated components of the wave.
• k is the wave number. It represents the
  propagating parts of the wave.
• So, for a propagating component of the wave, the
  solution can be expressed as

stands for the traveling direction (to the
right or left)
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Propagating speeds

• Consider the propagating component
• The time response is then in the form as

j represents the number of characteristic
constant ?
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• The time response is then in the form as
• For each ?j, we can see the response corresponds
  to (infinite) sinusoids traveling with a speed of
• cj is called the phase speed corresponding to ?j

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• So, for a specific ?j with only components
  traveling towards one direction (say -kx), we
  have the wave response expressed as
• Consider the interaction between two propagating
  wave components, the resultant response is thus

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• The first sinusoid is the average response called
  carrier wave . It travels at the average speed of
  the two interacting wave components, c ? /
  k.
• The second term represents the modulated effect
  between the interacting components. This is
  called group wave traveling at a speed

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• It can be expected when there are many waves
  interacting together, the overall effect would be
  a carrier wave modulated by a group wave.
• In reality, the individual sinusoids is hard to
  be observed unless through an FFT scheme.
• The wave energy and varying amplitude of the wave
  envelope travel at group speed.

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Transfer function

• Lets exam again the displacement function

• In a displacement force relationship, transfer
  function is then the inverse of dynamic stiffness
  function.

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Summary of wave terms

• Angular frequency (rad/s) ?
• Cyclic frequency (Hz) f ? / 2?
• Period (sec) T 1/f 2? / ?
• Wave number (1/length) k 2? / ? ? / c
• Wave length (length) ? 2? c / ? 2? / k
• Phase (rad) ? (kx - ?t)
• Phase velocity (length/s) c ? / k ?? / 2?
• Group speed (length/s) cg d? / dk

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Specific terms

• Spectrum relation ? vs k
• Dispersion relation ? vs c
• Non-dispersion phase velocity is constant for
  all frequency
• Evanescent wave the attenuated non-propagating
  components of waves
• Carrier wave main zero-crossing sinusoid waves
• Group wave modulation of wave groups

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Simple wave examples

• Wave equation of the 1-D axial member
• Non-dispersion
• Flexural wave in a beam
• dispersion