Relationship between Probability density and Spectrum

1
Relationship between Probability density and
Spectrum

• An interconnection of the statistical properties
  of random wave field

2
Motivations

• We have seen the proposed spectral forms mostly
  consisted of products of a power function and an
  exponential terms.
• How can we be sure that the functional form
  indeed is of such type?
• From probability properties of the wave field, we
  have various distribution of frequency,
  amplitude. Are there any connections between the
  probability density and spectral function?

3
Joint Amplitude and Period Density Functions

• If we know the joint amplitude-period
  distribution, we could derive a quantity that
  would have the physical property of a spectrum

4
Joint density Longuet-Higgins (1983)
5
Verification
6
Joint density Yuan (1982)
7
Verification
8
Verification
9
Verification
10
Marginal Probability Longuet-Higgins
11
Marginal Probability Yuan
12
Verification
13
Verification
14
Verification
15
Conclusion

• So, there is a relationship between probability
  density function and spectral function based on
  Fourier analysis.
• This relationships exists only from the energy
  containing point of view. In other words, it
  works only for energy containing range of the
  spectra.
• Fourier spectra works for wider bands than the
  probability density functions, which are all
  derived under narrow or near narrow band
  assumption.

16
A turning point in my research

• My effort to study the nonlinear Schrödinger
  equation

17
Types of Waves

• Water wave motion was amongst the first fluid
  mechanics treated successfully by mathematics.
• John Scott Russell observed a solitary wave in a
  barge canal in 1834 Shallow Water Waves.
• George Gabriel Stokes derived deep water
  periodic wave of permanent shape in 1847 Deep
  Water Waves.

18
Shallow Water Waves

• Solitary Waves

19
Solitary waves

• His experimental observations were viewed with
  skepticism by George Airy and George Stokes
  because their linear water wave theories were
  unable to explain them.
• Joseph Boussinesq (1871) and Lord Rayleigh (1876)
  published mathematical theories justifying Scott
  Russells observations.
• In 1895, Diederik Korteweg and Gustav de Vries
  formulated the KdV equation to describe shallow
  water waves.
• The essential properties of this equation were
  not understood until the work by Kruskal and his
  collaborators in 1960s.

20
Solitary wave and soliton

• The name soliton was coined by Zabusky and
  Martin Kruskal. However, the name solitary wave,
  used in the propagation of non-dispersive energy
  bundles through discrete and continuous media,
  irrespective of whether the KdV, sine-Gordon,
  non-linear Schrödinger, Toda or some other
  equation is used, is more general.
• Kruskal received the National Medal of Science in
  1993 for his influence as a leader in nonlinear
  science for more than two decades as the
  principal architect of the theory of soliton
  solutions of nonlinear equations of evolution.

21
Other soliton equations

• Sine-Gordon equation
• Nonlinear Schrödinger equation
• Kadomstev Petviashvili (KP) equation

22
KdV solitary wave solution

• Korteweg and de Vries (1895) discovered the
  equation possesses the solitary wave solution
• KdV equation
• Traveling wave solution (Kruskal)

23
Solitary Wave 2 solitons

• Two solitons travel to the right with different
  speeds and shapes

24
Solitary Waves
25
Solitary Waves
26
Solitary Waves
27
Shallow Water WavesThe governing equations

• What are the assumptions?
• Are they reasonable?

28
Shallow Water Waves Governing Equations I
Starting from the KdV equation, We can derive
the Nonlinear Schrödinger Equation, Provided
that the wave number and frequency are constant
to the third order.
29
Governing Equations II
We will have a different Nonlinear Schrödinger
Equation, Provided that the wave number and
frequency are constant to the second order.
30
Governing Equations III
We will have still another form of
NSE, Provided that the wave number and
frequency are constant to the first order.
31
What if the frequency is constantly changing?

• The frequency of all nonlinear waves are
  intra-wave modulated.

32
Deep Water Waves

• Periodic waves

33
Stokes

• George Gabriel Stokes 1819-1903
• Lucasian Professor
• President of Royal Society
• Navier-Stokes equations
• Stokes theorem
•   

34
The Classical Stokes Waves

• Stokes waves are periodic waves of permanent
  shape.
• The solution was derived in 1847 by Stokes using
  perturbation method.

35
The Classical Stokes Waves

• Stokes waves was treat as the standard solution
  for more than hundred years. It describes the
  strong nonlinear effects that caused the wave
  form distortions.

36
The Classical Stokes Waves

• Phillipss (1960) theory on the dynamics of
  unsteady gravity waves of finite amplitude opened
  a new paradigm through 3rd order resonant
  interactions (a much weaker nonlinear effects),
  the wave envelopes could change over a long time
  compared to the wave period.
• At that time, the Ship Division of the National
  Physical Laboratory built a wave tank, but the
  wave maker could not generate periodic wave of
  permanent shape.

37
Unstable Wave Train National Physical
Laboratory
38
The Classical Stokes Waves

• There was almost a law suit for the incompetence
  of the wave maker contractor.
• Then, Brook-Benjamin and Feir (1967) found that
  the Stokes wave was inherently unstable.
• Later, it was found that this instability was a
  special case of Phillipss 3rd order resonant
  interactions.
• A new era was down for deep water periodic wave
  studies.

39
Phillips Resonant Theory I
40
Phillips Resonant Theory II
41
In terms of Spectral RepresentationHasselmanns
Formula by Zakharov
42
In terms of Spectral RepresentationHasselmanns
Formula by Zakharov
43
In terms of Spectral RepresentationHasselmanns
Formula by Zakharov
44
NASA Wind-Wave Experimental Facility
45
Data
46
What should be the governing equations

• Based on Phillipss resonant theory, Hasselmann
  (1962, 1963, 1966) formulate the resonant
  interaction in spectral form.
• Meanwhile, the TRW group (Lake and Yuan etc.
  1975, 1878) formulated the wave evolution in
  nonlinear Schrödinger equation, and claimed
  Fermi-Ulam-Pasta recurrences They are the most
  successful wave research group in the US.

47
Unstable Wave Train Su
48
What should be the governing equations

• Unknown to the west, Zakharov (1966, 1968) had
  already derived the nonlinear Schrödinger
  equation from Hamiltonian approach.
• The nonlinear Schrödinger equation is in terms of
  envelope.
• The carrier should be water waves the envelope,
  a soliton.

49
What should be the governing equations

• This form is not just a group formed by beating
  of two independent free wave trains.
• An example of the Sech envelope soliton is given
  below

50
Deep Water WavesDysthe, K. B., 1979 Note on
a modification to the nonlinear Schrodinger
equation for application to deep water waves.
Proc. R. Soc. Lond., 369, 105-114.

• Equation by perturbation up to 4th order.
• But ? constant.

51
Governing Equations I
52
Governing Equations II
53
Governing Equations III
54
Governing Equations IV The 4th order
Nonlinear Schrödinger Equation
55
Observations

• All the published governing equations are in
  terms of envelope, which is governed by a cubic
  nonlinear Schrödinger equation.
• The carriers are assumed to be of constant
  frequency.
• The constant carrier frequency assumption is
  untenable.

56
Conclusions

• Most people studying waves are actually studying
  mathematics rather than physics.
• But it is physics that we should understand.
• We need new paradigm for wave studies.

57
My Path

• I started to explore the envelopes.
• A natural way was to turn to Hilbert Transform.
• Once I used Hilbert transform, I found solutions
  as well as problems it open a new view point not
  only of water waves but also the whole world.

58
Reminiscence

• By the time (1990) I finished these studies, I
  thought I had found the key to ocean wave study
  the significant slope, S.
• Then, I studied the governing equations of water
  wave motion, the nonlinear Schrödinger equation,
  and conducted laboratory experiments to compare
  with the theoretical results.
• In that effort, I used the Hilbert Transform to
  analyze the laboratory data the results shocked
  me. Fortunately, I made a mistake in the
  processes. HHT, born through my efforts to
  correct that mistake. The rest is history. and
  the subject of this course.