Seismic wave hyperbolic PDE ?

We have gotten the band structure of a seismic matematerial. Now, we want to distinguish the surface wave and body wave. How to extract the Rayleigh wave modes from the band structure of the seismic matematerials in COMSOL Multiphysics ? Thanks !

Practically it is known that seismic velocity increases as rock density increases, why then does the wave velocity equation involving elastic moduli and density indicate that velocity is inversely related to density?

The wave diagram has been attached herewith.What is the reason of such wave?Is it noise or signal?

I working on radiation pattern to seismology ,please help me about how plot it for seismic wave at an earthquake?

I have attached a paper below which describes a multi-region method for solving for the wave characteristics around a submerged porous structure. I am struggling to solve the last four equations, (23) - (26), (32) and (33). My equations are indeterminate because of the starting points A(M1+1, q) and B(0, q). If anyone could please give me some insight into how to solve these four equations I would be very appreciative.

Thanks for your time.

Kind Regards,

Alex Wylie.

I selected some seismic waves from the peer, but only the information related to the fracture surface. May I ask how to query the epicenter depth and epicenter distance?

I think it is clear physicists do not understand the symmetry of time independence on the string.

Time independence is a marvelous property that can be stated in a large number of equivalent ways. I am hoping one of these statements will trigger a physicist into understanding what string rigidity means.

1) The string obeys the Hamiltonian principle of least action.

2) The string conserves potential energy in a closed system.

3) The string is rigid, not elastic.

4) The boundary condition on the string is 𝜟x = 0, not 𝜟x ≧ 0.

5) String symmetry is time independence, and the standing wave does not depend on time.

6) String mechanics are naturally symplectic.

7) The string manifold is smooth with a natural tangent-cotangent vector field on the real coordinates of the string line.

8) The fundamental mode is the 1-periodic Hamiltonian solution, and frequency is an extremal that measures the volumetric capacity of the string manifold.

9) The motion of n points on the string creates n world lines that are holomonically constrained to those lines in Euclidean space along which energy is conserved.

10) Let x:ℝ → ℝ3 be a motion in ℝ3.The graph of this mapping is a curve in ℝ x ℝ3.

11) The string submanifold is a complex toroidal disc.

12) The string has a natural Liouville integral on the submanifold.

13) Perturbation theory applies to string mechanics.

All of the above statements are true.

The following statements are all equivalent to the assumption of time dependence which I am intentionally stating in a way that can be seen to be false.

1) The direct observation of trigonometric wave forms on an oscilloscope based on the transduction of sound waves and reflected light emitted by the strings into an electrical current proves the string action is explained by a partial differential equation (PDE) where the world lines of motion are embedded in the arbitrary plane of the oscilloscope screen.

2) The equation of motion on the oscilloscope is given by the PDE is “Let x:ℝ → ℝ2 be the planar motion and the graph of this mapping is a curve in ℝ x ℝ2 which parameterizes a planar graph by time like the oscilloscope screen but is not symplectic and cannot form a smooth manifold.

3) The boundary condition on the string are fixed endpoints and there is no limit on how much the elastic string can stretch (without an external force).

4) A string can vibrate in many, possible an infinite number of modes in the space of simultaneous events without violating energy conservation.

5) Potential energy is not defined on elasticity.

6) Modes of string vibration are sine waves equipped with an addition function on displacement allowing nodes and waves to add without violating natural law.

7) The string can bend and stretch into any shape during vibration without external force or internal constraints.

8) The standing wave is not really standing because it alternates phase with each cycle.

9) Kinetic and potential energy are exchanged like a pendulum.

10) Frequency is proportional to velocity, not potential.

11) Waves on the string propagate left and right, reflect at boundary endpoints and combine to make a standing wave that moves.

12) The standing wave stands down when the string comes to rest like a pendulum where potential and kinetic energy have run down to zero.

13) String frequency and amplitude are not determined by the same equation.

Amplitude decay seems proof the string is time independent. But decay is not independent of time if it follows a curve that is cycloidal (tautochrone) because then the interval of amplitude decay is always the same regardless of how much the string is accelerated.

The equation of motion controls the minimization of kinetic energy.

It might seem that we are required to know the acceleration of the string but in classical mechanics this is not true. How hard the string is plucked does not affect motion.

Suppose I have a non-self adjoint operator in a PDE which, being nonself adjoint, has deficiency indices in its spectrum and therefore requires an extension. Say that I solve this PDE numerically and want to analyze the graphical functions. Can these functions make some physical sense, although the inherent operator of the PDE is non-self adjoint?

I am really asking this because many rogue wave phenomena *appear* to be described by non self adjoint operators, and after transformation of their related PDE, physically sensible analytical and numerical solutions can always be gained. For instance, the Darboux transformation or the generation of a LAX pair is necessary to solve the NLSE, but is it because the NLSE is non-self adjoint and would otherwise give meaningless numerical results?

As an example:

Suppose I were to analyze the quartic Schrödinger equation:

(D²-V⁴)*Psi=E*Psi

where the quartic well potential replaces the quantum potential V² from the regular Schrödinger equation. Here the operator (D²-V⁴) acting on the wavefunction Psi is non-self adjoint, and has deficiency indices in its spectrum (requires an extension).

However, if I analyze this equation as it is numerically, and try to generate its solutions for arbitrary values of E, I get plots as attached. These plots show decaying oscillatory behavior further and further away from the origin, which is similar to the standard Schrödinger under an electromagnetic field for particles. But can these results be used to describe a phenomenon described by the quartic equation or does one have to transform it first to a form that yields a self adjoint operator on Psi first?