1. Hamiltonian formalism, seismic waves, and seismic barriers
Sergey V. Kuznetsov
Institute for Problems in Mechanics
Russian Academy of Sciences
web:

2. A unified theory of surface acoustic waves based on the generalized Hamiltonian formalism

3. Equation of motion and representation for Surface Acoustic Waves with plane wave-frontν
Representation for the wave
Splitting the equation of motion

4. Matrix ordinary differential equation of motion of the second order
Where
(will be needed further)

5. 6D Hamiltonian formalism
New function
Equation of motion
System of two ODEs
System in a matrix form

6. Exponential fundamental solution and boundary conditions (for Lamb waves)
Traction-free:
Clamped:

7. Dispersion equations for a single layer (conditions for existing non-trivial solutions)
Traction-free plate:
Clamped plate:

8. Generalization: Dispersion equation for a N-layered plate (Modified Transfer Matrix Method)
Traction-free n-layered plate:

9. Numerical results Dispersion curves for SH waves
in 31-layered composite plate
(phase speed vs. circular frequency)
Variation of the lower dispersion curve at varying depth of the median layer

10. Analytical results: A long-wave limit (soliton-like waves)
A condition for obtaining the limiting wave speed
where
The limiting Lamb wave speed for a N-layered isotropic plate

11. The main types of seismic waves

12. The main types of Acoustic Waves initiated at earthquakes: I
The main types of Acoustic Waves initiated at earthquakes: I. Bulk waves
Definition:
Bulk waves are either plane, spherical, or cylindrical (harmonic) waves that propagate in an infinite space.
Described:
Both longitudinal (L) and shear (S) waves were described by Siméon Denis Poisson (1828)
Remarks
In isotropic materials cL>cS . That is why longitudinal waves
are also called as P-waves (Primary-waves)
In anisotropic materials longitudinal waves can have smaller
speed than shear waves (e.g. TeO2)

13. The main types of Acoustic Waves initiated at earthquakes: I
The main types of Acoustic Waves initiated at earthquakes: I. Bulk waves
Representation for bulk wave with plane wave-front:
Christoffel equation:
Another form of Christoffel equation:
Remarks
In isotropic medium:

14. The main types of Acoustic Waves initiated at earthquakes: II
The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves n ν
Definition:
A wave that
(i) propagates on a homogeneous elastic halfspace with traction-free boundary conditions;
(ii) attenuates exponentially with depth;
is called the Rayleigh wave.
Described:
These waves were theoretically discovered
by Lord Rayleigh in 1885

15. The main types of Acoustic Waves initiated at earthquakes: II
The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves
Remark concerning “forbidden” directions
For a long time it was supposed that there can be anisotropic materials (may be artificial), that possesses specific directions along which Rayleigh waves cannot propagate. These hypothetical directions (and materials) where called “forbidden”.
Barnett and Lothe ( ) proved a theorem on existence of Rayleigh wave propagating on a free surface of any anisotropic halfspace and any direction.
Later on Ting (1998) and Kuznetsov (2001) considered a case of the non-semisimple degeneracy of the Jacobian resulting in a wave of the “non-Rayleigh” type.

16. The main types of Acoustic Waves initiated at earthquakes: II
The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves
A principle for creating horizontal barriers against Rayleigh waves n ν
Theorem of nonexistence for Rayleigh waves in a clamped anisotropic halfspace /Chadwick and Smith (1977)/.
If the halfspace surface is clamped, then no Rayleigh wave can propagate

17. The main types of Acoustic Waves initiated at earthquakes: III
The main types of Acoustic Waves initiated at earthquakes: III. Stoneley waves n ν
Definition:
A wave that
(i) propagates on an interface between homogeneous elastic halfspaces in a contact;
(ii) attenuates exponentially with depth in both halfspaces;
is called the Stoneley wave.
Described:
These waves were theoretically discovered
by Robert Stoneley in 1924

18. The main types of Acoustic Waves initiated at earthquakes: IV
The main types of Acoustic Waves initiated at earthquakes: IV. Love waves n ν
Definition:
A wave that
(i) propagates on a homogeneous elastic halfspace in a contact with a layer with the traction-free boundary conditions;
(ii) attenuates exponentially with depth in a halfspace;
is called the Love wave.
Described:
These waves were theoretically discovered
by Augustus Love in 1911

19. The main types of Acoustic Waves initiated at earthquakes: IV
The main types of Acoustic Waves initiated at earthquakes: IV. Love waves
Theorem of non-existence for Love waves propagating in an isotropic layer and a halfspace /actually Love/.
If speed of the transverse bulk wave in a layer is greater than in a halfspace:
then Love wave cannot propagate.
Corollary.
Since the condition of nonexistence becomes:

20. Seismic waves: experimental data

21. Frequency range Wave nature Frequency range, Hz Natural seismic waves
Most dangerous for civil structures: 10÷30Hz
Anthropogenic seismic waves Hz
Most dangerous for civil structures: 10-50Hz

22. Velocity range (Longitudinal bulk waves)
Material
Speed m/sec
air
Bullet in the air
~800
soil
sand
water
slate (shale)
limestone
granite

23. Wavelength range (Rayleigh waves)
Material
Wavelength range, meters
soil
5 - 30
sand
4 - 20
slate (shale)
limestone
granite
Remark

24. Seismic waves and barriers: FEM simulation

25. Seismic barriers,
Where are they needed?

26. Two types of seismic barriers
Longitudinal barrier
(“thin and long”)
The main principle:
to prevent surface waves from propagation
Transverse barrier
(“thin and deep”)
The main principle:
To reflect most of the wave energy 2

27. 2D FEM simulation of the seismic wave propagation near clamped surface (harmonic surface loading)
Magnified:

28. 2D FEM simulation of the seismic wave interaction with horizontal barrier (harmonic surface loading; a quarter of plane)

29. 2D FEM simulation of the seismic wave interaction with composite vertical barrier (harmonic surface loading; quarter of a plane)

30. 3D FEM simulation of seismic waves near the hypocenter

31. Longitudinal circular barrier against Rayleigh waves (Chadwick’s theorem)

32. Longitudinal circular barrier near the epicenter (Chadwick’s theorem)

33. Transverse vertical barrier against Rayleigh waves (principle of maximum reflection)

34. Another type of seismic barriers
A barrier composed of piles optimized for best scattering
A known result for scattering of bulk waves
Hasimoto, , Scattering of viscous fluid flow by spheres
Datta, , Scattering of elastic waves by ellipsoidal inclusions
Bose, Mal, , Scattering shear bulk waves by fibers
Gubernatis, 1979, Scattering of elastic waves by inclusions
Willis, , Long-wave limit for scattering of elastic waves
Kuznetsov, , Scattering of elastic waves in anisotropic media
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The best scattering (the largest scattering cross-section) is achieved by voids

35. Hollow piles

36. Clamped piles: single row

37. Clamped piles: two rows

38. Where to choose a building site?

39. Possible geological profile

40. FEM modeling of seismic wave propagation