1. Abnormal Amplification of Long Waves in the Coastal Zone
Ira Didenkulova & Efim Pelinovsky
Department of Nonlinear Geophysical Processes,
Institute of Applied Physics,
Nizhny Novgorod, Russia
Wave Engineering Laboratory,
Institute of Cybernetics, Tallinn, Estonia

3. Which bottom profile provides maximal amplification?
Motivation: Role of Each Factor
Which bottom profile provides
maximal amplification?

4. Simplified Linear Theory of 1D Shallow Water Waves
- Wave Speed
(x,t) – Water Displacement
h(x) – Water Depth

5. “Non-Reflected” Beach
Seek Solution of Wave Equation
Two unknown Functions: A and 

6. Exact Separation
- wavenumber
where

7. Energy Flux Conservation
One equation is integrated exactly
Energy Flux Conservation

8. No simple than Initial Wave Equation
Second Equation can not be integrated generally
It is a Variable-Coefficient 2d Order Equation
No simple than Initial Wave Equation

9. If Depth varies smoothly – WKB Approach
eikonal
together

10. Asymptotic Solution for h(x)
Described slowly varied propagated wave
Reflection – beyond asymptotic method
As exp(-1/)
Mathematics: Theory of catastrophes, caustics,
Maslov operator, ray approach….
Arnold, Maslov, Berry, Dobrokhotov, ……

11. Is it Propagated Wave???? To solve:
Second Equation can not be integrated generally
To solve:
1. Existing Analytical Solutions from Books
2. Find h(x) through A(x) – 1st order equation
Is it Propagated Wave????

12. Is it Propagated Wave????
V. Ginzburg – Nobel Laureate in Physics (2003)
Wave propagation in plasma (1968)-YES
L.Brekhovskikh – Leader and Head of Russian
Oceanography and Acoustics
Wave propagation in layer media-NO

13. Overdetermined System
Try to keep Features of Pure Propagated Wave
Overdetermined
System

14. “Non-Reflected” Beach
together with
gives
h(x) ~ x4/3

15. “Non-Reflected” Beach
Propagated Wave
Impulse posses a shape
But it is a singular solution
at x = 0 (h = 0)

16. Velocity Field
WKB amplitude

17. Non-bounded velocity
Big depth

18. It is a Solution, but is it a Wave???
Physical Solution Vanishing on the Ends
Sign-variable pulse
It is a Solution, but is it a Wave???

19. Reduction to constant-coefficient wave equation
The solution
reduces

20. If It proves uniqueness of exact travelling wave solutions
in inhomogeneous media

21. As a result, the general solution (Cauchy problem) can be founded
Natural condition on the shoreline –
boundness of water displacement
As a result, the general solution
(Cauchy problem) can be founded

22. where

23. Piston Model of Wave Generation

24. If initial disturbance is
sign-variable

25. No current

26. Sign – constant initial disturbance

27. current

28. Zoom

29. Runup on beach x4/3
Bounded on shore x = 0 (runup)

30. Velocity Field on Shoreline
But discharge

32. Soliton Runup
Plane Beach

33. “Non-Reflected” Beach
with Reflection
Coefficient:

34. Reflection
from
“Non-Reflected”
Beach

35. “Non-Reflected” Beach
Pulse Reflection from
“Non-Reflected” Beach
i = d/dt
Operator form

36. Impulse Reflection from “Non-reflected” Beach

37. Impulse Reflection from “Non-reflected” Beach

39. Tail -
Distributed
Reflection

40. For any Pulses with vanishing ends
“Pure” Distributed Reflection
“Non-reflected” Pulse

41. From
follows
From boundary condition on jog
follows
as obtained early

42. Nonlinear Traveling Waves in Strongly Inhomogeneous Media
Example: inclined channel of parabolic cross-section

43. Basic Equations - water displacement, u - depth-averaged flow,
S - variable water cross-section of the channel
For parabolic channel

44. Linear Waves If

45. Nonlinear Theory

46. And final linear system
Legendre (Hodograph) Transformation
New variables
And final linear system

48. Nonlinear Traveling Wave

49. Deformation of the wave shape in approaching wave:
blue dashed and red solid lines correspond to an incident wave
and the wave near the shoreline respectively

50. Variation of the positive (red) and negative (blue) amplitudes
with distance; black solid line corresponds to the linear Green’s law

51. Shapes of water displacement (red) and velocity (black)
near the shoreline

53. “Non-Reflected” Potential allows
Conclusions:
“Non-Reflected” Potential allows
To be benchmarks
2. To give simple algorithm to compute wave
propagation above complicated relief,
3. To understand better the wave physics
regimes

54. E. Pelinovsky Hydrodynamics of Tsunami Waves
Nonlinear Dynamics
of Tsunami Waves
E. Pelinovsky
Hydrodynamics
of Tsunami Waves

55. Springer, 2008
Springer, 2009