How Does Tsunami Wavelength Affect Particle Displacement and Wave Equation?

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SUMMARY

The discussion focuses on the relationship between tsunami wavelength and particle displacement, specifically addressing the horizontal particle displacement and the derivation of the wave equation in terms of amplitude (zeta) and nonconstant equilibrium depth (h_0). Participants emphasize the importance of understanding the ratio of vertical to horizontal particle velocity components (uz/ux) to validate the assumption of horizontal water momentum. The conversation highlights the need for a clear approach to deriving the wave equation, indicating that foundational knowledge in wave mechanics is essential for solving these problems.

PREREQUISITES
  • Understanding of sinusoidal plane waves
  • Knowledge of wave mechanics and particle displacement
  • Familiarity with the wave equation and its components
  • Concept of equilibrium depth in fluid dynamics
NEXT STEPS
  • Study the derivation of the wave equation in fluid dynamics
  • Learn about particle velocity components in wave motion
  • Research the effects of wavelength on tsunami behavior
  • Explore the implications of nonconstant equilibrium depth on wave propagation
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Students and professionals in physics, oceanography, and engineering who are studying wave mechanics, particularly those interested in tsunami dynamics and fluid behavior.

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Homework Statement


Wavelength of a tsunami:
upload_2015-9-19_16-35-28.png


Propagates toward shore as a sinusoidal plane wave.
- determine the peak value of the horizontal particle displacement. Is your answer consistent with the assumption that the water momentum is mainly horizontal? Explain by considering the magnitude of the ratio uz/ux (vertical/horizontal components of particle velocity)

- rederive the wave equation in terms of zeta (amplitude) allowing for nonconstant equilibrium depth h_0

Homework Equations



upload_2015-9-19_21-19-29.png
[/B]
Wave equation
upload_2015-9-19_21-7-31.png

The Attempt at a Solution


[/B]
I thought the horizontal displacement would just be the wavespeed times the period but that would just give the wavelength so that couldn't possibly be right. As for deriving the wave equation, I don't even know where to start on that one. Any help would be appreciated
 
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