Solving Shallow Water Equations with Kelvin Wave Solutions

  • Thread starter groovayness
  • Start date
  • Tags
    Wave
In summary, the shallow water equations are used to describe various phenomena such as wave and tidal motion, and can be modified to account for the Earth's spherical geometry. Eta, u, and v are expressions that solve these equations for a Kelvin wave when the wave speed is equal to the square root of gh. The task at hand involves differentiating and plugging the given formulas into the equations.
  • #1
groovayness
6
0
someone please help

The shallow water equations are used to describe, among other things, wave motion in coastal areas, and tidal motions (even across the entire globe, when suitably modified to account for the Earth's spherical geometry). As discussed in class these equations are
(attachment 1)


Eta is the local height of fluid above the equilibrium depth, h.

Show that the expressions for eta, u, and v, below, solve the shallow water equations when the wave speed, omega/k, is (gh)1/2. This is called a Kelvin wave.
(attachment 2)
 

Attachments

  • 1.jpg
    1.jpg
    4.8 KB · Views: 397
  • 2.jpg
    2.jpg
    5.3 KB · Views: 343
Physics news on Phys.org
  • #2
Welcome to PF.
Looks like you just have to differentiate and plug the given formulas into the given equations. Where is the problem?
 

1. What are shallow wave equations?

Shallow wave equations are mathematical equations that are used to model the propagation of water waves in shallow bodies of water. They are derived from the Navier-Stokes equations and are valid for small-amplitude waves in shallow water.

2. What is the importance of shallow wave equations?

Shallow wave equations are important because they allow scientists and engineers to predict the behavior of water waves in shallow bodies of water, such as coastal areas and lakes. This information is crucial for designing structures, such as seawalls and breakwaters, and for understanding the impacts of storms and tsunamis.

3. How are shallow wave equations used in real-world applications?

Shallow wave equations are used in a variety of real-world applications, including coastal engineering, marine navigation, and oceanography. They are also used in the design and analysis of wave energy converters, which harness the energy of ocean waves.

4. How do shallow wave equations differ from deep water wave equations?

Shallow wave equations differ from deep water wave equations in that they are valid for small-amplitude waves in shallow water, while deep water wave equations are valid for large-amplitude waves in deep water. This means that shallow wave equations are more accurate for modeling waves in coastal areas and other shallow bodies of water.

5. What are the limitations of shallow wave equations?

Shallow wave equations have limitations in that they are only valid for small-amplitude waves in shallow water. They also do not account for factors such as wave breaking and turbulence, which can significantly affect the behavior of water waves. Additionally, they are only applicable in certain situations and may not accurately predict the behavior of waves in extreme conditions.

Similar threads

I Shallow Water Wave Equation Simulation: Reverse Shoaling?
    • Classical Physics
Replies
1
Views
1K
Fluid mechanics - Linearized shallow water equations
    • Mechanics
Replies
2
Views
964
Finding the correct formula for the speed of waves in shallow water
    • Introductory Physics Homework Help
Replies
1
Views
1K
Refraction of Ocean waves: Wide Headland vs Narrow
    • Optics
Replies
22
Views
7K
A Wave equation in free space
    • Differential Equations
Replies
17
Views
2K
MHB Solving the Wave Equation with Initial Data: Ex. 5 Solution and Derivation
    • Differential Equations
Replies
11
Views
2K
Shallow Water Equations and Initial Conditions
    • Differential Equations
Replies
1
Views
2K
Fortran 1D Shallow Water Wave in FORTRAN using LAX WENDROFF Method
    • Programming and Computer Science
Replies
3
Views
2K
Water waves travelling from deep to shallow water
    • Introductory Physics Homework Help
Replies
4
Views
1K
B Traveling wave solution notation
    • Classical Physics
Replies
4
Views
986

Hot Threads

Recent Insights

Back
Top