Fluid mechanics - Linearized shallow water equations

In summary, the conversation is about finding a solution to linearized shallow water equations. The equations involve a steady state of velocity and height of free surface, as well as disturbances u' and h'. The goal is to solve for h'(x,t). The governing equations are the momentum equation and mass continuity equation, which can be combined to arrive at a wave equation. However, the person is having trouble eliminating the term including u' in the wave equation and is looking for a step-by-step guide on how to do so. They also mention that they have found a solution and provide intermediate steps.
  • #1
kmot
7
1
Hi,

In a text describing solution to linearized shallow water equations, I am not able to move forward.

It's a 1 dimensional shallow water setup. There is a steady state
1608989852741.png
(velocity) and
1608989862952.png
(height of free surface). On top of this steady state there are u' and h' as disturbances. The goal is to solve for h'(x,t)

1608989771671.png


The governing equations are:
(1)
1608989324189.png
(momentum)
(2)
1608989337300.png
(mass continuity)

Subscripts denote partial derivatives.

The text says these two can be combined to arrive at wave equation, without providing further details of how:
(3)
1608989399570.png


But with anything I tried I wasn't able to eliminate u' completely. I'm always left with a term including u' in (3). Perhaps the term that can't be eliminated can be neglected, but I don't see why.

Would somebody be willing to provide a step by step guide how to go from (1) & (2) to (3), including reasoning of why something can be neglected (if neglecting is needed)?

Thank you very much
 
Last edited:
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  • #2
I found the solution. Here are the intermediate steps
1609003715700.png
 
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  • #3
kmot said:
I found the solution. Here are the intermediate steps
View attachment 275166 8 ball pool
Okay, that makes sense. Thanks for the answers.
 

Related to Fluid mechanics - Linearized shallow water equations

1. What is fluid mechanics?

Fluid mechanics is the branch of physics that deals with the study of fluids (liquids and gases) and their behavior under various conditions. It involves the study of how fluids move, interact with their surroundings, and how they respond to external forces.

2. What are the linearized shallow water equations?

The linearized shallow water equations are a set of simplified equations used to describe the behavior of shallow water waves. They are derived by linearizing the full Navier-Stokes equations, assuming small amplitude and long wavelength waves, and neglecting terms related to viscosity and turbulence.

3. What is the significance of the linearized shallow water equations?

The linearized shallow water equations are important in the study of oceanography, meteorology, and coastal engineering. They provide a simplified model for understanding the behavior of shallow water waves and can be used to predict the movement and behavior of these waves in various scenarios.

4. How are the linearized shallow water equations solved?

The linearized shallow water equations can be solved using various numerical methods, such as finite difference, finite volume, or finite element methods. These methods involve breaking the equations into smaller discrete parts and using iterative calculations to approximate the solution.

5. What are some applications of the linearized shallow water equations?

The linearized shallow water equations have many practical applications, including predicting storm surges, studying ocean currents and tides, designing coastal structures, and understanding the behavior of tsunamis. They are also used in the simulation of fluid dynamics in computer graphics and animation.

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