How Do You Simplify a Wave Equation in a Non-Uniform Pipe?

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Homework Help Overview

The discussion revolves around simplifying a wave equation that describes sound waves in a pipe with a varying cross-section. The equation involves derivatives with respect to both space and time, and participants are exploring how to manipulate it effectively.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to substitute specific expressions into the wave equation and are discussing the application of the chain rule. There are questions about the next steps in the simplification process and comparisons to general forms of second-order hyperbolic equations.

Discussion Status

The discussion is active, with participants providing suggestions and exploring different methods for simplification. Some guidance has been offered regarding the use of the chain rule, but there is no explicit consensus on the correctness of the attempts made so far.

Contextual Notes

There is a focus on simplifying the equation rather than solving it, and participants are comparing their results to a general form of the equation to identify parameters. The original poster's approach and the assumptions made about the variable A are under scrutiny.

andrey21
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Wave Equation (urgent)

Sounds waves in a pipe of varying cross-section are described by the wave equation

v2 d/dx .(1/A.dAu/dx) = d2u /dt2

Where A = 0.2 +0.3x simplify the equation


My attempt at a solution

Sub in A:

v2 d/dx ( 1/(0.2+0.3x) . d(0.2+0.3x)u/dx) =d2u/dt2

Not to sure where to go from here
 
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Are you trying to solve the equation? If so, you should perhaps check out the method of separation of variables.
 


No just simplify the equation then compare with general second order hyperbolic equation to find a,b,c, and e
 


I'm seeing some chain rule in your future...
 


So by applying the chain rule this is my solution:

V2 (1/0.2+0.3x) .(0.3u + (0.2+0.3x)du/dx)

Is this correct?
 

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