Energy flow in the wave equation (PDE)

[Brian4455]

Homework Statement

I have a problem that I'm trying to make sense of. Note y_t is the partial derivative of y with respect to t and y_tt is the second order partial derivative of y with respect to t, etc. The complete problem statement is the following:

Show that for the equation y_tt - c^2 y_xx = 0
the quantity E = 1/2(y_t^2 + c^2 y_x^2)
satisfies a conservation law dE/dt + dJ/dx = 0

Homework Equations

The Attempt at a Solution

I calculated dE/dt to = y_t * y_tt + c^2 y_x * y_xt so dJ/dx must equal the negation of dE/dt. But I'm not sure where J comes from. I'm guessing that it is somehow related to y through the wave equation but I'm not sure how. Also it is unclear to me how I could integrate the negation of dE/dt to arrive at J. I gave the complete problem statement. In that chapter of the book the function J is used but I don't think it applies to this problem. It involves a function J in terms of other variables. Hoping what I gave makes sense.

Brian

 

[dsrobertson]

I think this is just a mathematical trick. After you have differentiated E with respect to t then you
can observe that the two terms may also be obtained by differentiating another function with respect
to x. This function turns out to be y_t y_x and we can call it -J. Making this definition gives the result.

Homework Statement

I have a problem that I'm trying to make sense of. Note y_t is the partial derivative of y with respect to t and y_tt is the second order partial derivative of y with respect to t, etc. The complete problem statement is the following:

Show that for the equation y_tt - c^2 y_xx = 0
the quantity E = 1/2(y_t^2 + c^2 y_x^2)
satisfies a conservation law dE/dt + dJ/dx = 0

Homework Equations

The Attempt at a Solution

I calculated dE/dt to = y_t * y_tt + c^2 y_x * y_xt so dJ/dx must equal the negation of dE/dt. But I'm not sure where J comes from. I'm guessing that it is somehow related to y through the wave equation but I'm not sure how. Also it is unclear to me how I could integrate the negation of dE/dt to arrive at J. I gave the complete problem statement. In that chapter of the book the function J is used but I don't think it applies to this problem. It involves a function J in terms of other variables. Hoping what I gave makes sense.

Brian

 

[hunt_mat]

You know E, you know the conservation relation that E and J satisfy, can't you use these two to try and compute what J has to be?

Mat