- #1

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_{e}(x) satisfies the wave equation? =(

i get that u

_{e}(x)=gx

^{2}/2c

^{2}+ ax + b where a and x are just constants but how does this satisfy the wave equation?

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- Thread starter tuan43
- Start date

- #1

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i get that u

- #2

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Generally to show that a function satisfies a DE, you'll need to show that its derivatives actually have the relationship in question. So I'd start by differentiating your definition of v twice with respect to each variable.

- #3

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- #4

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Ah, I see. :) So you've got a string with a uniform loading (which you've called Q) along the x-direction, sagging in the shape of a parabola because of that (note that in reality, if this loading were due to the weight of the string, the equilibrium shape would be a catenary, not a parabola).

And you're trying to prove that if u(x,t) is a solution to the wave equation on an equivalent (same boundary conditions) unloaded string, then:

[tex]u(x,t) - u_e(x)[/tex]

Will be a solution to the wave equation on the loaded string.

Did I understand the question correctly?

If so, then look up the superposition theorem for linear differential equations (such as the wave equation). This states that the sum of two solutions to a DE will also be a solution to the DE - So in this case, if both u

If you also need to show that it's the solution that you're looking for, then you'll need to check it satisfies the appropriate boundary/initial conditions.

- #5

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Are we supposed to be psychic? Why don't you give us the equation you are trying to satisfy? How does Q enter in to it?

- #6

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LCKurtz: Q(x,t) is just force acting on the string, so gravity in most cases. i got the answer now, sorry for not being more precise.

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