Wave Eqn: Show v(x,t)=u(x,t)-ue(x) Satisfies Wave Eqn

[tuan43]

how do i show that v(x,t)=u(x,t)-u_e(x) satisfies the wave equation? =(

i get that u_e(x)=gx²/2c² + ax + b where a and x are just constants but how does this satisfy the wave equation?

 

[Jmf]

Can you clarify your question? I don't understand what u_e(x) actually is in terms of the function u(x,t)? Is u(x,t) an arbitrary function, or one that itself satisfies the wave equation? Perhaps I'm missing something obvious. :)

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.

 

[tuan43]

thanks for taking a look. still stumped. Ue(x) is the sagged equilibrium position ( when Q(x,t)=-g and the boundary conditions are u(0)=0 and u(L)=0 or fixed boundary/ends of the string). i hope that clarifies it a bit?

 

[Jmf]

thanks for taking a look. still stumped. Ue(x) is the sagged equilibrium position ( when Q(x,t)=-g and the boundary conditions are u(0)=0 and u(L)=0 or fixed boundary/ends of the string). i hope that clarifies it a bit?

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:

$$u(x,t) - u_e(x)$$

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_e(x) and u(x,t) are solutions then it immediately follows that v(x,t) is a solution.

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.

 

[LCKurtz]

thanks for taking a look. still stumped. Ue(x) is the sagged equilibrium position ( when Q(x,t)=-g and the boundary conditions are u(0)=0 and u(L)=0 or fixed boundary/ends of the string). i hope that clarifies it a bit?

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

 

[tuan43]

wow you totally got the question. thanks a lot. i see it clearly now :)

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.