Solutions to 2D wave equation using 1D equation solution.

In summary, the question is asking to integrate given ψ(x), which must be done over x and y, but there is no explicate form for ψ(x) so you will have to know how to integrate ψ(x) for the integral to be valid.
  • #1
Greger
46
0
http://imageshack.us/a/img824/1121/asdasdaw.png [Broken]

I am having trouble completely understanding what the question wants. I know it is quite clear but the part I am having trouble is the following.

It says 'pretend' w(x,t) is a solution to the 2D equation, just independent of y, then to substitute it into the integrate,

That means that u_t(x,y,0) = w_t(x,0) = ψ(x)

But I am trouble seeing how you can perform this integration without knowing what ψ(x) actually is as it would be,

∫∫ψ(x)/(stuff) dx dy over D.

But ψ(x) is unknown, so how can you integrate?
 
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  • #2
Hi Greger! :smile:
Greger said:
But ψ(x) is unknown …

No, ψ(x) is given, as an initial boundary condition! :wink:
 
  • #3
Thanks for replying,

Yea u_t(x,y,0) = w_t(x,0) = ψ(x),

But what I mean is there is no explicate form of ψ(x), for example ψ(x) = x, ψ(x) = cos(x), its juts left arbitrarily as ψ(x).

What would ∫ψ(x)dxdy be if there is no explicate form for ψ(x)?
 
  • #4
Greger said:
… But ψ(x) is unknown, so how can you integrate?

it's just a formula for you to use in the future, when you do know ψ(x), isn't it? :confused:

not an actual problem?
 
  • #5
Yea it's an actual problem, haha I see what you mean, it looks like instructions.

The question asks to actually preform the integration given,

And in the integral you have u_t(x,y,0) and the question says substitute:

u_t(x,y,0) = w_t(x,0) = ψ(x)

so now the integral becomes ∫∫ ψ(x)/stuff dx dy.

Then asks to actually compute the integral,

The integration is over x and y and ψ(x) is a function of x, so you will have to know how to integrate ψ(x) right? Which means you need to have an explicit form of ψ(x) right?

I understand what you are saying, that if its left arbitrarily as ψ(x) you perform the integration when you know ψ(x), but the question is actually asking to integrate as if you already know what ψ(x) is.

There must be some kind of trick that I'm just not seeing to be able to perform the integration, or a way to find the expression for ψ(x), or maybe there is just information missing from the question?
 
  • #6
Have you tried doing the integration over y first? Since the initial data doesn't depend on y you may be able to do that, and then perhaps it will be clearer what to do with the x integral.
 
  • #7
Greger said:
There must be some kind of trick that I'm just not seeing to be able to perform the integration …

if there is, i'm not seeing it either :redface:
 

What is the 2D wave equation and how is it related to the 1D equation?

The 2D wave equation is a mathematical equation that describes the motion of waves in a two-dimensional space. It is related to the 1D equation through the principle of superposition, which states that the overall solution to a problem can be found by combining individual solutions from simpler problems.

What are the challenges in solving the 2D wave equation using the 1D equation solution?

One of the main challenges in solving the 2D wave equation using the 1D equation solution is the need for additional boundary conditions. In the 1D case, one boundary condition is sufficient, but in 2D, at least two boundary conditions are needed to fully define the solution.

What are some applications of using the 1D equation solution for the 2D wave equation?

The 1D equation solution for the 2D wave equation has many practical applications, such as in predicting the behavior of sound waves in a two-dimensional space. It can also be used in the study of electromagnetic waves, ocean waves, and other types of waves.

What assumptions are made when using the 1D equation solution for the 2D wave equation?

One of the main assumptions made when using the 1D equation solution for the 2D wave equation is that the waves are propagating in a homogeneous medium with a constant wave speed. This means that the properties of the medium, such as density and elasticity, do not vary with position.

Are there any limitations to using the 1D equation solution for the 2D wave equation?

Yes, there are some limitations to using the 1D equation solution for the 2D wave equation. For example, it may not accurately describe the behavior of waves in highly non-uniform media or in situations where the boundary conditions are complex. In these cases, a more sophisticated approach may be needed.

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