Thermo-ish diffusion/wave equation - metal plate and temperature difference

[samee]

Homework Statement

The edges of a thin plate are held at the temperature described below. Determine the steady-state temperature distribution in the plate. Assume the large flat surfaces to be insulated.

If the plate is lying along the x-y plane, then one corner would be at the origin. The height of the plate would be 1m along the y-axis and the length would be 2m along the x-axis. The edge along the y-axis is being held at 0 C. The edge along the x-axis is being held at 0 C. The edge parallel to the x-axis is being held at 0 C. The edge parallel to the y-axis is being held at 50sin(pi*y) C.

Homework Equations

So I'm assuming this question is actually just a diffusion equation or a wave equation, because that's what the rest of our homework was on. Alpha²u_xx=u_t
and
u(x,t)=X(x)T(t)=(C₁coskx+C₂sinkx)e^-K2alpha2t+C₃+C₄x

The Attempt at a Solution

So I tried to solve this like the wave equations and it seems to just be blowing out of proportion and not making sense... Also... I think we need to consider a thrid position variable here, we need x,y AND t. I don';t know how to do this at all :(

 

[KataKoniK]

Thank you for posting your question! I would approach this problem by first identifying the relevant physical principles and equations that apply to this scenario. In this case, the relevant principle is heat transfer, and the relevant equation is the heat equation, which is a type of diffusion equation.

The heat equation can be written as α∇^2u = ∂u/∂t, where α is the thermal diffusivity, u is the temperature, and ∇^2 is the Laplace operator. This equation describes the rate of change of temperature over time and space.

In order to solve this problem, we need to apply the boundary conditions given in the question. These boundary conditions are the temperatures at the edges of the plate, which are held at 0 C along the x and y axes, and 50sin(πy) C along the edge parallel to the y-axis. We also need to consider the initial temperature distribution, which is not specified in the question.

To solve this problem, we can use a method called separation of variables, which involves separating the variables in the heat equation (in this case, x, y, and t) and solving each part separately. This will give us a general solution for the temperature distribution in the plate.

Once we have the general solution, we can apply the boundary conditions and the initial temperature distribution to determine the specific solution for this particular scenario. This will give us the steady-state temperature distribution in the plate, which is what the question is asking for.

I hope this helps you understand how to approach this problem. If you need further assistance, please don't hesitate to ask for clarification. Good luck with your homework!