Steady State Temperature Distribution inside a plate of infinite length

[Matt Blood]

Homework Statement

Consider a rectangular metal plate 20 cm wide along y-axis and infinitely long in x
as shown in the Figure below. The two long sides and the far end are held at 0ºC and
the short side along y-axis has the temperature distribution T (y) = y, that is at a point
y in cm between 0 and 20 cm the temperature is y degrees. Find the steady state
temperature distribution inside the plate. Describe in detail all steps leading to the
solution.

Homework Equations

I think Fourier transforms are involved but I don't know how to use them on something like this.

The Attempt at a Solution

I can't think where to start...

 

[anathema]

Thank you for your question. I can offer you some guidance on how to approach this problem. First, let's define the problem and the variables involved. We have a rectangular metal plate that is 20 cm wide along the y-axis and infinitely long in the x-axis. The temperature distribution along the y-axis is given by T(y) = y, where y is the distance from the short side of the plate along the y-axis. The two long sides and the far end of the plate are held at 0ºC.

To find the steady state temperature distribution inside the plate, we need to solve the heat equation, which describes how heat is distributed in a material. In this case, we can use the one-dimensional heat equation, which is given by:

∂T/∂t = α∂^2T/∂y^2

Where T is the temperature, t is time, y is the distance along the y-axis, and α is the thermal diffusivity of the material. In our case, we can assume that α is constant.

To solve this equation, we will use the method of separation of variables. This involves assuming that the solution can be written as a product of two functions, one depending only on time and the other depending only on distance. We can write the solution as:

T(y,t) = X(y)T(t)

Substituting this into the heat equation, we get:

X(y)T'(t) = αX''(y)T(t)

Dividing both sides by X(y)T(t), we get:

T'(t)/T(t) = αX''(y)/X(y)

Since the left side depends only on time and the right side depends only on distance, both sides must be equal to a constant, which we will call -λ^2. This gives us two ordinary differential equations:

T'(t)/T(t) = -λ^2

X''(y)/X(y) = -λ^2/α

The first equation can be solved easily, giving us T(t) = Ce^-λ^2t, where C is a constant of integration. The second equation can be solved using standard techniques for solving differential equations. The general solution is given by:

X(y) = Acos(λy) + Bsin(λy)

Where A and B are constants of integration. Now, we