Question

Solve the semi-infinite plate problem if the bottom edge of width 30 is held at \(T=\left\\{\begin{array}{rr}x, & 0

Short answer

The temperature distribution is \[ T(x, y) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{30}\right)e^{-\frac{n\pi y}{30}} \]

Step-by-step solution

Understand the Boundary Conditions

The semi-infinite plate has its bottom edge of width 30 and held at different temperatures. For the bottom edge, the temperature is defined as follows: For 0 < x < 15, the temperature T = x For 15 < x < 30, the temperature T = 30 - x Other sides are held at 0 degrees.

Set Up the Problem

Visualize the plate and label the edges. The bottom edge of the plate with coordinates (0,0) to (30,0) has the varying temperature. The other sides are at zero temperature.

Use the Heat Equation

Since the problem involves a semi-infinite plate, use the heat equation in two dimensions: \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \]

Apply Separation of Variables

Attempt a solution of the form \( T(x, y) = X(x)Y(y) \). Substituting into the heat equation, we get: \[ X''(x)Y(y) + X(x)Y''(y) = 0 \]

Solve the Ordinary Differential Equations

Divide by \( T(x, y) = X(x)Y(y) \) to separate variables, yielding: \[ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = -\lambda \] This leads to two ordinary differential equations: \[ X''(x) + \lambda X(x) = 0 \] \[ Y''(y) - \lambda Y(y) = 0 \]

Solve for X(x)

Solve the equation for X(x): \[ X''(x) + \lambda X(x) = 0 \] General solution: \[ X(x) = A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x) \]

Solve for Y(y)

Solve the equation for Y(y): \[ Y''(y) - \lambda Y(y) = 0 \] General solution: \[ Y(y) = Ce^{\sqrt{\lambda}y} + De^{-\sqrt{\lambda}y} \] Given the plate is semi-infinite, the solution must decay as y -> infinity, so D must be zero. Thus, \[ Y(y) = Ce^{-\sqrt{\lambda}y} \]

Form the General Solution

Combine both solutions: \[ T(x, y) = (A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x))Ce^{-\sqrt{\lambda}y} \]

Apply Boundary Conditions

Using the boundary conditions, For y = 0, \[ T(x, 0) = (A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x)) = f(x) \] where f(x) is the piecewise function defined by the boundary conditions.

Fourier Series Representation

Express f(x) by a Fourier series in sine terms (since the problem is odd about x = 15): \[ f(x) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{30}\right), \text{for } 0 < x < 30 \]

Combine Results

The general solution combining the boundary conditions and the Fourier series representation: \[ T(x, y) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{30}\right)e^{-\frac{n\pi y}{30}} \]

Key concepts

heat equation

Understanding the heat equation is key to solving many thermal problems, including the semi-infinite plate problem. The heat equation in two dimensions is given by \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0. \] It describes how temperature, T, changes over space in the x and y directions. This partial differential equation assumes that the plate is in a steady-state, meaning the temperature distribution does not change over time. By solving this equation, we can determine the temperature at any point on the plate given the boundary conditions.

boundary conditions

Boundary conditions are essential to solving the heat equation because they describe the temperature at the edges or surfaces of the material. For our semi-infinite plate problem, the bottom edge of width 30 is held at a specific temperature pattern. Specifically:

• For 0 < x < 15, the temperature T = x.
• For 15 < x < 30, the temperature T = 30 - x.

The other sides of the plate are held at 0 degrees. These conditions must be satisfied by the solution to ensure the model reflects the physical scenario accurately.

Fourier series

A Fourier series breaks down complex periodic functions into simpler sine and cosine components. For the bottom boundary condition of our plate, the temperature pattern T(x) is a piecewise function. To handle this, we express the function using a sine Fourier series since the problem is odd about the midpoint (x = 15): \[ f(x) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{30}\right), \text{for } 0 < x < 30. \] By transforming the piecewise function into a series, we can more easily solve the heat equation and apply boundary conditions.

separation of variables

Separation of variables is a technique to solve partial differential equations, such as the heat equation. We assume the solution can be written as a product of functions, each depending on only one coordinate, like so: \[ T(x, y) = X(x)Y(y). \] Substituting this into the heat equation and dividing through by T(x, y), we obtain separate ordinary differential equations for X(x) and Y(y):

• \[ X''(x) + \lambda X(x) = 0 \]
• \[ Y''(y) - \lambda Y(y) = 0 \]

These equations can then be solved individually, and their solutions combined to find the overall temperature distribution. This method greatly simplifies solving complex problems like the semi-infinite plate.