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Chapter 13: Problem 1

Find the steady-state temperature distribution for the semi-infinite plate problem if the temperature of the bottom edge is \(T=f(x)=x\) (in degrees; that is, the temperature at \(x \mathrm{~cm}\) is \(x\) degrees), the temperature of the other sides is \(0^{\circ}\), and the width of the plate is \(10 \mathrm{~cm}\).

Short Answer

Expert verified
The steady-state temperature distribution is \ u(x,y) = \sum_{n=1}^\infty \left( \frac{2}{n \pi} (-1)^{n+1} \sin \( \frac{n \pi x}{10} \right) \sinh \( \frac{n \pi y}{10} \right). \

Step by step solution

01

Define the Problem

We need to find the steady-state temperature distribution of a semi-infinite plate where the temperature at the bottom edge \( T=f(x)=x \) and the temperature of the other sides is \( 0^{\circ} \). The plate extends from x=0 to x=10 cm.
02

Set Up the Problem with Boundary Conditions

Given that \( u(x,0)=f(x)=x \), \( u(0,y)=0 \), and \( u(10,y)=0 \, \) the steady-state temperature distribution satisfies Laplace's equation: \ \ \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] with appropriate boundary conditions.
03

Solve Laplace's Equation Using Separation of Variables

Using separation of variables method, assume that \ u(x,y) = X(x)Y(y) \. Substitute this form into Laplace's equation: \ \ \[ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 \] This is separable into two ordinary differential equations.
04

Solve the Ordinary Differential Equations

Solve the equations \ X''(x) + \lambda X(x) = 0 \ and \ Y''(y) - \lambda Y(y) = 0 \ with appropriate boundary conditions. The solution is in the form of: \ \ \[ X_n(x) = A_n \sin \left( \frac{n \pi x}{10} \right) \] and \ \[ Y_n(y) = B_n \sinh \left( \frac{n \pi y}{10} \right) \]
05

Apply the Boundary Conditions

Given \ u(x,0) = x \, expand the solution using Fourier series coefficients: \ \ \[ u(x,0) = \sum_{n=1}^{\infty} A_n \sin \left( \frac{n \pi x}{10} \right) \] Match this to the boundary condition to determine \ A_n \ coefficients.
06

Construct the Solution

The final steady-state temperature distribution is given by summing over the series for all coefficients: \ \ \[ u(x,y) = \sum_{n=1}^{\infty} \left( \frac{2}{n \pi} \times \left(-1\right)^{n+1} \sin \left( \frac{n \pi x}{10} \right) \right) \sinh \left( \frac{n \pi y}{10} \right) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laplace's Equation
To understand steady-state temperature distribution, we first need to talk about Laplace's equation. This equation models the way heat (temperature) spreads out in a stable way over time. Mathematically, Laplace's equation is written as:

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Most popular questions from this chapter

Show that the gravitational potential \(V=-G m / r\) satisfies Iaplace's equation, that is, show that \(\nabla^{2}(1 / r)=0\) where \(r^{2}=x^{2}+y^{2}+z^{2}, r \neq 0\). (Also see Chapter 15, Section 8.)

A sphere initially at \(0^{\circ}\) has its surface kept at \(100^{\circ}\) from \(t=0\) on (for example, a frozen potato in boiling water !). Find the time- dependent temperature distribution. Hint : Subtract \(100^{\circ}\) from all temperatures and solve the problem; then add the \(100^{\circ}\) to the answer. Can you justify this procedure ? Show that the Legendre function required for this problem is \(P_{0}\) and the \(r\) solution is \((1 / \sqrt{r}) J_{1 / 2}\) or \(j_{0}\) [see (17.4) in Chapter 12]. Since spherical Bessel functions can be expressed in terms of elementary functions, the series in this problem can be thought of as cither a Bcssel series or a Fouricr series. Show that the results arc identical.

The following two \(R(r)\) equations arise in various separation of variables problems in polar, cylindrical, or spherical coordinates: $$ \begin{aligned} &r \frac{d}{d r}\left(r \frac{d R}{d r}\right)=n^{2} R \\ &\frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=N(l+1) R \end{aligned} $$ There are various ways of solving them: They are a standard kind of equation (often called Euler or Cauchy equations-see Chapter 8, Section 7d); you could use power series methods; given the fact that the solutions are just powers of \(r\), it is easy to find the powers. Choose any method you like, and solve the two equations for future reference. Consider the case \(n=0\) separately. Is this necessary for \(l=0 ?\)

Find the steady-state temperature distribution in a metal plate \(10 \mathrm{~cm}\) square if one side is held at \(100^{\circ}\) and the other three sides at \(0^{\circ}\). Find the temperature at the center of the plate.

A cube is originally at \(100^{\circ}\). From \(t=0\) on, the faces are held at \(0^{\circ}\). Find the timedependent temperature distribution. Hint : This problem leads to a triple Fourier series; see the double Fourier series in Problem 9 and generalize it to three dimensions.
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