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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

Water at \(100^{\circ}\) is flowing through a long pipe of radius 1 rapidly enough so that we may assume that the temperature is \(100^{\circ}\) at all points. At \(t=0\), the water is turned off and the. surface of the pipe is maintained at \(40^{\circ}\) from then on (neglect the wall thickness of the pipe). Find the temperature distribution in the water as a function of \(r\) and \(t\). Note that you need only consider a cross section of the pipe.

Solve the semi-infinite plate problem if the bottom edge of width 20 is held at $$ T= \begin{cases}0^{\circ}, & 0

Find the steady-state temperature distribution in a rectangular plate covering the area \(0

The Klein-Gordon equation is \(\nabla^{2} u=\left(1 / v^{2}\right) \partial^{2} u / \partial t^{2}+\lambda^{2} u .\) This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string.

Find the steady-state temperature distribution in a rectangular plate \(30 \mathrm{~cm}\) by \(40 \mathrm{~cm}\) given that the temperature is 0 along the two long sides and along one short end; the other short end along the \(x\) axis has temperature $$ T=\left\\{\begin{array}{lc} 100^{\circ}, & 0
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