Chapter 13: Problem 3

Solve the semi-infinite plate problem if the bottom edge of width \(\pi\) is held at \(T=\cos x\). and the other sides are at \(0^{\circ}\).

Short Answer

Expert verified

The temperature distribution is \[T(x,y) = \sin( x)e^{- y}\below the value of \pi of the semi-plate

Step by step solution

- Understand the Boundary Conditions

The semi-infinite plate problem specifies that the temperature at the bottom edge is given by the function: \[T(x, 0) = \cos x \quad \text{for} \quad 0 \leq x \leq \pi \quad \text{and} \quad T(x, 0) = 0 \quad \text{otherwise}\].The other sides, extending into infinity, are held at a temperature of 0°C: \[T(0, y) = 0, \quad T(\pi, y) = 0 \quad \text{for} \quad y \geq 0\].

- Set Up the Laplace Equation

Since we are dealing with steady-state heat flow, we use Laplace's equation for temperature distribution: \[abla^2 T = 0\].In two dimensions, it simplifies to: \[\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0\].

- Apply Separation of Variables

Assume a solution of the form: \[T(x,y) = X(x)Y(y)\].Substitute this form into the Laplace equation to get: \[X''(x)Y(y) + X(x)Y''(y) = 0\].Dividing both sides by \[X(x)Y(y)\] yields: \[\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0\].Each term must be equal to a constant, \[- \lambda\]: \[X''(x) + \lambda X(x) = 0\], \[Y''(y) - \lambda Y(y) = 0\].

- Solve the Equation for X(x)

Solve the equation for \[X(x)\]: \[X''(x) + \lambda X(x) = 0\].The solution is: \[X(x) = A\sin(\sqrt{\lambda} x) + B\cos(\sqrt{\lambda} x)\].Using the boundary conditions: \[T(0, y) = 0\] implies \[X(0) = 0\], so \[B = 0\].Using \[T(\pi, y) = 0\], implies \[A\sin(\sqrt{\lambda}\pi) = 0\], thus \[\sqrt{\lambda} = n\] for \[ n \in \{1,2,3,...\} \].Then, \[X(x) \equiv A_n \sin(nx)\].

- Solve the Equation for Y(y)

Solve the equation for \[Y(y)\]: \[Y''(y) - \lambda Y(y) = 0\].The solution is: \[Y(y) = C_n e^{n y}\].Using the boundary condition as \[T(x, \infty) = 0\], implies \[Y(y) = C_n e^{-n y}\] thus \[C_n = 0\].

- Combine Solutions and Apply Initial Conditions

Combine the solutions: \[T(x,y) = \sum_{n=1}^{\infty}A_n \sin(nx) e^{- n y}\].Apply the initial condition \[T(x,0) = \cos x\].This is expressed as the Fourier series: \[T(x,0) = \sum_{n=1}^{\infty} A_n \sin(nx) = \cos x\].Since \cos x = \frac{e^{i x} + e^{- i x}}{2}, we see that can be expressed as the Fourier series sum (only the first term is non-zero), so now fit the Fourier series.

- Final Temperature Distribution

The only non-zero coefficient we have is the first one corresponding to \[n=1\] in the Fourier series. The temperature distribution is: \[T(x,y) = \sin( x)e^{- y}\].

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Key Concepts

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

Laplace Equation

The Laplace equation is fundamental in understanding the steady-state heat distribution within a semi-infinite plate. The general form in two dimensions is written as: \[ \frac{abla^2 T}{abla x^2} + \frac{abla^2 T}{abla y^2} = 0 \]. Here, \( T \) represents the temperature at any point \( (x, y) \) on the plate. Since the plate should not store heat over time, the heat flow is in a steady-state condition where it is evenly distributed.

Boundary Conditions

Boundary conditions are key to solving the Laplace equation. They define how the function behaves at the edges of the domain. In the semi-infinite plate problem, the bottom edge of the plate (the edge along the x-axis from \(0 \leq x \leq \pi \)) is held at a temperature \(T = \cos x \). This is expressed as: \[ T(x, 0) = \cos x, \ 0 \leq x \leq \pi \] For other parts where x is greater than \( \pi \), the temperature is held at \( 0^{\circ} \). The sides extending into infinity on both y-axis and parts of x-axis are also held at \(0^{\circ} \). These boundary conditions ensure proper constraints while solving the Laplace equation.

Separation of Variables

Separation of variables is a method employed to simplify partial differential equations (PDEs) like the Laplace equation. By assuming a solution for temperature \( T(x, y) \) in the form \( X(x)Y(y) \), the equation transforms into two ordinary differential equations which can be more easily solved. On substituting into the Laplace equation, we get: \[ X''(x)Y(y) + X(x)Y''(y) = 0 \]. Dividing through by \( X(x)Y(y) \) separates into: \[ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 \], leading to each term equaling a constant \( - \lambda \).

Fourier Series

The Fourier series expresses a function as a sum of sines and cosines. It's particularly useful in the semi-infinite plate problem to match complicated boundary conditions. For the bottom edge condition \(T(x, 0) = \cos x\): We express temperature as a sum: \[ T(x, y) = \sum_{n=1}^{\infty} A_n \sin(nx) \ e^{-ny} \]. By applying the initial condition, we align the series to match the given temperature distribution, which simplifies our calculation to find the coefficients \( A_n \).

Steady-State Heat Flow

Steady-state heat flow implies that the temperature distribution in the semi-infinite plate does not change over time. The temperature field \( T(x, y) \) hence depends only on the spatial coordinates and adheres to the Laplace equation \( \abla^2 T = 0 \). In our problem, the solution: \[ T(x, y) = \sin(x) e^{- y} \] shows the steady-state condition, where heat flow stabilizes and no longer fluctuates.