Coefficients of the Fourier Sine Series

Oct 13, 2025 by talynt.org 129 views

Introduction

Heat Transfer is a fundamental concept in physics that deals with the transfer of thermal energy from one body to another due to a temperature difference. In this article, we will explore the temperature function of a rod of length $$l$$ with insulated ends, which is initially at 0 degrees Celsius. The rod is suddenly heated at one end to $$0^{\circ} C$$ and cooled at the other end. We will use the Heat Equation to derive the temperature function $$U(x, t)$$.

The Heat Equation

The Heat Equation is a partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. It is given by:

\frac{\partial U}{\partial t} = c^2 \frac{\partial^2 U}{\partial x^2}

where $$U(x, t)$$ is the temperature at point $$x$$ and time $$t$$, and $$c^2$$ is the thermal diffusivity of the material.

Boundary and Initial Conditions

We are given the following boundary and initial conditions:

The rod is initially at 0 degrees Celsius, so $$U(x, 0) = 0$$ for all $$x$$.
The rod is heated at one end to $$0^{\circ} C$$, so $$U(0, t) = 0$$ for all $$t$$.
The rod is cooled at the other end, so $$U(l, t) = 0$$ for all $$t$$.

Solution of the Heat Equation

To solve the Heat Equation, we will use the Method of Separation of Variables. We assume that the solution can be written as:

U(x, t) = X(x)T(t)

Substituting this into the Heat Equation, we get:

\frac{\partial U}{\partial t} = c^2 \frac{\partial^2 U}{\partial x^2}

X(x)T'(t) = c^2 X''(x)T(t)

Dividing both sides by $$c^2 X(x)T(t)$$, we get:

\frac{T'(t)}{c^2 T(t)} = \frac{X''(x)}{X(x)}

Since the left-hand side depends only on $$t$$ and the right-hand side depends only on $$x$$, both sides must be equal to a constant, say $$-\lambda$$.

Eigenvalue Problem

We now have the following eigenvalue problem:

X''(x) + \lambda X(x) = 0

with boundary conditions $$X(0) = X(l) = 0$$.

Solution of the Eigenvalue Problem

To solve the eigenvalue problem, we will use the Trigonometric Form. We assume that the solution can be written as:

X(x) = A \cos (\sqrt{\lambda} x) + B \sin (\sqrt{\lambda} x)

Applying the boundary conditions, we get:

X(0) = A \cos (0) + B \sin (0) = A = 0

X(l) = A \cos (\sqrt{\lambda} l) + B \sin (\sqrt{\lambda} l) = 0

Since $$A = 0$$, we have:

B \sin (\sqrt{\lambda} l) = 0

This implies that either $$B = 0$$ or $$\sin (\sqrt{\lambda} l) = 0$$.

Non-trivial Solution

Since we want a non-trivial solution, we must have $$B \neq 0$$. Therefore, we must have:

\sin (\sqrt{\lambda} l) = 0

This implies that:

\sqrt{\lambda} l = n \pi

where $$n$$ is an integer.

Eigenvalues and Eigenfunctions

We now have the following eigenvalues and eigenfunctions:

\lambda_n = \frac{n^2 \pi^2}{l^2}

X_n(x) = \sin \left( \frac{n \pi x}{l} \right)

Solution of the Heat Equation

We now have the following solution of the Heat Equation:

U(x, t) = \sum_{n=1}^{\infty} A_n \sin \left( \frac{n \pi x}{l} \right) e^{-\frac{n^2 \pi^2 c^2 t}{l^2}}

where $$A_n$$ are the coefficients of the Fourier sine series.

Coefficients of the Fourier Sine Series

To find the coefficients $$A_n$$, we will use the Orthogonality of the eigenfunctions. We have:

\int_{0}^{l} X_n(x) X_m(x) dx = 0

for $$n \neq m$$.

Coefficients of the Fourier Sine Series

We now have the following coefficients of the Fourier sine series:

A_n = \frac{2}{l} \int_{0}^{l} U(x, 0) \sin \left( \frac{n \pi x}{l} \right) dx

Temperature Function

We now have the following temperature function:

U(x, t) = \sum_{n=1}^{\infty} A_n \sin \left( \frac{n \pi x}{l} \right) e^{-\frac{n^2 \pi^2 c^2 t}{l^2}}

where $$A_n$$ are the coefficients of the Fourier sine series.

Conclusion

In this article, we have derived the temperature function of a rod of length $$l$$ with insulated ends, which is initially at 0 degrees Celsius. The rod is suddenly heated at one end to $$0^{\circ} C$$ and cooled at the other end. We have used the Heat Equation to derive the temperature function $$U(x, t)$$.

References

Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press.
Horton, C. W., & Lomax, H. (1956). Fundamentals of Heat Transfer. Addison-Wesley.
Incropera, F. P., & Dewitt, D. P. (2002). Fundamentals of Heat and Mass Transfer. John Wiley & Sons.

Discussion

The temperature function $$U(x, t)$$ is a fundamental concept in physics that deals with the transfer of thermal energy from one body to another due to a temperature difference. The Heat Equation is a partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

The Method of Separation of Variables is a powerful technique for solving partial differential equations. It involves separating the variables and solving the resulting ordinary differential equations.

The Eigenvalue Problem is a fundamental concept in mathematics that deals with the solution of a differential equation with boundary conditions. The Trigonometric Form is a powerful technique for solving the eigenvalue problem.

The Fourier Sine Series is a fundamental concept in mathematics that deals with the representation of a function as a sum of sine functions. The Coefficients of the Fourier Sine Series are the coefficients of the sine functions in the Fourier sine series.

The Temperature Function is a fundamental concept in physics that deals with the transfer of thermal energy from one body to another due to a temperature difference. The Heat Equation is a partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

The References section provides a list of references that were used in the derivation of the temperature function $$U(x, t)$$.

The Discussion section provides a discussion of the temperature function $$U(x, t)$$ and its applications in physics.

Q&A

Q: What is the temperature function of a rod of length $$l$$ with insulated ends, which is initially at 0 degrees Celsius?

A: The temperature function of a rod of length $$l$$ with insulated ends, which is initially at 0 degrees Celsius, is given by:

U(x, t) = \sum_{n=1}^{\infty} A_n \sin \left( \frac{n \pi x}{l} \right) e^{-\frac{n^2 \pi^2 c^2 t}{l^2}}

where $$A_n$$ are the coefficients of the Fourier sine series.

Q: What are the boundary and initial conditions of the rod?

A: The boundary and initial conditions of the rod are:

The rod is initially at 0 degrees Celsius, so $$U(x, 0) = 0$$ for all $$x$$.
The rod is heated at one end to $$0^{\circ} C$$, so $$U(0, t) = 0$$ for all $$t$$.
The rod is cooled at the other end, so $$U(l, t) = 0$$ for all $$t$$.

Q: How do we solve the Heat Equation?

A: We solve the Heat Equation using the Method of Separation of Variables. We assume that the solution can be written as:

U(x, t) = X(x)T(t)

Substituting this into the Heat Equation, we get:

\frac{\partial U}{\partial t} = c^2 \frac{\partial^2 U}{\partial x^2}

X(x)T'(t) = c^2 X''(x)T(t)

Dividing both sides by $$c^2 X(x)T(t)$$, we get:

\frac{T'(t)}{c^2 T(t)} = \frac{X''(x)}{X(x)}

Since the left-hand side depends only on $$t$$ and the right-hand side depends only on $$x$$, both sides must be equal to a constant, say $$-\lambda$$.

Q: What is the eigenvalue problem?

A: The eigenvalue problem is:

X''(x) + \lambda X(x) = 0

with boundary conditions $$X(0) = X(l) = 0$$.

Q: How do we solve the eigenvalue problem?

A: We solve the eigenvalue problem using the Trigonometric Form. We assume that the solution can be written as:

X(x) = A \cos (\sqrt{\lambda} x) + B \sin (\sqrt{\lambda} x)

Applying the boundary conditions, we get:

X(0) = A \cos (0) + B \sin (0) = A = 0

X(l) = A \cos (\sqrt{\lambda} l) + B \sin (\sqrt{\lambda} l) = 0

Since $$A = 0$$, we have:

B \sin (\sqrt{\lambda} l) = 0

This implies that either $$B = 0$$ or $$\sin (\sqrt{\lambda} l) = 0$$.

Q: What are the eigenvalues and eigenfunctions?

A: The eigenvalues and eigenfunctions are:

\lambda_n = \frac{n^2 \pi^2}{l^2}

X_n(x) = \sin \left( \frac{n \pi x}{l} \right)

Q: How do we find the coefficients of the Fourier sine series?

A: We find the coefficients of the Fourier sine series using the Orthogonality of the eigenfunctions. We have:

\int_{0}^{l} X_n(x) X_m(x) dx = 0

for $$n \neq m$$.

Q: What is the temperature function?

A: The temperature function is:

U(x, t) = \sum_{n=1}^{\infty} A_n \sin \left( \frac{n \pi x}{l} \right) e^{-\frac{n^2 \pi^2 c^2 t}{l^2}}

where $$A_n$$ are the coefficients of the Fourier sine series.

Q: What are the applications of the temperature function?

A: The temperature function has many applications in physics, including:

Heat transfer
Thermal energy
Temperature distribution
Thermal conductivity

Q: What are the limitations of the temperature function?

A: The temperature function has some limitations, including:

It assumes a uniform temperature distribution
It assumes a constant thermal conductivity
It assumes a one-dimensional heat transfer

Q: How do we extend the temperature function to two-dimensional and three-dimensional heat transfer?

A: We extend the temperature function to two-dimensional and three-dimensional heat transfer by using the Laplace Equation and the Heat Equation in multiple dimensions.

Q: What are the challenges of solving the temperature function?

A: The challenges of solving the temperature function include:

Finding the coefficients of the Fourier sine series
Solving the eigenvalue problem
Applying the boundary and initial conditions

Q: How do we use the temperature function in real-world applications?

A: We use the temperature function in real-world applications by:

Modeling heat transfer in engineering systems
Analyzing temperature distribution in materials
Designing thermal systems

Q: What are the future directions of research in temperature function?

A: The future directions of research in temperature function include:

Developing new methods for solving the temperature function
Investigating new applications of the temperature function
Improving the accuracy and efficiency of the temperature function.