Question

These exercises explore some additional aspects of the radiative transport model presented in Section 11.1. Suppose we modify the radiative transport model solved in Example 4 to allow for a partial reflection of energy at the slab edge at \(x=l\). A portion of the energy arriving at \(x=l\) from within the slab is reflected backwards, while the rest exits the slab. To model this phenomenon, we adopt the boundary condition $$ I^{(-)}(l)=\Gamma I^{(+)}(l), $$ where \(\Gamma\), a positive constant satisfying \(0 \leq \Gamma \leq 1\), is often called a reflection coefficient. (Note that \(\Gamma=0\) is the case solved in Example 4, while the other extreme, \(\Gamma=1\), corresponds to placing a reflecting wall at \(x=l\).) The new boundary value problem becomes $$ \begin{aligned} &\frac{d}{d x}\left[\begin{array}{l} I^{(+)} \\ I^{(-)} \end{array}\right]=\beta\left[\begin{array}{ll} -1 & 1 \\ -1 & 1 \end{array}\right]\left[\begin{array}{l} I^{(+)} \\ I^{(-)} \end{array}\right], \quad 0

Short answer

$$ \det(A - \lambda I) = \det \begin{bmatrix} -1 - \lambda & 1 \\ -1 & 1 - \lambda \end{bmatrix} = \lambda^2 - 2\lambda. $$ The eigenvalues are then found by solving the characteristic equation, giving us: $$ \lambda_1 = 0 \quad \text{and} \quad \lambda_2 = 2. $$ Now we find the corresponding eigenvectors of these eigenvalues: For \(\lambda_1 = 0\), we have: $$ \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} v_{11} \\ v_{12} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, $$ which gives us the eigenvector \(v_1 = \begin{bmatrix}1 \\ 1\end{bmatrix}\). For \(\lambda_2 = 2\), we have: $$ \begin{bmatrix} -1 - 2 & 1 \\ -1 & 1 - 2 \end{bmatrix} \begin{bmatrix} v_{21} \\ v_{22} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, $$ which gives us the eigenvector \(v_2 = \begin{bmatrix}1 \\ -1\end{bmatrix}\). Now that we have the eigenvalues \(\lambda_1, \lambda_2\) and their corresponding eigenvectors \(v_1, v_2\), we can form the general solution as: $$ \left[\begin{array}{l} I^{(+)} \\ I^{(-)} \end{array}\right] = C_1 e^{\lambda_1 \beta x} v_1 + C_2 e^{\lambda_2 \beta x} v_2 $$ Substituting the eigenvalues and eigenvectors, we get: $$ \left[\begin{array}{l} I^{(+)} \\ I^{(-)} \end{array}\right] = C_1 \begin{bmatrix} 1 \\ 1 \end{bmatrix} + C_2 e^{2 \beta x} \begin{bmatrix} 1 \\ -1 \end{bmatrix}. $$ Now we need to find the constants \(C_1\) and \(C_2\) by applying the given boundary conditions. For \(I^{(+)}(0) = I^{\text {inc }}\): $$ I^{(+)}(0) = C_1 + C_2 = I^{\text {inc }}. $$ For \(I^{(-)}(l) = \Gamma I^{(+)}(l)\): $$ I^{(-)}(l) = C_1 - C_2 e^{2 \beta l} = \Gamma(C_1 + C_2 e^{2 \beta l}), $$ Using the first boundary condition, we can rewrite this equation as: $$ I^{(-)}(l) = \Gamma I^{(+)}(0) - (\Gamma - 1) C_2 e^{2 \beta l} $$ and solve for \(C_2\): $$ C_2 = \frac{I^{(+)}(0) - I^{(-)}(l)}{\Gamma - 1}. $$ Finally combining all the results, we get the solution: $$ \left[\begin{array}{l} I^{(+)}(x) \\ I^{(-)}(x) \end{array}\right] = \left[\begin{array}{l} I^{\text {inc }} \\ 0 \end{array}\right] + \left(\frac{I^{(\text{inc})}(0) - I^{(-)}(l)}{\Gamma - 1}\right)e^{2\beta x} \begin{bmatrix} 1 \\ -1 \end{bmatrix}. $$ Now we will analyze the case of \(\Gamma=1\) and find the dependence of reflected radiation intensity on the reflection coefficient \(\Gamma\).

Step-by-step solution

Solve the given system of differential equations for I^{(+)}(x) and I^{(-)}(x)

To solve the given system of differential equations, let's rewrite them in matrix notation. The given boundary value problem is $$ \frac{d}{d x}\left[\begin{array}{l} I^{(+)} \\\ I^{(-)} \end{array}\right]=\beta\left[\begin{array}{ll} -1 & 1 \\\ -1 & 1 \end{array}\right]\left[\begin{array}{l} I^{(+)} \\\ I^{(-)} \end{array}\right], \quad 0<x<l , $$ with the boundary conditions: \(I^{(+)}(0)=I^{\text {inc }}\) and \(I^{(-)}(l)=\Gamma I^{(+)}(l)\). First, we find the eigenvalues of the matrix \(\beta\begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix}\). We have the characteristic equation,

Key concepts

Radiative Transport Model

The radiative transport model is a mathematical framework used to describe the transport of energy, usually in the form of light or radiation, through a medium. This model is crucial for understanding how light or other forms of energy behave when they encounter different materials, such as clouds, gases, or even spaces like water bodies. In this context, we specifically analyze a slab of material where radiation is partially absorbed, scattered, or reflected.

• The model focuses on the direction and amount of energy flowing in different directions within the slab.
• It considers two main streams of energy: the component moving forward, denoted by \( I^{(+)} \), and the component moving backward, \( I^{(-)} \).

When modeling the slab, boundary conditions must be set to describe how energy enters and exits the system. This includes considering the reflection at the edges and whether some of the radiation exits or is partially reflected back into the medium, leading to the development of phenomena like the reflection coefficient.

Reflection Coefficient

The reflection coefficient, denoted as \( \Gamma \), is a parameter that describes how much of the incident energy on a surface or boundary is reflected back. In the context of the boundary value problem, this parameter is crucial for determining the behavior of radiation at the boundary, particularly at \( x=l \) where reflection can occur.

• \( \Gamma \) ranges between 0 and 1, where 0 means no reflection (all energy exits) and 1 means total reflection (the surface acts like a perfect mirror).
• The boundary condition \( I^{(-)}(l) = \Gamma I^{(+)}(l) \) mathematically models this behavior by linking the backward radiation at the slab edge to the forward radiation.

The reflection coefficient is not only key in solving the model mathematically but also aids in our physical understanding. When \( \Gamma = 1 \), the system behaves as if there's a perfectly reflective wall, which can dramatically affect the energy distribution within the slab.

Differential Equations

Differential equations are the mathematics that allow us to understand how things change over time or space. In this exercise, they are used to model how the intensity of radiation changes as it moves through the slab. The equations are given in a form that describes how the energy's flow changes with respect to the position \( x \).

• The system of differential equations in this case takes into account both the forward and backward flow of radiation.
• Solving these equations helps us predict the distribution of radiation intensity \( I^{(+)}(x) \) and \( I^{(-)}(x) \) throughout the slab.

These equations, often first presented in a general form, can be turned into specific solutions by applying boundary conditions, like the values given at \( x=0 \) and \( x=l \). Using mathematical methods, such as finding eigenvalues and eigenvectors, we can solve these to get explicit formulas for \( I^{(+)}(x) \) and \( I^{(-)}(x) \).

Matrix Notation

Matrix notation is a compact, efficient way to express systems of equations, which is particularly beneficial when dealing with multiple variables and their interactions. In this scenario, we use matrix notation to express the coupling between \( I^{(+)} \) and \( I^{(-)} \) through the differential equations.

• The matrix \( \beta \begin{bmatrix} -1 & 1 \ -1 & 1 \end{bmatrix} \) represents the interaction between forward and backward moving radiation.
• This matrix relates the rate of change of each component to the presence of the other through a linear transformation, simplifying the calculation process.

Matrix notation not only simplifies notation but allows for the application of linear algebra techniques such as eigenvalue decomposition. This is crucial as these techniques can simplify solving differential equations and yield insights into the system's behavior, such as stability and the impact of varying \( \Gamma \).