Question

Find a general solution for the electromagnetic wave propagation in a resonant cavity, a rectangular box of sides \(0 \leq x \leq a, 0 \leq y \leq b\), and \(0 \leq z \leq d\) with perfectly conducting walls. Discuss the modes the cavity can accommodate.

Short answer

The general solution for electromagnetic wave propagation in a rectangular resonant cavity with perfectly conducting walls is given by \(\Psi(x,y,z) = A\sin(k_x x)\sin(k_y y)\sin(k_z z) \) with the wavevector components defined by \(k_x = \frac{n\pi}{a}\), \(k_y = \frac{m\pi}{b}\), \(k_z = \frac{p\pi}{d} \). These solutions represent the different modes that the cavity can accommodate. Each mode corresponds to a different resonance frequency of the cavity.

Step-by-step solution

Define the Wave Equation

To start, define the wave equation in a cavity. The Helmholtz equation represents wave propagation, and in this case, represents the propagation of an electromagnetic wave in a cavity. This can be written as: \[ \nabla^2 \Psi + k^2 \Psi = 0 \] where \(\nabla^2\) is the Laplacian operator, \(\Psi\) is the potential or field (E-field or H-field), and \(k\) is the wave number.

Define Boundary Conditions

Since the cavity consists of perfectly conducting walls, the tangential component of the E-field is zero at the walls. This sets up the boundary conditions for the problem. These conditions can be represented mathematically as: \[ \Psi(x=0) = \Psi(x=a) = 0 \] \[ \Psi(y=0) = \Psi(y=b) = 0 \] \[ \Psi(z=0) = \Psi(z=d) = 0 \]

Expressing the Solution

Substituting the boundary conditions into the wave equation, a possible solution to the wave equation can be written in the form: \[ \Psi(x,y,z) = A\sin(k_x x)\sin(k_y y)\sin(k_z z) \] where A is the amplitude of the wave, and \(k_x, k_y, k_z\) are the wavevector components along the x, y, and z directions respectively.

Calculating The Modes

The modes of the cavity can be determined by setting the wavevector components equal to the resonant condition for each dimension. This comes out to be: \[ k_x = \frac{n\pi}{a} \] \[ k_y = \frac{m\pi}{b} \] \[ k_z = \frac{p\pi}{d} \] where n, m, p are the mode numbers which can take on integer values (including zero). The overall wave number \(k^2 = k_x^2 + k_y^2 + k_z^2\). The resonance frequencies of the cavity are then \(\omega = c k\) where c is the speed of light.

Discussing the Modes

The modes of the resonant cavity are characterized by the mode numbers n, m, p. Each set of (n, m, p) denotes a mode and corresponds to a specific resonance frequency. The fundamental mode is the mode with smallest non-zero resonance frequency. Higher order modes can be determined by increasing the values of n, m, p. The mode structure allows for various ways to propagate through the resonant cavity.

Key concepts

Wave Equation

The wave equation is fundamental in describing how electromagnetic waves propagate through different media and is crucial in understanding resonant cavities. In general, the wave equation helps to characterize how fields like the electric field (E-field) and magnetic field (H-field) change over space and time.
This equation is typically expressed as:

• \( abla^2 \Psi + k^2 \Psi = 0 \)

Here, \( \Psi \) represents the field potential, and \( abla^2 \) is the Laplacian operator, which calculates the rate at which the potential changes over space. The term \( k \), known as the wave number, connects the spatial variation in the wave to its frequency. In essence, solving the wave equation allows us to understand how electromagnetic waves behave inside the resonant cavity.

Boundary Conditions

Boundary conditions are a critical concept in solving the wave equation for specific problems like the resonant cavity. They define how the electromagnetic fields interact with the boundaries, which in this case are the perfectly conducting walls of the resonant cavity.
These conditions enforce the fact that along the surfaces of these perfect conductors, the tangential components of the E-field must be zero. Mathematically, for a resonant cavity of size \( a \times b \times d \), this can be expressed as:

• \( \Psi(x=0) = \Psi(x=a) = 0 \)
• \( \Psi(y=0) = \Psi(y=b) = 0 \)
• \( \Psi(z=0) = \Psi(z=d) = 0 \)

These conditions help to restrict the solution of the wave equation to configurations that can naturally occur in the cavity, setting the stage for identifying possible modes.

Resonant Cavity

A resonant cavity is essentially a structure that confines electromagnetic waves, allowing only specific wave patterns, known as modes, to persist. By reflecting waves back and forth, the cavity supports certain wavelengths that fit perfectly, akin to standing waves on a string.
In practice, resonant cavities are designed in various shapes, with the rectangular box shape being a common and straightforward example. The length of the box (defined by sides \(a, b, d\)) determines the allowable modes, and thus the frequencies that can resonate within the cavity. The walls of the cavity act as mirrors, perfectly reflecting the wave until a constructive interference pattern or mode is established, amplifying the wave.

Modes

Modes in a resonant cavity are the distinct patterns and frequencies at which electromagnetic waves can persist. Each mode is characterized by a combination of resonance conditions in the spatial dimensions of the cavity.
Mathematically, this is depicted as:

• \( k_x = \frac{n\pi}{a} \)
• \( k_y = \frac{m\pi}{b} \)
• \( k_z = \frac{p\pi}{d} \)
• where \( n, m, p \) are integers representing the mode numbers.

These integers indicate how many half-wavelengths fit into each dimension of the cavity. The fundamental mode, or lowest-frequency mode, is usually \((n, m, p) = (1, 1, 1)\), and higher modes involve higher integers. Each set of \( (n, m, p) \) modes corresponds to a resonance frequency, which can greatly affect how a cavity operates in practical applications.

Helmholtz Equation

The Helmholtz equation is essentially a version of the wave equation that is adapted specifically for analyzing phenomena like electromagnetic waves in bounded systems, such as resonant cavities. By focusing on the spatial variation of the fields at a fixed frequency, the Helmholtz equation facilitates easier analysis.

• It's given by: \( abla^2 \Psi + k^2 \Psi = 0 \),

where \( k \) is again the wave number associated with the length scale set by the cavity's dimensions. Solving the Helmholtz equation, given the appropriate boundary conditions, allows for identifying all possible modes that can exist within a cavity. It is a critical mathematical tool for physicists and engineers working with wave-based systems, providing insights into how waves can be manipulated and controlled.