Consider two unbounded plane waves whose vector wave numbers \(k_{1}\) and \(\kappa_{2}\left(\left.\right|_{1,2} \mid=\right.\) \(\omega / c)\) define a plane and whose electric fields are polarized normal to the plane. (a) Show that the superposition of these two plane waves is a wave traveling in the direction bisecting the angle \(\alpha\) between \({ }_{1}{ }_{1}\) and \({ }_{k}\) and that the \(\mathbf{E}\) field vanishes on a set of nodal planes spaced \(a=\) \(\lambda_{0} / 2 \sin \frac{1}{2} \alpha\) apart. \((b)\) Show that plane conducting walls can be placed at two adjacent nodal planes without violating the electromagnetic boundary conditions and likewise that a second pair of conducting walls of arbitrary separation \(b\) can be introduced to construct a rectangular waveguide of cross section \(a\) by \(b\), propagating the \(T E_{10}\) mode. Thus establish that the TE \(_{10}\) mode (more generally, the TE \(_{10}\) modes) may be interpreted as the superposition of two plane waves making the angle \(\frac{1}{2} \alpha\) with the waveguide axis and undergoing multiple reflections from the sidewalls. Note: The situation is directly analogous to that discussed in Sec. 2.4. Higherorder TE modes \((m>0)\) and TM modes may be described similarly as a superposition of four plane waves.

Step by step solution

01

Consider two unbounded plane waves with vector wave numbers k1 and k2. Their electric fields, E1 and E2, are polarized normal to the plane defined by k1 and k2. The superposition of these two plane waves can be represented as E_total = E1 + E2, and the resulting wave will travel in a direction that bisects the angle α between k1 and k2. #Step 2: Nodal Planes and Electric Field Vanishing Conditions#

The electric field vanishes on a set of nodal planes when the superposition of the two electric fields results in destructive interference. The spacing between the nodal planes can be determined by finding the condition for which E_total = 0. The spacing 'a' between these planes can be given by: a = λ0 / (2 * sin(α/2)), where λ0 is the wavelength of the plane wave. #Step 3: Conducting Walls at Nodal Planes#

02

A conducting wall can be placed at a nodal plane without violating the electromagnetic boundary conditions because the tangential components of the electric field (E-field) will be zero at the surface of the wall. Thus, no charges will accumulate at these surfaces, and the boundary conditions are preserved. #Step 4: Rectangular Waveguide and TE10 Mode Propagation#

A rectangular waveguide with cross-section 'a' by 'b' can be constructed by placing two pairs of conducting walls at adjacent nodal planes with arbitrary separation 'b'. The waveguide will support the TE10 mode when the cross-sectional dimensions a and b meet specific requirements, and the wave will propagate with multiple reflections along the waveguide axis. When these conditions are met, the TE10 mode can be interpreted as the superposition of two plane waves making an angle (α/2) with the waveguide axis, undergoing multiple reflections from the sidewalls. Higher-order TE modes (with m > 0) and TM modes can also be described similarly by considering the superposition of four plane waves.

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Key Concepts

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

Wave Superposition in Electromagnetic Theory

Wave superposition is a fundamental principle in physics, specifically in electromagnetism. It describes the phenomenon where two or more waves overlap and combine to form a new wave pattern. The total displacement of the electromagnetic field at any point is the vector sum of the individual wave displacements. This concept is especially pertinent when exploring the formation of modes in waveguides, such as the TE10 mode in rectangular waveguides.

Considering two plane waves with their electric fields polarized perpendicularly to the plane they define, their superposition results in a wave traveling in a new direction. In the textbook exercise, we're looking at waves whose vector wave numbers, denoted as \( k_1 \) and \( k_2 \), define such a plane. By applying the principle of superposition, these combined waves generate a pattern where the wavefronts travel in the direction that bisects the angle between \( k_1 \) and \( k_2 \). This concept is central to understanding how the TE10 mode can be seen as a result of superposition.

Nodal Planes in Wave Propagation

Nodal planes are critical to understanding wave behavior within waveguides. In the context of electromagnetic waves, these are regions where the electric field (E-field) intensity is consistently zero due to destructive interference of two overlapping waves. In the given exercise, when two plane waves are combined, their electric fields cancel out at specific intervals, giving rise to these nodal planes.

The spacing between nodal planes \( a \) can be determined by the expression \( a = \frac{\lambda_0}{2 \sin(\frac{\alpha}{2})} \), where \( \lambda_0 \) is the wavelength of the plane wave, and \( \alpha \) is the angle between the wave vectors of the two plane waves. These nodal planes play a crucial role when incorporating conducting walls to create a waveguide, as they are strategically placed where the electric field is zero to satisfy electromagnetic boundary conditions.

Understanding the Rectangular Waveguide

A rectangular waveguide is a structure used to guide electromagnetic waves from one point to another. It confines the wave energy to its linear axis and allows for efficient transfer without significant losses. The waveguide acts as a conduit, controlling the wave's path due to reflections off the conducting walls.

In our exercise, conducting walls placed at adjacent nodal planes ensure that we aren't violating electromagnetic boundary conditions. This creates a section of a waveguide with particular dimensions 'a' and 'b,' which supports specific wave modes like the TE10. The TE10, or Transverse Electric mode, boasts a single half-wavelength variation across the width of the waveguide and no variation along its length, making it the dominant mode in rectangular waveguides. The dimensions required for the TE10 mode ensure that only specific wavelengths resonate, which correlates to the superposition and reflection of plane waves described in the problem.

Electromagnetic Boundary Conditions in Waveguides

Electromagnetic boundary conditions are pivotal when designing waveguides. These conditions dictate how electric and magnetic fields behave at the boundary surfaces - such as the conducting walls of the waveguide. For instance, the tangential component of an electric field must be zero at the surface of a perfect conductor, and the magnetic fields must satisfy a similar boundary condition.

In the scenario where conducting walls are introduced at nodal planes, the condition for the electric field is naturally met, as the field's intensity is already zero at these planes. By cleverly using this approach, we can contain certain wave modes, like TE10, within a rectangular waveguide without disrupting the electromagnetic wave pattern. Multiple reflections are inherently part of this confinement, as waves bounce between the conductive walls, reinforcing the desired mode within the guide. Meeting these boundary conditions ensures that the waves within the waveguide remain undisturbed and coherent, allowing for the transport and manipulation of electromagnetic signals with minimal loss of energy.