Using the relation \(\langle S\rangle=\frac{1}{2} \operatorname{Re}\left\\{\mathbf{E}_{s} \times \mathbf{H}_{s}^{*}\right\\}\) and Eqs. (106) through (108), show that the average power density in the \(\mathrm{TE}_{10}\) mode in a rectangular waveguide is given by $$ \langle S\rangle=\frac{\beta_{10}}{2 \omega \mu} E_{0}^{2} \sin ^{2}\left(\kappa_{10} x\right) \mathbf{a}_{z} \mathrm{~W} / \mathrm{m}^{2} $$

When studying the concept of waveguides, it's crucial to understand the various modes of electromagnetic wave propagation within them. The TE10 mode is the fundamental transverse electric mode in a rectangular waveguide. This mode is characterized by having no electric field component in the direction of wave propagation, known as the z-direction in a standard coordinate system.

The 'TE' signifies 'Transverse Electric,' indicating that the electric field lines are purely transverse to the direction of wave propagation, with no longitudinal electric field component. The '10' denotes that there is one half-wave variation of the electric field in the 'b' dimension (the wider side of the waveguide) and zero half-wave variations in the 'a' dimension (the narrow side).

The TE10 mode is significant as it has the lowest cutoff frequency among all the modes that can propagate in a rectangular waveguide. The frequency must be greater than this cutoff frequency for wave propagation to occur. This unique feature of TE10 makes it highly efficient for microwave frequency ranges, leading to its widespread use in various applications such as satellite communications and radar systems.

Electromagnetic wave propagation within a waveguide involves complex interactions between electric and magnetic fields. In a rectangular waveguide operating in the TE10 mode, the wave propagates in the z-direction, and the electric and magnetic fields are confined within the waveguide walls.

As a wave travels, its shape and intensity can be described mathematically by field components. These components depend on the waveguide's physical dimensions and the frequency of the electromagnetic wave. The propagation constant \(\beta_{10}\) identifies how the phase of the wave changes along the waveguide, while the wave number \(\kappa_{10}\) describes the variation across the waveguide's width.

Understanding the fundamentals of wave propagation aids in determining how waves behave under different circumstances and the implications for power distribution and signal strength. Moreover, this knowledge is essential for designing waveguides that are optimal for certain frequencies and applications, ensuring maximum efficiency and minimal signal loss.

The Poynting vector, represented by \(\mathbf{S}\), is a vital concept in electromagnetics that provides the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field.

To calculate the average power density, which is the average energy flow per unit area, we use the Poynting vector. The equation \(\langle S\rangle=\frac{1}{2}\operatorname{Re}\{\mathbf{E}_{s} \times \mathbf{H}_{s}^{*}\}\) is critical for this calculation. In this context, \(\mathbf{E}_{s}\) refers to the electric field, and \(\mathbf{H}_{s}^{*}\) is the complex conjugate of the magnetic field.

To obtain the average power density in the TE10 mode of a rectangular waveguide, one takes the real part of the cross product of these two field components. This real part calculation assumes time-harmonic fields and results in the average power density over a wave cycle. The Poynting vector highlights an essential aspect of electromagnetic wave propagation, indicating how much power is actually being transmitted through a given area, and is a fundamental tool for engineers and scientists working with waveguides and other electromagnetic systems.