6.2: The Rectangular Wave Equation

Maxwell’s equations will be put in a form that can be used in establishing the field descriptions in parallel-plate and rectangular waveguides. The EM fields in these structures vary sinusoidally with respect to both position and time so the first simplification of Maxwell’s equations is to use phasors. Boundary conditions are established by the metal walls, and these walls match the Cartesian coordinate system. So Maxwell’s equations are put in Cartesian coordinate form. Simplifications of the fields can be made that relate to the positions of the metal walls. Another simplification is made by assuming that there can only be propagation in the \(±z\) direction. When the wave propagates in the \(+z\) direction it is called the forward-traveling wave, and when it propagates in the \(−z\) direction it is called the reverse-traveling wave.

The development begins with Maxwell’s equations (Equations (1.5.1)–(1.5.4)) in a source-free region (\(\rho = 0\) and \(J = 0\)). A simplification comes from assuming a linear, isotropic, and homogeneous medium so that \(\varepsilon\) and \(\mu\) are independent of signal level and are independent of the field direction and of position, thus

\[\begin{align}\label{eq:1}\nabla\times\overline{\mathcal{E}}&=-\frac{\partial\overline{\mathcal{B}}}{\partial t}=-\mu\frac{\partial\overline{\mathcal{H}}}{\partial t} \\ \label{eq:2}\nabla\cdot\overline{\mathcal{D}}&=0=\nabla\cdot\overline{\mathcal{E}} \\ \label{eq:3}\nabla\times\overline{\mathcal{H}}&=\frac{\partial\overline{\mathcal{D}}}{\partial t}=\varepsilon\frac{\partial\overline{\mathcal{E}}}{\partial t} \\ \label{eq:4} \nabla\cdot\overline{\mathcal{B}}&=0=\nabla\cdot\overline{\mathcal{H}} \end{align} \]

Taking the curl of Equation \(\eqref{eq:1}\) leads to

\[\label{eq:5}\nabla\times\nabla\times\overline{\mathcal{E}}=-\nabla\times\mu\frac{\partial\overline{\mathcal{H}}}{\partial t}=-\mu\frac{\partial(\nabla\times\overline{\mathcal{H}})}{\partial t} \]

Applying the identity \(\nabla\times\nabla\times\overline{\mathcal{A}}=\nabla(\nabla\cdot\overline{\mathcal{A}}) −\nabla^{2}\overline{\mathcal{A}}\) to the left-hand side of Equation \(\eqref{eq:5}\), and replacing \(\nabla\times\overline{\mathcal{H}}\) with the right-hand side of Equation \(\eqref{eq:3}\), the equation above becomes

\[\label{eq:6}-\nabla^{2}\overline{\mathcal{E}}+\nabla(\nabla\cdot\overline{\mathcal{E}})=-\mu\frac{\partial}{\partial t}\left(\varepsilon\frac{\partial\overline{\mathcal{E}}}{\partial t}\right)=-\mu\varepsilon\frac{\partial^{2}\overline{\mathcal{E}}}{\partial t^{2}} \]

Using Equation \(\eqref{eq:2}\) this reduces to

\[\label{eq:7}\nabla^{2}\overline{\mathcal{E}}=\mu\varepsilon\frac{\partial^{2}(\overline{\mathcal{E}})}{\partial t^{2}} \]

where

\[\label{eq:8}\nabla^{2}\overline{\mathcal{E}}=\frac{\partial^{2}\overline{\mathcal{E}}}{\partial x^{2}}+\frac{\partial^{2}\overline{\mathcal{E}}}{\partial y^{2}}+\frac{\partial^{2}\overline{\mathcal{E}}}{\partial z^{2}}=\nabla_{t}^{2}\overline{\mathcal{E}}+\frac{\partial^{2}\overline{\mathcal{E}}}{\partial z^{2}} \]

and

\[\label{eq:9}\nabla_{t}^{2}\overline{\mathcal{E}}=\frac{\partial^{2}\overline{\mathcal{E}}}{\partial x^{2}}+\frac{\partial^{2}\overline{\mathcal{E}}}{\partial y^{2}} \]

is used for fields propagating in the \(±z\) direction and the subscript \(t\) indicates the transverse plane (the \(x–y\) plane here). Equation \(\eqref{eq:9}\) can be put into the form of its components. Since

\[\label{eq:10}\overline{\mathcal{E}}=\mathcal{E}_{x}\hat{\mathbf{x}}+\mathcal{E}_{y}\hat{\mathbf{y}}+\mathcal{E}_{z}\hat{\mathbf{z}} \]

then

\[\begin{align} \nabla^{2}\overline{\mathcal{E}}&=\left(\frac{\partial^{2}\mathcal{E}_{x}}{\partial x^{2}}\hat{\mathbf{x}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial x^{2}}\hat{\mathbf{y}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial x^{2}}\hat{\mathbf{z}}\right) + \left(\frac{\partial^{2}E_{x}}{\partial y^{2}}\hat{\mathbf{x}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial y^{2}}\hat{\mathbf{y}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial y^{2}}\hat{\mathbf{z}}\right) \nonumber \\ \label{eq:11} &\quad +\left(\frac{\partial^{2}E_{x}}{\partial z^{2}}\hat{\mathbf{x}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial z^{2}}\hat{\mathbf{y}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial z^{2}}\hat{\mathbf{z}} \right) \\&=\left(\frac{\partial^{2}E_{x}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{x}}{\partial y^{2}}+\frac{\partial^{2}\mathcal{E}_{x}}{\partial z^{2}}\right)\hat{\mathbf{x}}+\left(\frac{\partial^{2}E_{y}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial y^{2}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial z^{2}}\right)\hat{\mathbf{y}} \\ \label{eq:12}&\quad +\left(\frac{\partial^{2}\mathcal{E}_{z}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial y^{2}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial z^{2}}\right)\hat{\mathbf{z}} \end{align} \]

and

\[\label{eq:13}\nabla_{t}^{2}\overline{\mathcal{E}}=\left(\frac{\partial^{2}\mathcal{E}_{x}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{x}}{\partial y^{2}}\right)\hat{\mathbf{x}}+\left(\frac{\partial^{2}\mathcal{E}_{y}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{y}}{\partial y^{2}}\right)\hat{\mathbf{y}}+\left(\frac{\partial^{2}\mathcal{E}_{z}}{\partial x^{2}}+\frac{\partial^{2}\mathcal{E}_{z}}{\partial y^{2}}\right)\hat{\mathbf{z}} \]

Invoking the phasor form, \(\partial /\partial t\) is replaced by \(\jmath\omega\), and with propagation only in the \(±z\) direction there is an assumed \(\text{e}^{(\jmath\omega t−\gamma z)}\) dependence of the fields. Development is now simplified by introducing the phasor \(\overline{\mathcal{E}}\) defined so that

\[\label{eq:14}\overline{\mathcal{E}}=\overline{E}\text{e}^{-\gamma z} \]

Now Equation \(\eqref{eq:7}\) further reduces to

\[\label{eq:15}\nabla^{2}\overline{E}=\left(\nabla_{t}^{2}\overline{E}+\frac{\partial^{2}\overline{E}}{\partial z^{2}}\right)=\nabla_{t}^{2}\overline{E}+\gamma^{2}\overline{E}=(\jmath\omega)^{2}\mu\varepsilon\overline{E}=-k^{2}\overline{E} \]

where \(k =\omega\sqrt{\mu\varepsilon}\) is the wavenumber (with SI units of \(\text{m}^{−1}\)). Rearranging Equation \(\eqref{eq:15}\) yields

\[\label{eq:16}\nabla_{t}^{2}\overline{E}=-(\gamma^{2}+k^{2})\overline{E} \]

A similar expression can be derived for the magnetic field:

\[\label{eq:17}\nabla_{t}^{2}\overline{H}=-(\gamma^{2}+k^{2})\overline{H} \]

Equations \(\eqref{eq:16}\) and \(\eqref{eq:17}\) are called wave equations, or Helmholtz equations, for phasor fields propagating in the \(z\) direction. Equations \(\eqref{eq:16}\) and \(\eqref{eq:17}\) are usually written as

\[\begin{align}\label{eq:18}\nabla_{t}^{2}\overline{E}=-k_{c}^{2}\overline{E} \\ \label{eq:19}\nabla_{t}^{2}\overline{H}=-k_{c}^{2}\overline{H}\end{align} \]

where the cutoff wavenumber is

\[\label{eq:20}k_{c}^{2}=\gamma^{2}+k^{2} \]

Equations \(\eqref{eq:18}\) and \(\eqref{eq:19}\) describe the transverse fields (the fields in the \(x–y\) plane) between the conducting plates of the parallel-plate as well as within the walls of the rectangular waveguide having a \(\text{e}^{(\jmath\omega t−\gamma z)}\) dependence. The general form of the solution of these equations is a sinusoidal wave moving in the \(z\) direction. For propagating waves in a lossless medium, \(\gamma = \jmath\beta\), where \(\beta\) is the phase constant:

\[\label{eq:21}\beta=\pm\sqrt{k^{2}-k_{c}^{2}} \]

If \(\beta\) is not real, which occurs when \(|k_{c}| < |k|\), then an EM wave cannot propagate and such modes are called evanescent modes. These are like fringing fields. If they are generated, say at a discontinuity, they will store reactive energy locally.

Boundary conditions, resulting from the charges and current on the plates, further constrain the solutions. Equation \(\eqref{eq:1}\) with Equation (1.A.39) becomes

\[\label{eq:22}\nabla\times\overline{E}=\jmath\omega\mu\overline{H} \]

In rectangular coordinates, \(\overline{E}= E_{x}\hat{\mathbf{x}}+E_{y}\hat{\mathbf{y}}+E_{z}\hat{\mathbf{z}}\) and \(\overline{H}= H_{x}\hat{\mathbf{x}}+H_{y}\hat{\mathbf{y}}+H_{z}\hat{\mathbf{z}}\), and Equation \(\eqref{eq:22}\) becomes

\[\label{eq:23}\left.\begin{array}{ll}{\frac{\partial E_{z}}{\partial y}+\gamma E_{y}=-\jmath\omega\mu H_{x}}&{-\frac{\partial E_{z}}{\partial x}-\gamma E_{x}=-\jmath\omega\mu H_{y}}\\{\frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}=-\jmath\omega\mu H_{z}}&{}\end{array}\right\} \]

Similarly for \(\nabla\times\overline{H}\jmath\omega\varepsilon\overline{E}\):

\[\label{eq:24}\left.\begin{array}{ll}{\frac{\partial H_{z}}{\partial y}+\gamma H_{y}=\jmath\omega\varepsilon E_{x}}&{-\frac{\partial H_{z}}{\partial x}-\gamma H_{x}=\jmath\omega\varepsilon E_{y}}\\{\frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}=\jmath\omega\varepsilon E_{z}}&{}\end{array}\right\} \]

Solving Equations \(\eqref{eq:23}\) and \(\eqref{eq:24}\) yields the rectangular wave equations for \(k_{c}\neq 0 (k_{c}^{2} = k^{2} +\gamma^{2}\) and if there is no loss \(k_{c}^{2} = \omega^{2}\mu\varepsilon −\beta^{2}\)):

\[\label{eq:25}\left.\begin{array}{ll}{E_{x}=\frac{-1}{k_{c}^{2}}\left(\gamma\frac{\partial E_{z}}{\partial x}+\jmath\omega\mu\frac{\partial H_{z}}{\partial y}\right)}&{E_{y}=\frac{1}{k_{c}^{2}}\left(-\gamma\frac{\partial E_{z}}{\partial y}+\jmath\omega\mu\frac{\partial H_{z}}{\partial x}\right)}\\{H_{x}=\frac{1}{k_{c}^{2}}\left(-\gamma\frac{\partial H_{z}}{\partial x}+\jmath\omega\varepsilon\frac{\partial E_{z}}{\partial y}\right)}&{H_{y}=\frac{-1}{k_{c}^{2}}\left(\gamma\frac{\partial H_{z}}{\partial y}+\jmath\omega\varepsilon\frac{\partial E_{z}}{\partial z}\right)}\\{E_{z}=\frac{-\jmath}{\omega\varepsilon}\left(\frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}\right)}&{H_{z}=\frac{\jmath}{\omega\mu}\left(\frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}\right)}\end{array}\right\} \]

The solution for \(k_{c} = 0\) is arrived at separately. Since \(k_{c} = 0\) there is no loss. Also propagation at DC is a solution and the phasor fields at \(\omega = 0\) will also be the field descriptions at any frequency. At \(\omega = 0,\:\gamma = \jmath\beta = 0\) and \(k = 0\). Equations \(\eqref{eq:23}\)–\(\eqref{eq:24}\) are now written as

\[\label{eq:26}\left.\begin{array}{ll}{\frac{\partial E_{z}}{\partial y}+0\cdot E_{y}=0\cdot H_{x}=0}&{-\frac{\partial E_{z}}{\partial x}-0\cdot E_{x}=0\cdot H_{y}=0}\\{\frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}=-0\cdot H_{z}=0}&{\frac{\partial H_{z}}{\partial y}+0\cdot H_{y}=0\cdot E_{x}=0}\\{-\frac{\partial H_{z}}{\partial x}-0\cdot H_{x}=0\cdot E_{y}=0}&{\frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}=0\cdot E_{z}=0}\end{array}\right\} \]

The only solutions to these with \(\partial /\partial x = 0\) that also satisfies boundary conditions are that \(E_{z} =0= H_{z} = E_{x} = H_{y}\), and \(H_{y}\) and \(E_{x}\) are constants.

Now that the fields are in the appropriate forms, classification of possible solutions (i.e. modes) can be developed for the parallel-plate and rectangular waveguides. At this stage the following simplifications have been made to Maxwell’s equations to get them into the form of Equation \(\eqref{eq:25}\)):

• Using phasors
• Restriction of propagation to the \(+z\) and \(−z\) directions
• Assuming that \(\varepsilon\) and \(\mu\) are constants
• Putting the wave equations in rectangular form so that boundary conditions established by the metal walls can be easily applied.