Wednesday, December 26, 2012

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium

  • Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium

We`ll now derive the Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium from the Maxwell`s equations.

The general Maxwell`s equations are

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium maxwell equation \center \overrightarrow{\triangledown }\cdot \overrightarrow{E} = \frac{\rho_{free}}{\varepsilon_0} \; \; \; \; \;\;\;\; (1)\\ \\ \\ \\ \overrightarrow{\triangledown }\cdot \overrightarrow{B} = 0 \; \; \; \; \;\;\;\; (2)\\ \\ \\ \\ \overrightarrow{\triangledown } \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t} \; \; \; \; \;\;\;\; (3) \\ \\ \\ \\ \\ \overrightarrow{\triangledown } \times \overrightarrow{B} = \mu\overrightarrow{J_{free}} + \frac{1}{c^2} \frac{\partial \overrightarrow{E}}{\partial t} \; \; \; \; \;\;\;\; (4)


For linearity and isotropy, we have

 Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 1 \overrightarrow{D} = \varepsilon \overrightarrow{E}\; , \; \overrightarrow{B} = \mu \overrightarrow{H}\; , \; \overrightarrow{J_{free}} = \sigma  \overrightarrow{E}

We will use the following identity to obtain the wave equations

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium curl curl identity \overrightarrow{\triangledown} \times (\overrightarrow{\triangledown} \times \overrightarrow{X})= \overrightarrow{\triangledown} (\overrightarrow{\triangledown}\cdot \overrightarrow{X}) - \overrightarrow{\triangledown}^2 \overrightarrow{X}


To derive the wave equation for E

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 5 \center \overrightarrow{\triangledown} \times (\overrightarrow{\triangledown} \times \overrightarrow{E})= \overrightarrow{\triangledown}(\overrightarrow{\triangledown}\cdot \overrightarrow{E}) - \overrightarrow{\triangledown}^2 \overrightarrow{E} \; \; \; \; \; \; \; \; (5)

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 6 \overrightarrow{\triangledown}^2 \overrightarrow{E} - \overrightarrow{\triangledown}(\overrightarrow{\triangledown}\cdot \overrightarrow{E}) = \frac{\partial}{\partial t}(\overrightarrow{\triangledown} \times \overrightarrow{B}) \; \; \; \; \; \; \; \; (6)

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 7 \overrightarrow{\triangledown}^2 \overrightarrow{E} - \overrightarrow{\triangledown}(\overrightarrow{\triangledown}\cdot \overrightarrow{E}) = \frac{\partial}{\partial t}(\mu\overrightarrow{J_{free}} + \varepsilon\mu \frac{\partial \overrightarrow{E}}{\partial t} ) \; \; \; \; \; \; \; \; (7)

replacing Jfree in the equation (7)

So, we get the wave equation for E

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 9 \overrightarrow{\triangledown}^2 \overrightarrow{E} -  \sigma\mu \frac{\partial \overrightarrow{E}}{\partial t} - \frac{1}{c^2} \frac{\partial^2 \overrightarrow{E}}{\partial t^2} =  \overrightarrow{\triangledown}(\frac{\rho_{free}}{\varepsilon_0})\; \; \; \; \; \; \; \; (9)


To obtain the wave equation for B in Harmonic Isotropic Linear Stationary (HILS) medium, we follow the same principle.

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 10 \overrightarrow{\triangledown} \times (\overrightarrow{\triangledown} \times \overrightarrow{B})= \overrightarrow{\triangledown}(\overrightarrow{\triangledown}\cdot \overrightarrow{B}) - \overrightarrow{\triangledown}^2 \overrightarrow{B} \; \; \; \; \; \; \; \; (10)

Taking the divergence of B =0 (Maxwell second equation above) and writing B in terms of H as appropriate for linearity and isotropy, we obtain

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 11 \mu\overrightarrow{\triangledown} \times (\overrightarrow{\triangledown} \times \overrightarrow{H})=  - \mu\overrightarrow{\triangledown}^2 \overrightarrow{H} \\ \Rightarrow \overrightarrow{\triangledown} \times (\overrightarrow{\triangledown} \times \overrightarrow{H})=  - \overrightarrow{\triangledown}^2 \overrightarrow{H}\; \; \; \; \; \; \; \; (11)

From equation (4) of the Maxwell`s equations, we obtain equation (12) below, and upon simplifying and applying curl on both sides, we obtain the following form of the equation

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 12 \mu\overrightarrow{\triangledown } \times \overrightarrow{H} - \frac{1}{c^2} \frac{\partial \overrightarrow{E}}{\partial t}= \mu\overrightarrow{J_{free}} \\ \\ \overrightarrow{\triangledown } \times(\overrightarrow{\triangledown } \times \overrightarrow{H}) - \frac{1}{\mu c^2} \frac{\partial (\overrightarrow{\triangledown } \times\overrightarrow{E})}{\partial t}= \sigma(\overrightarrow{\triangledown } \times\overrightarrow{E})  \; \; \; \; \;\;\;\; (12)

Then, we replace the right-hand side of equation (11) and the curl of E (from the 3rd Maxwell`s equation) in equation (12)

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 13 \overrightarrow{\triangledown } \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t} = -\mu\frac{\partial \overrightarrow{H}}{\partial t} \\ \\ - \overrightarrow{\triangledown}^2 \overrightarrow{H} + \frac{1}{c^2} \frac{\partial^2\overrightarrow{H}}{\partial t^2}= -\sigma\mu\frac{\partial\overrightarrow{H}}{\partial t} \\ \\ \overrightarrow{\triangledown}^2 \overrightarrow{H} -\sigma\mu\frac{\partial\overrightarrow{H}}{\partial t} - \frac{1}{c^2} \frac{\partial^2\overrightarrow{H}}{\partial t^2}= 0  \; \; \; \; \;\;\;\; (13)
Hence, we have obtain the wave equation for H in a HILS medium.

So, we have finally obtained the wave equations for both E and H. That`s what we wanted.



Now, let`s try an additional simplification for the wave equation for E.

let E be in the x-direction,
Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 13-1 \overrightarrow{\triangledown }\cdot \overrightarrow{E} = \frac{\rho_{free}}{\varepsilon_0} \; \; \; ; \;\frac{\overrightarrow{E}}{\left |\overrightarrow{E} \right |}=\widehat{i}  \; \; \; \; ; \; \overrightarrow{\triangledown }\cdot \overrightarrow{E} = \frac{\partial E_x}{\partial x}

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 13-2 \overrightarrow{\triangledown}(\frac{\rho_{free}}{\varepsilon_0}) = \overrightarrow{\triangledown}(\overrightarrow{\triangledown}\cdot \overrightarrow{E}) = \frac{\partial^2 \overrightarrow{E}}{\partial x^2} \widehat{i} = \overrightarrow{\triangledown}^2 \overrightarrow{E}

We can therefore replace the right-hand side of equation (9) with the result of the above equation

Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 14-15 \center\overrightarrow{\triangledown}^2 \overrightarrow{E} -  \sigma\mu \frac{\partial \overrightarrow{E}}{\partial t} - \varepsilon\mu \frac{\partial^2 \overrightarrow{E}}{\partial t^2} =  \overrightarrow{\triangledown}^2 \overrightarrow{E} \\ \\ \\ \Rightarrow \sigma \frac{\partial \overrightarrow{E}}{\partial t} - \varepsilon \frac{\partial^2 \overrightarrow{E}}{\partial t^2} = 0 \; \; \; \; \; \; \; \; (14)
\\ \\ \\ E_x = E_1 +E_2 e^{-\frac{\sigma}{\varepsilon}t} \; \; \; \; \; \; \; \; (15)

Equation (15) is the solution for Ex that would be obtained from (14).

As E and H are transverse, an example of a solution would be
Plane Harmonic Wave Equations in Harmonic Isotropic Linear Stationary (HILS) medium equation 16 \center\overrightarrow{E} = E_0 e^{i(\omega t-kx)} \widehat{j}\\ \\ \overrightarrow{H} = \frac{k}{\omega\mu}E_0 e^{i(\omega t-kx)} \widehat{k}\\ \\ Z = \left | \frac{E}{H}\right | = \frac{\omega\mu}{k} = \mu c = \frac{1}{\varepsilon c} =(\frac{\mu}{\varepsilon})^{\frac{1}{2}}

where Z is the characteristic impedance.

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