# Question on wave equation of plane wave.

• yungman

#### yungman

For plane wave travel in +ve z direction in a charge free medium, the wave equation is:

$$\frac{\partial^2 \widetilde{E}}{\partial z^2} -\gamma^2 \widetilde E = 0$$

Where $\gamma^2 = - k_c^2 ,\;\; k_c= \omega \sqrt {\mu \epsilon_c} \hbox { and } \epsilon_c = \epsilon_0 \epsilon_r -j\frac{\sigma}{\omega}$ .

$$\widetilde E = E_0^+ e^{-\gamma z} \;+\; E_0^- e^{\gamma z} \;=\; E_0^+ e^{-\alpha z}e^{-j \beta z} \;+\; E_0^- e^{\alpha z}e^{j \beta z}$$

Notice the second term is the reflected wave AND is growing in magnitude as it move in -ve z direction! That cannot be true. It should still decay at rate of $e^{-\alpha z }$.

What am I missing?

Last edited:
erm as we move towards -ve z direction z gets smaller and smaller hence $$e^{az}$$ gets smaller too for a>0.

Why didn't I think of that!

Thanks

Happens to me too, when i focus my mind on something i miss some relatively simple things.

## 1. What is the wave equation of a plane wave?

The wave equation of a plane wave is a mathematical equation that describes the behavior of a traveling wave in a given medium. It is written as y = A sin(kx - ωt), where y is the displacement of the wave, A is the amplitude, k is the wave number, x is the position, ω is the angular frequency, and t is the time.

## 2. What is the significance of the wave equation of a plane wave?

The wave equation of a plane wave is significant because it allows us to predict the behavior of a wave in a given medium. It helps us understand how the position and time affect the displacement of the wave, and how the amplitude and frequency of the wave affect its intensity.

## 3. How is the wave equation of a plane wave derived?

The wave equation of a plane wave is derived from the general wave equation, which describes the propagation of any type of wave. By assuming that the wave is traveling in a straight line and has a constant amplitude and frequency, we can simplify the general wave equation to the specific form of a plane wave.

## 4. Can the wave equation of a plane wave be applied to all types of waves?

No, the wave equation of a plane wave is specifically applicable to transverse waves, which are waves that vibrate perpendicular to the direction of propagation. It cannot be applied to longitudinal waves, which vibrate parallel to the direction of propagation.

## 5. How does the wave equation of a plane wave relate to wave interference?

The wave equation of a plane wave can be used to understand wave interference, which occurs when two or more waves interact with each other. By analyzing the displacement, amplitude, and frequency of each wave, we can predict how they will interfere with each other and create a resultant wave.