A continuous traveling wave with amplitude Ais incident on a boundary. The continuous reflection, with a smaller amplitude B, travels back through the incoming wave. The resulting interference pattern is displayed in Fig. 16-51. The standing wave ratio is defined to be$\mathit{S}\mathit{W}\mathit{R}\mathbf{=}\frac{\mathbf{A}\mathbf{+}\mathbf{B}}{\mathbf{A}\mathbf{-}\mathbf{B}}$

The reflection coefficient Ris the ratio of the power of the reflected wave to thepower of the incoming wave and is thus proportional to the ratio ${\mathbf{\left(}\frac{\mathbf{B}}{\mathbf{A}}\mathbf{\right)}}^{\mathbf{2}}$. What is the SWR for (a) total reflection and (b) no reflection? (c) For SWR = 1.50, what is expressed as a percentage?

1. Interference pattern is given as:$SWR\mathit{=}\frac{\mathit{A}\mathit{+}\mathit{B}}{\mathit{A}\mathit{-}\mathit{B}}$
2. The reflection coefficient,$R\infty {\left(\frac{B}{A}\right)}^2$

Two waves that are travelling in opposite directions but have the same amplitude and frequency are combined to form a standing wave, also known as a stationary wave. We can find the Standing Wave Ratio for total reflection and no reflection by inserting the values of A and B in the given formula for Standing Wave Ratio.We can write an expression for R in terms of the Standing Wave Ratio using the given formulae. Then inserting values in it we can easily get R in percentage.

In case of the total reflection, amplitude of the incident wave is equal to the amplitude of reflected wave, i.e, A = B

$$\begin{gathered} SWR=\frac{A+B}{A-B} \\ =\frac{A+A}{A-A} \\ =\frac{2A}{0} \\ =\infty \end{gathered}$$

Therefore, SWR for total reflection is$\infty$

In case of no reflection, amplitude of the reflected wave is zero.

i.e, B = 0

$$\begin{gathered} SWR=\frac{A+B}{A-B} \\ =\frac{A+0}{A-0} \\ =\frac{A}{A} \\ =1 \end{gathered}$$

Therefore, SWR for no reflection is 1

We have been given that the reflection coefficient as:

$$\begin{gathered} R\infty {\left(\frac{B}{A}\right)}^2 \\ \sqrt{R}=\frac{B}{A} \end{gathered}$$

Again, the standing waveratiois given as:

$$\begin{gathered} SWR=\frac{A+B}{A-B} \\ =\frac{1+\left({\displaystyle \frac{B}{A}}\right)}{1-\left({\displaystyle \frac{B}{A}}\right)} \\ =\frac{1+\sqrt{R}}{1-\sqrt{R}} \end{gathered}$$

By componendo-dividendo theorem, we can write

$$\begin{gathered} \frac{SWR+1}{SWR-1}=\frac{\left(1+\sqrt{R}\right)+\left(1-\sqrt{R}\right)}{\left(1+\sqrt{R}\right)-\left(1-\sqrt{R}\right)} \\ \frac{SWR+1}{SWR-1}=\frac{2}{2\sqrt{R}}=\frac{1}{\sqrt{R}} \\ \sqrt{R}=\frac{SWR-1}{SWR+1} \\ R={\left(\frac{SWR-1}{SWR+1}\right)}^2 \end{gathered}$$

Ifthen the value of reflection coefficient is given as:

$$\begin{gathered} R={\left(\frac{1.50-1}{1.50+1}\right)}^2 \\ ={\left(\frac{0.50}{2.50}\right)}^2 \\ =0.04 \\ \%R=0.04\times 100 \\ =4.0\% \end{gathered}$$

Therefore, R (reflection coefficient) for SWR = 1.50 is 4.0 % .