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 beSWR=A+BA-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 (BA)2. What is the SWR for (a) total reflection and (b) no reflection? (c) For SWR = 1.50, what is expressed as a percentage?

Short Answer

Expert verified
  1. Standing Wave Ratio for total reflection is
  2. Standing Wave Ratio for no reflection is 1
  3. R (reflection coefficient) for Standing Wave Ratio = 1.50 is 4.0%.

Step by step solution

01

Identification of given data

  1. Interference pattern is given as:SWR=A+BA-B
  2. The reflection coefficient,RBA2
02

Significance of standing waves

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.

03

(a) Determining the Standing Wave Ratio for total reflection

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

SWR=A+BA-B=A+AA-A=2A0=

Therefore, SWR for total reflection is

04

(b) Determining the Standing Wave Ratio for no reflection

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

i.e, B = 0

SWR=A+BA-B=A+0A-0=AA=1

Therefore, SWR for no reflection is 1

05

(c) Determining the reflection coefficient for the SWR= 1.50

We have been given that the reflection coefficient as:

RBA2R=BA

Again, the standing waveratiois given as:

SWR=A+BA-B=1+BA1-BA=1+R1-R

By componendo-dividendo theorem, we can write

SWR+1SWR-1=1+R+1-R1+R-1-RSWR+1SWR-1=22R=1RR=SWR-1SWR+1R=SWR-1SWR+12

Ifthen the value of reflection coefficient is given as:

R=1.50-11.50+12=0.502.502=0.04%R=0.04×100=4.0%

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

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