Question

A sinusoidal wave is reflected at the surface of a medium whose properties are such that half the incident energy is absorbed. Consider the field that results from the superposition of the incident and the reflected wave. An observer stationed somewhere in this field finds the local electric field oscillating with a certain amplitude E. What is the ratio of the largest such amplitude noted by an observer to the smallest amplitude noted by any observer

Answer 1

The ratio of the largest to smallest oscillation amplitude noted by any observer in the given scenario is approximately 5.9, due to constructive and destructive interference of the incident and reflected waves where half the energy is absorbed.

Step-by-step explanation:

The superposition of the incident and reflected wave in a medium where half the incident energy is absorbed results in local electric field amplitudes that vary depending on the observer's position. Given that energy in a wave is related to the amplitude squared, and when a wave is reflected with half the incident energy being absorbed, the reflected wave has an amplitude that is /\-1/2, or about 0.71 times the amplitude of the incident wave. At certain points, the incident and reflected waves will constructively interfere resulting in a maximum amplitude equal to the sum of the amplitudes of the two waves (E + 0.71E = 1.71E), and at other points, they will destructively interfere, resulting in a minimum amplitude equal to the difference of their amplitudes (E - 0.71E = 0.29E). Therefore, the ratio of the largest to smallest amplitude noted by observers is 1.71E/0.29E, which simplifies to approximately 5.9.

Answer 2

`(Ei + Er)/(Ei-Er)`

Step-by-step explanation:

Ratio of the largest amplitude to the smallest amplitude is called voltage standing wave ratio ( along a transmission line )

Determine the ratio of the largest such amplitude noted by an observer tot the smallest amplitude

Expression for VSWR

VSWR =
`\frac{1 + \sqrt{(Pr)/(Pi) } }{1 - \sqrt{(Pr)/(Pi) } }` where ; Pr = reflected power , Pi = Incident power

hence the ratio of the largest amplitude to the smallest amplitude

=
`(Emax)/(Emin)` =
`(Ei + Er)/(Ei-Er)` where ; Ei = amplitude of incident wave, Er = amplitude of reflected wave