Question

Physics of waves

Problem 9 Consider a voltage wave of the form V Vocos (w t -k2) that is incident on a short circuit a located at coordinate 0. A) Find the form of the reflected wave, V B) Show that the total wave in the line (the sum of the incident and reflected waves) can be written in the standing wave form, V V V 2Vo sin (kz) sin (wt) C) Is there a voltage node or antinode at the short circuit. Justify your answer using the form in B).

Answer 1

The revlected wve will have the same amlitude and frequency but traveling in oppoiste direction and hecn can be written as

V^- = V₀ cos($\omega$t + kz +$\pi$) , the reflection will add an additional phase diferenc of 180 deg.

    = - V₀ cos($\omega$t + kz)

The resultant wave

V = V⁺ + V^-   = V₀ (cos($\omega$t - kz) - cos($\omega$t + kz) )

                     = V₀ ([ cos($\omega$t)cos(kz) + sin($\omega$t) sin(kz) ] - [ cos($\omega$t)cos(kz) - sin($\omega$t) sin(kz) ] )

                      = 2V₀ sin($\omega$t) sin(kz)

C) at the short circuit z=0, sin(kz) =0 and V=0 hence it will be node at the short circuit.