Question

370_3_1

Problem 1: The first two problems in this assignment will help you to understand how a planar mirror resonator (Fabry-Perot resonator) works. Consider two monochromatic waves travelling in z-direction, one wave traveling forward , u,(3,t) = a cos(@t- k z), the other backward, u,(z,t), both having the same amplitude. Determine the total real-valued wave u(z,t)=u,(z,t)+uz(z,t) proceeding in two different ways: (i) using only the real-valued wave functions for the forward and backward travelling waves together with the well-known cosine addition formulas, and (ii) using the complex wave functions for the forward and backward travelling waves and deducing from its sum the real-valued total wave. Do you get the same answer in both cases?

Answer 1

The forward travelling wave function in z- directionis given as

u_{​​​​​​1}​​​​​(z, t)=a cos( $\omega$ ​​​t-kz) and for backward travelling, the wave function is

u_{​​​​​​2}​​​( z, t)= a cos( $\omega$ t+kz ). Adding using Cos addition formula we get the result explained in the solution below.

(ii) Using The complex valued forward travelling wave function, u_{​​​​​​1}​​​​​(z,t)= a e^i($\omega$^{t- kz)} and for backward travelling, u_{​​​​​​2}​​​​​(z, t)= a e^{i($\omega$t+kz),} we have calculated the real valued total wave function.

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