Question

(III) A right-moving harmonic wave with angular frequency wo and wavenumber k passes from a medium in which its speed is c wo/k into another medium in which its speed is doubled. I.e., d wo/k' 2 c. (a) Determine the relative fractions of the amplitude of the wave which are (i) transmitted into the new medium, and (ii) reflected back to the left. (b) Determine the relative intensities of the transmitted and reflected waves. [HINT: The (energy) intensity of a wave is proportional to its amplitude-squared.] Page 1 of 2

Answer 1

For a harmonic wave, the reflection and transmission co-efficient are siven as r = Ar/Ai = v_2 - v_1/v_2 + v_1 t = At/Ai = 2v_2/v_1 + v_2 A_r = amplitude of radiated wave A_t = Amplitude of transmitted wave A_i = Amplitude of incident wave v_1 = velocity of wave in 1st medium v_2 = velocity of wave in 2nd medium \r\n r = 2c - c/2c + c = c/3c = 1/3 reflection co-efficient t = 2 (2 (c)/2c + c = 4c/3c = 4/3 transmission co-efficient Intensity alpha Vertical-bar A Vertical-bar ^2 n (n = reflection index) Therefore R = Vertical-bar Ar Vertical-bar Ai Vertical-bar n^2_1/n_1 = Vertical-bar r^2 Vertical-bar = 1/g T = It/Ii = Vertical-bar A_t/A_i Vertical-bar ^2 n_2/n_1 \r\n = Vertical-bar A_t/A_i Vertical-bar ^2 times v_1/v_2 = (4)^2/g times 1/2 = 16/g times 1/2 = 8/g Therefore R + T = 1/g + 8/g = 1