A wave of amplitude A i and wave number k i is incident on the junction of two s

Question

A wave of amplitude Ai and wave number ki is incident on the junction of two strings. If the wave number in the second string is kt = 2ki , find the amplitudes of the transmitted and reflected waves. By what fraction is the speed of the wave reduced as it crosses the junction? What fraction of the incident power is transmitted? (Power is proportional to amplitude square but also depends on wavespeed, so it is easiest to compare the power reflected to that of the incident wave, and then find the transmitted power by requiring energy be conserved across the boundary.)

At = (2/3)Ai , Ar = (-1/3)Ai , vt/vi = 1/2, Pt/Pi = 8/9

Explanation / Answer

At = [kt/( kt+ki ) ]*Ai

use kt = 2*ki

= [ 2*ki/(2*ki+ki) ] * Ai

= [ 2*ki/3*ki ] *Ai

At = (2/3)*Ai

Ar = [(Ki -kt) /( kt+ki ) ]*Ai

use kr = 2*ki

= [(ki- 2*ki)/(2*ki+ki) ] * Ai

= [ -ki/3*ki ] *Ai

Ar = (-1/3)*Ai

Velocity = wavelenght*frequency = (2*pi/k)*f

frequency doesnt change.

So, Vt/Vi = [ (2*pi/kt)*f ]/[ (2*pi/ki)*f] = ki/kt = ki/2ki = 1/2

means speed reduces by fraction of 1/2

Power is proportional to (amplitude^2)*(frequency^2)*velocity

for reflected wave, frequency and velocity doesnt change.

So, Pr/Pi = (Ar/Ai)^2 = (1/3)^2 = 1/9

conserving energy at boundary, Pt + Pr = Pi i.e. Pt = Pi-Pr

So, Pt/Pi = 1- (Pr/Pi) = 1 - 1/9 = 8/9

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