Week 7 Travelling Waves 2

Question 1

what is the KE for a string δx and mass ρδx oscillating in y direction

Answer 1

KE = 1/2 (ρδx)(dy/dt)^2

Question 2

what is the potential energy equation of the string

Answer 2

U = 1/2 (dy/dx)^2 δx

Question 3

what is the elemental total energy of a string

Answer 3

δE = KE + U
δE = 1/2 (ρδx)(dy/dt)^2  +  1/2 (dy/dx)^2 δx
δE = 1/2 μδx [ (dy/dt)^2 + v^2(dy/dx)^2

Question 4

how do you obtain the energy stored in a length of wire between x=a and x=b

Answer 4

integrate the elemental total energy wrt x from a to b

Question 5

define a boundary

Answer 5

point where the line density of the string changes

Question 6

what are the incident and reflected wave equations

Answer 6

yi = Icos(k1x - ωt)
yr = Rcos(k1x + ωt)

Question 7

what is the transmitted wave equation

Answer 7

yt = Tcos(k2x - ω’t)

Question 8

what is the refractive index equation

Answer 8

n = v1 / v2 = k2 / k1

k is the wavenumber

Question 9

what are the two boundary conditions

Answer 9

1) y is continuous across the boundary ω = ω’

2) dy/dx is continuous across the boundary

Question 10

what is the transmission coefficient equation

Answer 10

Tc = 2/1+n = 2k1/(k1+k2) = T/I

Question 11

what is the reflection coefficient equation

Answer 11

Rc = (1-n)/(1+n) = (k1-k2)/(k1+k2) = R/I

Question 12

what is the condition for the transmitted wave to always be in phase with the incident wave

Answer 12

0 < Tc < 2

Question 13

what are the two possible conditions for the reflected wave to only be in phase with the incident wave

Answer 13

-1 < Rc < 1

or Rc < 0 if k2 > k1

Question 14

what is the condition for Rc to be very small so that there is hardly any reflection

Answer 14

k1 ~ k2