Reflections Flashcards
Characteristic Impedance Z =
√Tρ
Characteristic impedance is an intrinsic property of
the stretched string and a measure of how hard it is to move the string up and down
Why is it hard to move the string up and down
You have to act against the transverse component of the tension
I.e. the driving force acts against the tension
The required driving force is proportional to
Tension x √ρ/T = √Tρ
If you join strings of non-matched impedance you will get
some power reflected back towards the source
Geometrical boundary conditions
The displacement at x=0 must be continuous
A+B = C
Dynamical boundary condition
Transverse force at x=0 must be continuous, otherwise there would be a non-zero net force acting on an infinitely small string element which would result in a non-physical infinite acceleration
if there is no boundary expect
no reflection
Z1 = Z2 => B/A = 0
if the boundary is fixed expect
inversion
Z2 = ∞ => B/A = -1
if the boundary is free expect
no inversion
Z2 = 0 => B/A = 1
Energy is proportional to the
square of the velocity
Reflected intensity coefficient
Z2/Z1 (C/A)^2
Transmitted intensity coefficient
(B/A)^2
reflection at boundaries for a massless boundary
∂yt/∂x | xo = ∂(yi+yr)/∂x | xo
reflection of a point mass
T ∂yt/dx | join - T ∂(yi+yr)/∂x | join = M ∂y^2t/∂t^2 | join
=> T ∂yt/dx | join - T ∂(yi+yr)/∂x | join = -Mω^2Ce^(iωt)
