Modes Flashcards
in the general case our nth harmonic is
yₙ (x,t) = sin {nπx/L}(Aₙcos{wₙt}+Bₙsin{wₙt})
Total energy of a mode is
Etotal = mₛwₙ^2/4 *(Aₙ^2+Bₙ^2)
Period Equation
kL/2 tan{k L/2} = ρL/M
wₙ =
nct/L
How to find An and Bn
evaluate the general case at t = 0
rewrite the general case
equate to velocity
then evaluate at t = 0 and multiply by a factor of sin(mπx/L)
integrate between limits - evaluating LHS then RHS
(where side with m and n is evaluated for m = n and m≠n )
put new integrals back together, simply and solve.
Fractional energy of a mode can be defined as
En/Etotal * 100 = %
Ekin =
ρ/2 L ∫ 0 (∂y/∂t)^2 dx
Epot =
T/2 L ∫ 0 (∂y/∂t)^2 dx
Loading the string - analysis
T ( ∂y2/∂x | join - ∂y1/∂x | join ) = M ∂^2y1/∂t^2 | join
1D longitudinal waves in bar
Newton’s Law
AY [ ε(x+ 𝛿x/2) - ε(x-𝛿x/2)] = [A𝛿x]ρ ∂^2Ψ/∂t^2
ε(x,t) = ∂Ψ(x,t)/ ∂x
Aρ ∂^2Ψ/∂t^2 = YA ∂ε/∂x = YA ∂/dx ∂Ψ/∂x = YA ∂^2Ψ/∂x^2
longitudinal waves in gases c =
√γP/ρo
Telegraph equation
∂^2y/∂x^2 = 1/c^2 [∂^2y/∂t^2 + Γ∂y/∂t + qy]
PV =
ρ =
nRT = M/mo RT
M/V
show that frequencies
= √Y/ρ (2n-1)/4l
use wave equation of form
∂^2Ψ/∂t^2 = 1/c^2 ∂^2Ψ/∂x^2
= Y/ρ ∂^2Ψ/∂x^2
evaluate at x = 0
balance forces
T = YA ∂Ψ/∂x
wn = ck = √Y/ρ 2n-1/2l π
T =
YA ∂Ψ/∂x
