Mode Coupling
When do modes couple?
- frequencies are close together
- radial and geographical shapes are similar
Equation of Motion of Isolated Pendulum
E.o.M:
θ
Lcosθ
L
For small displacement:
Solution:
m
Displacement:
Velocity:
Kinetic Energy:
Potential Energy:
Assume solution:
Two Coupled Pendula
Kinetic Energy:
Potential Energy:
Motion is linear combination
of two solutions:
2x2 eigenvalue problem:
2 solutions:
Now Include Attenuation for One Pendulum
Define Q based on energy loss:
This makes frequencies complex:
instead of:
now have:
And for coupled pendula:
Complex frequencies for coupled pendula:
Asymptotic minimum of ∆Q:
Lagrangian Form (rotation for isolated multiplet)
potential
Coriolis coupling
kinetic
isolated mode: s is linear combination of multiplets -> general eigenvalue problem
coupled modes: s is linear combination of a singlets -> quadratic eigenvalue problem
when modes span small frequency range:
to get standard linear form:
For Complete Solution
rewrite
using
to get standard linear form:
this is twice as big!
Spheroidal-Toroidal Mode
Pairs coupled through rotation
and ellipticity
Strong coupling exists for
low-l mode pairs when
|l-l’|=1
Complete Splitting Matrix showing
the two self-coupling
blocks for 0S4 and 0T3
as well as the cross-coupling
blocks, for rotation and ellipticity
(strong coupling through rotation).
Both, rotation and ellipticity
contribute to the imaginary
part only.
Coupling of Radial Modes
It turns out that the coupling of
radial modes to l=2 modes is
strongest
(coupling through ellipticity and
structure)
Coupling of Radial Modes
with l=2 Modes
Predictions for Individual Spectra without and with Coupling
Frequencies repell each other. Also,the amplitude of the radial
mode is often diminished.
Predictions for Receiver Strips for Two Earthquakes
Rotation, ellipticity and
structure change the
receiver strips somewhat.
Coupling changes the
receiver strips a lot.
frequency
What Happens to the Frequencies?
The radial mode and the
l=0 line of 27S2 repell each
other. A certain structure
can cause a W-shape in
the frequencies of the l=2
mode, a feature that’s
often observed.
m