Energy Flux of Elastic Waves – Bill Menke – Oct 22, 2015
Fundamental relations:
Equation of Motion:
Energy density is sum of strain energy and kinetic energy:
Energy flux (rate of energy transport per unit area) in direction (Synge, 1956-1957):
Conservation of energy
Proof that
and
obey conservation of energy (for constant
Apply chain rule:
Insert equation of motion:
Apply symmetry of
and rename summation indices:
):
For a real displacement, the positive and negative frequency components are complex conjugate
pairs:
Here the overbar indicates complex conjugation and
of , respectively. The corresponding real stress is:
and
are the real and imaginary parts
Note that these quantities satisfy the stress-strain relation
motion
. Inserting into the flux equation yields:
and the equation of
We time-average the trigonometric functions
and
, where
signified the average over one cycle. The time averaged flux
is then:
=
In the isotropic case,
For
and
, so the average flux is:
and
(for all ):
Now suppose:
The vertically integrated horizontal flux
can be calculated as:
Reference
Synge, J.L., Flux of energy for elastic waves in anisotropic media, Proceedings of the Royal Irish
Academy: Section A, Mathematical and Physcial Sciences, Volume 58, 1956-1957, 12-21.