Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

acoustic equations by Zienkiewicz and Newton, 6 and further ®nite element models
by Craggs. 7 A more comprehensive survey of the development of the method is
given by Astley. 8 Provided that the dielectric constant, ", and the permeability, ,
are constant, then Maxwell's equations for electromagnetics can be reduced to the
form
r 2 ÿ "
c 2
@ 2
ˆÿ4 2 @t "
and r 2 A ÿ "
c 2
@ 2 A J
ˆÿ4 2 @t c
…8:4†
where is the charge density, J is the current, and and A are scalar and vector
potentials, respectively. When and J are zero, which is a frequent case, and the
time dependence is harmonic, Eqs (8.4) reduce to the Helmholtz equations. More
details are given by Morse and Feshbach. 9
For surface waves on water when the wavelength, ˆ 2 =k, is small relative to the
depth, H, the velocities and the velocity potential vary vertically as cosh kz. 1;2;10;11 The
full equation can now be written as
r T …cc gr †‡ !2
g
ˆ 0 or
Waves in closed domains ± ®nite element models 243
@
@x i
@
ccg @xi ‡ !2
g
ˆ 0 …8:5†
where the group velocity, c g ˆ nc, n ˆ…1 ‡…2kH= sinh 2kH††=2 and the dispersion
relation
! 2 ˆ gk tanh kH …8:6†
links the angular frequency, !, and the water depth, H, to the wavenumber, k.
8.2 Waves in closed domains ± ®nite element models
We now consider a closed domain of any shape. For waves on water this could be
a closed basin, for acoustic or electromagnetic waves it could be a resonant cavity.
In the case of surface waves we consider a two-dimensional basin, with varying
depth. In plan it can be divided into two-dimensional elements, of any of the
types discussed in Volume 1. The wave elevation, , at any point … ; † within the
element, can be expressed in terms of nodal values, using the element shape function
N, thus
^ ˆ N~g …8:7†
Next Eq. (8.2) is weighted with the shape function, and integrated by parts in the
usual way, to give
…
@N
H
@xi @N
ÿ N
@xi T ! 2
g N
!
d ~g ˆ 0 …8:8†
The integral is taken over all the elements of the domain, and ~g represents all the
nodal values of .
The natural boundary condition which arises is @ =@n ˆ 0, where n is the normal to
the boundary, corresponding to zero ¯ow normal to the boundary. Physically this
corresponds to a vertical, perfectly re¯ecting wall. Equation (8.8) can be recast in