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

8
Waves
Peter Bettess
8.1 Introduction and equations
The main developments in this chapter relate to linearized surface waves in water, but
acoustic and electromagnetic waves will also be mentioned. We start from the wave
equation, Eq. (7.23), which was developed from the equations of momentum balance
and mass conservation in shallow water. The wave elevation, , is small in comparison
with the water depth, H. If the problem is periodic, we can write the wave elevation, ,
quite generally as
…x; y; t† ˆ …x; y† exp…i!t† …8:1†
where ! is the angular frequency and may be complex. Equation (7.23) now
becomes
r T … Hr †‡ !2
g
or, for constant depth, H,
ˆ 0 or
@
@x i
@ 2
H @
@x i
‡ !2
g
ˆ 0 …8:2†
r 2 ‡ k 2 ˆ 0 or
‡ k
@xi@xi 2 ˆ 0 …8:3†
p
where the wavenumber k ˆ != gH.
The wave speed is c ˆ !=k. Equation (8.3) is
the Helmholtz equation (which was also derived in Chapter 7, in a slightly di€erent
form, as Eq. (7.23)) which models very many wave problems. This is only one form
of the equation of surface waves, for which there is a very extensive literature. 1ÿ4
From now on all problems will be taken to be periodic, and the overbar on will
be dropped. The Helmholtz equation (8.3) also describes periodic acoustic waves.
The wavenumber k is now given by !=c, where as in surface p waves ! is the angular
frequency and c is the wave speed. This is given by c ˆ K= , where is the density
of the ¯uid and K is the bulk modulus. Boundary conditions need to be applied to
deal with radiation and absorption of acoustic waves. The ®rst application of ®nite
elements to acoustics was by Gladwell. 5 This was followed in 1969 by the solution of
Professor, Department of Civil Engineering, University of Durham, UK.