Dynamic Analysis - The Boundary Element Method with Programming - page 381

C0= CMPLX (0,0.25)
UW= C0* Hankel0 (k*r)
CASE (3) ! Three-dimensional solution
C0= CMPLX (0,k*r)
UW= 1/(4.0*k*r*Pi)*EXP(C0)
CASE DEFAULT
UW=0.0
WRITE (11,*)'Cdim not equal 2 or 3 in Function UW(...)'
END SELECT
END FUNCTION UW

14.3 SCALAR WAVE EQUATION, TIME DOMAIN

For the case where u=u ( Q,t ) is not harmonic but transient, the scalar wave equation is
given by

1

' u u F 0

(14.17)

2

c

where overdots mean differentiation with respect to time t , c is the wave velocity and F
is a specified body source. The assumption is of an isotropic and homogeneous medium.
For a well posed problem we must have initial and boundary conditions. The initial
conditions at time 0 are specified as

uQ ( , 0) u Q ( ) ; ( uQ , 0) v Q ( )

(14.18)

0 0

The condition

uQ vQ () () 0 is termed initial rest or quiescent past ,

0 0

14.3.1 Fundamental solutions

A fundamental solution of the differential equation can be found by assuming an
impulsive point source at P applied at time t W

, in an infinite domain. Therefore we

seek the solution of

1

(14.19)

' UUPQt

c

G ( ) G W ( ) 0

2

where a Dirac Delta function has been introduced for the time and space. The Dirac
Delta function for the space has been discussed previously; the one for the time is
defined as

GW ( t ) 0 when t z W

f

(14.20)

³

GWW ( t ) d 1

f