Radiation Boundary Conditions

where the explicit form of (pa, ua, va, pa) may be found in Eq. (6.1). The functions X, f, and F are entirely arbitrary. It is observed that the total pressure fluctuation comes only from the acoustic component of the outgoing disturbances. Thus, the appropriate outflow boundary condition for p is the same as that of Eq. (6.2). On writing out Cartesian coordinates in two dimensions, it is

d p.9 p p

+ cos в — + sin в — + ^~ d x dy 2r

(6.4)

0,

where в is the angular coordinate of the boundary point.

By differentiating the expression for p in Eq. (6.3) with respect to t and x, the following equation is found:

dp dp dp dp„

— + U0—— = — + u0—. 91 0 dx 91 0 dx

(6.5)

But pa = pa/a2 = p/a0 andp is known from Eq. (6.4). By eliminating pa, the outflow boundary condition for p becomes

du 9 u 1 9 p 91 0 dx p0 dx

9 v 9 v 1 9 p

dt ^ U° 9x p0 9y ’

In summary, the outflow boundary conditions are as follows:

(6.11)

(6.12)

(6.13)