Recall that the evaluation of the pressure requires the normalized modes. The normalization constant is
The integral term can be evaluated by the trapezoidal rule. That is,
where
In problems with density discontinuities, the trapezoidal rule is
applied separately within each smooth region. Finally, the
derivatives 
can be either evaluated analytically
or by a simple centered-difference formula depending on their
complexity. In order to accommodate a wide variety of boundary conditions
without over-complicating the code,
KRAKEN
uses the difference approach.