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,
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.