Reducing aircraft noise is one of the greatest challenges facing the aviation industry. A promising trend in aircraft design is to take inspiration from owls: amongst other features, it is believed that the porous trailing edge of an owl's wing is largely responsible for their almost-silent flight. Unfortunately, although porosity may reduce wing noise, it may also impair the corresponding aerodynamic performance. Therefore, aircraft designers have the difficult task of balancing the deleterious aerodynamic effects of porosity with its favourable aeroacoustic advantages. We present several perspectives on this problem, with an emphasis on "cascade" geometries that may be found within turbomachinery. To ensure a holistic understanding, we consider both aerodynamic and aeroacoustic studies. Our analysis is primarily theoretical: we solve the relevant elliptic partial differential equations exactly by leveraging recent advances in complex function theory. The ensuing solutions show that even small amounts of porosity have a significant effect on the mathematical structure of the problem, which leads to an improved understanding of the underlying physical mechanisms. In addition to offering physical insight, the solutions we present are analytic and rapid to compute, and could therefore be implemented in aircraft design tools for noise-reduction technologies.