A sonic black hole of a rectangular cross-section

In this study, a rectangular cross-section sonic black hole (SBH) is explored and a one-dimensional model equation for the acoustic field inside it is presented. The model equation's applicability is verified through a comparison with the corresponding two-dimensional Helmholtz equation. A general analytical solution, which takes into account the SBH's quadratically varying internal cross-section, is derived using linearly independent confluent Heun functions. To investigate the acoustic wave behavior in the SBH, the wave splitting method is utilized to obtain forward and backward acoustic pressure waves, and the reflection coefficient is calculated. To confirm the validity of the analytical solution, a comparison is made with numerical results obtained from the Riccati equation. Within the context of this study, a detailed discussion is provided regarding the applicability of the analytical solution.

Automated summary

This study explores the acoustic field inside a rectangular cross-section sonic black hole (SBH) and presents a one-dimensional model equation. The model equation's applicability is verified through comparison with the corresponding two-dimensional Helmholtz equation. A general analytical solution, taking into account the SBH's quadratically varying internal cross-section, is derived using linearly independent confluent Heun functions. The study investigates the acoustic wave behavior in the SBH using the wave splitting method and calculates the reflection coefficient. The validity of the analytical solution is confirmed through comparison with numerical results obtained from the Riccati equation.