Generalized acoustic Helmholtz equation and its boundary conditions in a quasi 1-D duct with arbitrary mean properties and mean flow

We derive the generalized Helmholtz equation governing the acoustic pressure field in a quasi one-dimensional duct with axially varying cross-section and arbitrary (axially inhomogeneous) mean properties such as the velocity, temperature, density and pressure. To express the Helmholtz equation exclusively in terms of the fluctuating pressure field (p) over cap, we developed an expression relating density and pressure fluctuations, which is a differential equation for the generalized case of a duct with arbitrary mean properties. The classical algebraic expression for the density fluctuation field, (p) over cap = (p) over cap/(c) over bar (2), is strictly valid for a constant cross-section duct with uniform mean properties and uniform or zero mean flow ((c) over bar is the mean sound speed). We show that using the classical (p) over cap-(p) over cap relation in deriving the Helmholtz equation may lead to significant errors in both the phase and amplitude of the acoustic field (p) over cap. These errors arise because the Helmholtz equation thus obtained fails to account for the boundary condition on density fluctuations at the duct inlet. Furthermore, a linearly-exact derivative boundary condition to the Helmholtz equation of the form d (p) over cap /dx = f ((p) over cap, (u) over cap, (rho) over cap; omega) is developed, where (u) over cap is the velocity fluctuation field and.. is the angular frequency. The (p) over cap field obtained by solving the generalized Helmholtz equation in conjunction with the derivative boundary condition shows excellent agreement with that obtained through the solution of the linearized Euler equations.

Automated summary

The study focuses on the generalized Helmholtz equation for the acoustic pressure field in a quasi one-dimensional duct, highlighting potential errors in using classical expressions and introducing a new boundary condition to address density fluctuations at the duct inlet. Results show that the new approach yields excellent agreement with solutions obtained using traditional methods.