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

• Published in: Journal of sound and vibration Vol. 512; p. 116377
• Main Authors: Basu, Sattik, Rani, Sarma L.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 10.11.2021; Elsevier Science Ltd
• Subjects: Acoustics; Angular velocity; Boundary conditions; Cross-sections; Density; Differential equations; Errors; Euler equations; Euler-Lagrange equation; Eulers equations; Helmholtz equation; Helmholtz equations; Linearly exact boundary condition; Pressure–density relation; Helmholtz equation; Euler equations; Pressure–density relation; Linearly exact boundary condition
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2021.116377

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 pˆ, 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, ρˆ=pˆ/c̄2, is strictly valid for a constant cross-section duct with uniform mean properties and uniform or zero mean flow (c̄ is the mean sound speed). We show that using the classical ρˆ–pˆ relation in deriving the Helmholtz equation may lead to significant errors in both the phase and amplitude of the acoustic field pˆ. 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 dpˆdx=f(pˆ,uˆ,ρˆ;ω) is developed, where uˆ is the velocity fluctuation field and ω is the angular frequency. The pˆ 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.