Transient acoustic wave propagation

The model consists of a column of fluid 1 m long with a constant cross-sectional area. The tube lies horizontally (along the x-axis), and the acoustic medium has a prescribed constant inward particle acceleration of $a_0=$ 1 m/s² at $x=$ 0 m. A nonreflective boundary is specified at $x=$ 1 m using nonreflective impedance conditions.

The acoustic material in the column is air with a bulk modulus $K_f=$ 1.424 × 10⁵ N/m² and a density ${\rho }_f=$ 1.21 kg/m³. The speed of sound is calculated as $c=\sqrt{\frac{K_f}{{\rho }_f}}$ = 343.05 m/s. The analytical result for the pressure is

The column is modeled using either 100 first-order or 50 second-order acoustic elements. For each acoustic element tested, the acceleration is specified in each of two ways:

1. There is no ASI element or tie constraint, and an inward volume acceleration is specified on degree of freedom 8 as a concentrated load (“afav” files).

2. In Abaqus/Standard an ASI element is placed at $x=$ 0 with its normal pointing into the fluid (this activates the displacement degree of freedom on the node at $x=$ 0), and in Abaqus/Explicit a structural element with a tie constraint is used to define the interaction between the fluid and structure. An acceleration is prescribed directly with a boundary condition (“afas” files). In these cases the first time interval in the analysis is performed using boundary impedances; the analysis continues in time using surface impedances

A transient dynamic analysis is performed for a period long enough to allow the wave to propagate past the nonreflective boundary.