The following is a series of test cases provided with your FEWaves installation media which you should follow through and both understand how each model is created and how to interpret the results.
2D Axi-symmetric Piston (piston.geo)
The geometry for a water loaded baffled circular piston is shown in the figure above. Due to the piston symmetry, this can be modeled as an axi-symmetric problem with the axis of symmetry being the vertical axis. The piston has a radius a = 0.23873 meters. To represent the water loading, a circular edge of radius a+L surrounds the piston. L is the wavelength at 3000hz which equals 0.5 meters. This outer edge has absorbing boundary conditions applied: the edge along the r axis and the edge along the z axis are planes of symmetry in which the normal velocity equals zero and hence require no boundary condition. The condition that the normal velocity equals zero at a pressure surface is a natural finite element boundary condition. The mesh created fits the rule of at least 16 first order elements per wavelength at the highest frequency of interest. A normal velocity of 1.0 e-06 is applied at the surface of the piston as shown. For this test case, we will be comparing the mechanical impedance, defined as the force/velocity at the piston surface, and the pressure distribution along the axis of symmetry from the piston head to theoretical predictions. A second order finite element mesh refined near the piston head is used to improve the accuracy of the mechanical impedance calculation. The model will be run from 500 to 4000 hz, which represents a normalized 2ka of 1 to 8.
The distribution of FEWaves includes the files piston.geo, piston.pts, piston.edg, piston.fem, piston.pre, and fewaves.mat. These files must be loaded to run this example. The loaded mesh should look like:
Note the highly refined mesh near the piston. The file piston.ref is included to accomplish this refinement. Running the solution results in the following plots from the out1.plt file, which contains the real and imaginary mechanical impedance variation with frequency:
One could take these values, normalize the mechanical impedance magnitude to 
and normalize the abscissa to 2ka to get the following plot:
These plots show excellent agreement with theory (ref Acoustics by A.D. Pierce, 1989 ASA, page 224).
We will also compare the pressure distribution along the piston symmetry axis. First access the solution through the View Solution pulldown and load the piston.pot file. Now choose the solution at 4000 Hz. The pressure along the axis is given by (see Acoustics by A.D. Pierce, 1989, ASA, page 233) (exp(-jwt) dependency):
For the material values and model values as given, the pressure at z = 0.2 meters is 
. Examining the following pressure real and imaginary line plots
from (0,0,0) to (0,0.45,0) at 4000 Hz yields exact agreement with theory.
