## 1. Free Vibrations of Volume of Air

Free vibration modes and eigenfrequencies of an unconstrained (unbounded) volume of air are calculated, demonstrating the acoustic elements solving the Helmholtz equation. The geometry consists of a cube 0.5m by 0.5m by 0.1m (see figure below). To resolve frequencies of about 1000[Hz], one needs about 10 elements over a wavelength, which is calculated with l = c/f. With the speed of sound c=340 m/s, and f=1000Hz, one obtains l=0.34 m, a tenth of which gives l=0.034 m. The cube is meshed with 20 by 20 by 5 HE8 acoustic elements:

To perform the vibration analysis, B2000++ is launched with a case defining the free vibration problem parameters:

cases
case 1
analysis free_vibration
nmodes 20
shift 100.0
end
end

Note that an arbitrary positive shift can be performed, requesting 20 eigenvalues around the shift of 100.0 [Hz]. The utility script print.py prints the frequencies (see Makefile):

Free vibration analysis, case 1 of cycle 0
Title: 'Free vibration with shift 100 Hz'
Number of computed modes: 20

Mode     Eigenvalue  Frequency        Omega
1   -4.267e-08
2    4.573e+06        340.3         2138
3    4.573e+06        340.3         2138
4    9.146e+06        481.3         3024
5    1.841e+07        682.8         4290
6    1.841e+07        682.8         4290
7    2.298e+07        762.9         4794
8    2.298e+07        762.9         4794
9    3.681e+07        965.6         6067
10    4.184e+07         1029         6468
11    4.184e+07         1029         6468
12    4.641e+07         1084         6813
13    4.641e+07         1084         6813
14    6.024e+07         1235         7762
15    6.024e+07         1235         7762
16    7.545e+07         1382         8686
17    7.545e+07         1382         8686
18    8.002e+07         1424         8946
19    8.002e+07         1424         8946 

To visualize the free vibration modes, make use of the Modes tool of baspl++ by selecting, from the Renderer part, Create Modes Tool...:

Note that, when displaying scalar modes, the scale factor has to be set to 0.0! Some of the modes (mode 1, 4, and 20) is displayed in the figures below, obtained with the modes tool.