The Helmholtz medium is air at 20 celcius and 1 atmosphere so that the speed of sound c is 344m/s. The program HMBEM2_T contains three test problems. In the first test the mode(s) of the pure Neumann problem is sought in the frequency range [40Hz,60Hz] with NK=3 (quadratic interpolation) whereas in the second test the same modes are sought in the range [40Hz,50Hz] with NK=4 (cubic interpolation). The mode exists in both of these frequency ranges and satisfies both the Helmholtz equation and the homogeneous boundary condition is of the form f(p) = sin(10pp1) sin(10pp2) with the corresponding resonant wavenumber of 10 Ö2 p which equals 44.4288 (Hz) to four decimal places. In the first test the resonant frequency is approximated by 44.2092 whereas in the second test problem the resonant frequency is approximated by 44.4804. In general the shorter the wavenumber range the more accurate the results are (up to a point) and this is confirmed by these examples.
In the third test problem the square is assigned the Dirichlet condition on two adjacent sides and a Neumann condition on the other two adjacent sides. Under these conditions the first mode is of the form f*(p) = sin(5pp1) sin(5pp2) with corresponding resonant wavenumber 5 Ö2 p which equals 22.2144 to four decimal places. In the test the solution is sought in the wavenumber range [20Hz,30Hz] and the approximation to the wavenumber is 22.2316. The exact and computed values of the mode shape at five points in the interior are listed in Table 6.A. Note that the modal analysis programs assign the value f = 1 to the maximum value of f at the selected interior points hence in this case the exact solution is f*(p) = sin(5pp1) sin(5pp2)/0.83553.
|Table 6.A: Results from HMBEM2_T
||point || exact solution || numerical solution
||(0.025,0.025) || 0.1716 || 0.1729 + i0.0028
||(0.025,0.075) || 0.4142 || 0.4138 + i0.0017
||(0.075,0.025) || 0.4142 || 0.4138 + i0.0017
||(0.075,0.075) || 1.0000 || 1.0000 + i0.0000
||(0.05,0.05) || 0.5858 || 0.5862 + i0.0018 |
The program HMBEM3_T contains two test problems. In the first test the mode(s) of the pure Dirichlet problem is sought in the wavenumber range [3,4] with NK=4. The approximation to the characteristic wavenumber p is 3.5567. This is equivalent to 194.7 Hz in the Helmholtz medium of air.
In the second test the Neumann mode is sought in the range [2,3]. The program returns three results 2.4068, 2.4071 and 2.3163. However, these are all approximations to the same characteristic wavenumber of 2.0816. The approximation occurs three times because this is a repeated eigenvalue.
The results show that the approximate wavenumber is significantly greater than the exact wavenumber in both cases. This can be explained simply by observing that the effect of the boundary approximation is to significantly lessen the size of the sphere and the larger eigenvalues reflect this.
In the first test problem the Dirichlet modes are sought in the range [3,5] with NK=2 (linear interpolation). The approximations 3.138 and 4.512 are found to the exact solutions of 3.142 and 4.493. In the Helmholtz medium of water the approximate eigenfrequencies are 761Hz and 1094Hz.
In the second test problem the Neumann modes are sought in the range [2,4] with NK=4 (cubic interpolation). The approximations 2.089 and 3.360 are found to the exact solutions of 2.082 and 3.342.