6  Concluding Discussion

The results in the previous section demonstrate that the method described in this paper is able to model the effect of acoustic loading on the resonant frequencies of a structure. The mode shapes of the acoustically-loaded structure can also be recovered from the eigenvectors, each mode shape being directly expressed as a weighted sum of the in vacuo mode shapes. However, in the tests given in the previous section the mode shapes were generally found to be only a slight perturbation of the in vacuo mode shapes.
In air, the real part of the resonant wavenumbers is generally only slightly less than its in vacuo frequency whereas, in water, the effect is to significantly reduce the resonant frequency. The results also demonstrate, in all cases, that as the thickness increases the damping is reduced.
In the tests, we see that the imaginary part for the second mode is much less than that of the first mode. It is also noticeable that the imaginary part of the second mode converges to a small number as the discretisation is improved. This can also be interpreted physically in that the first mode requires a change of volume in the structure whereas the second mode does not. Hence it would be expected that damping would be much greater on the first mode.
Further work is required for an efficient choice of subrange [kA, kB]. The degree of the interpolant (25) can also be increased (see [9]) though the larger matrices that would arise in the ensuing eigenvalue problem is generally likely to make this unattractive. Comparison of exact results or measured results from a physical experiment with results from the numerical method would also be useful.
In many situations integral equation methods are preferred to finite element methods for the modelling of acoustic fields. This is particularly true in exterior problems such as the test problems in the previous section. The method in this paper is thus a useful development in the vibratory analysis of acoustic-structure problems.


Acknowledgement. This work was funded by a SERC/Admiralty Research Establishment Grant GR/GO1416.