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Comments on “ On ‘ hearing the shape of drums ’ : An experimental study using vibrating smectic films ”
| Content Provider | Semantic Scholar |
|---|---|
| Author | Bassey, Antigha Okon |
| Copyright Year | 2000 |
| Abstract | – We point out that one of a pair of isospectral shapes subjected to a certain intentionally altered building scheme would be expected to lose isospectrality because of the mathematical form of a relevant spectral function expansion which involves angles. A higher “calibration” mode which has common exactly specified eigenfrequency for all three shapes is mentioned. A recent paper [1] by Even and Pieranski presented the results of an experimental investigation, using smectic films, of the isospectrality of pairs of shapes built from seven triangles. They also constructed an altered shape (A′ in ref. [1]) related to their shape A (90◦) but utilizing a different symmetry for one of the constituent isosceles rectangular triangles, and drew attention to the consequent loss of isospectrality. We should like to point out that their a contrario-type experiment might a priori be expected to yield a different frequency spectrum because of the nature of a mathematical asymptotic spectral function expansion involving a sum over the eigenvalues λ of the negative two-dimensional Laplacian (Helmholtz equation with Dirichlet boundary conditions) which for shapes such as those under discussion takes the form [2–4] ∑ exp[−λs] ∼ a/(4πs) − /[8 √ (πs)] + ∑ (π − θ)/(24πθ) + · · · (1) as s → 0, where a is the area of the domain, is the length of its perimeter, and the θ are the internal angles. Whilst the altered shape A′ in ref. [1] does have the same area and perimeter as shape A, the internal angles are different, such that the angle sum in the 3rd term of eq. (1) is different. Thus the set of eigenvalues {λ}, or eigenfrequencies-squared {f2}, could not be the same. Shape A′ was found experimentally in ref. [1] to have a different overall frequency spectrum and different mode patterns, in particular for the important 9th mode. For the isospectral pair of shapes A and B (90◦), this mode involved the fundamental patterns for the basic building triangles. For unit cell length, its eigenvalue can be shown mathematically (cf. refs. [5, 6]) to equal exactly 5π. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://research-repository.griffith.edu.au/bitstream/handle/10072/3202/epl_50_2_280.pdf;jsessionid=075105AE587F7902AAE8E5982642833A?sequence=1 |
| Alternate Webpage(s) | https://research-repository.griffith.edu.au/bitstream/handle/10072/3202/epl_50_2_280.pdf?isAllowed=y&sequence=1 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
