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So-called Rayleigh–Bloch waves can release enormous amount of energy that can damage technical systems under certain circumstances. They only exist below a precisely defined cut-off frequency; above this they disappear abruptly. Strangely enough, however, there are isolated high frequencies at which they can also be detected. Mathematicians from the Universities of Augsburg and Adelaide have recently proposed an explanation for this puzzling phenomenon. Together with researchers from the University of Exeter, they have now been able to prove experimentally that their theory is indeed correct. The study has just been published in the journal Nature communications physics.
Prof Dr Malte Peter with the grating that the researchers used to test their predictions. The small black loudspeaker that generated the sound waves can be seen on the right-hand side of the grating's frame, while the microphone stage that travelled along the grating during the experiment to measure the propagation of the waves can be seen on the far right. © Gregory Chaplain/Universität Exeter
Suppose you had a gigantic barbecue grill that could easily accommodate several hundreds of sausages. Then, you could not only use it to invite your children's entire school to a barbecue. The numerous stainless steel struts aligned parallel to each other are also ideal for generating Rayleigh–Bloch waves.
These are vibrations that propagate from gap to gap between the struts and do not lose any energy in the process. Above or below the grating, on the other hand, they quickly decay. ‘In principle, it doesn't matter whether they are sound, light or water waves,’ explains Prof Dr Malte Peter from the Institute of Mathematics at the University of Augsburg. ‘In order to develop, however, they always require a grid consisting of as many regularly recurring struts as possible, along which they can propagate.’
It all depends on the frequency
Rayleigh–Bloch waves are feared, among other things, because they can release very large amounts of energy. If, for example, pylons are anchored to the sea floor at regular intervals, there is a theoretical risk that they will be destroyed by these waves. However, some aspects of their formation are still poorly understood. ‘For example, we know that only vibrations with a low frequency can generate Rayleigh–Bloch waves,’ says Peter. ‘Above a certain cut-off frequency, they disappear abruptly. Interestingly, however, there are isolated high frequencies at which they can be detected again. For a long time, we didn't know what the connection was and where they disappear to at the frequencies in between.’
Together with his colleague Prof Dr Luke Bennetts from the University of Adelaide, the Augsburg scientist set about cracking this puzzle a few years ago. At that time, the two succeeded in mathematically characterising the Rayleigh–Bloch waves above the cut-off. According to this, they transform into a kind of phantom above the cut-off frequency. In mathematical terms, the distance between two wave crests can then suddenly only be calculated using imaginary numbers (imaginary numbers are those that - when multiplied by themselves - result in a negative value; normally, square numbers are always positive).
However, there are frequencies above the cut-off frequency at which this imaginary component becomes very small. And it is precisely in these cases that the Rayleigh–Bloch waves can suddenly be detected again. ‘Until now, however, we didn't know whether this phenomenon only appears in our formulas or is also significant in reality,’ explains Peter. This has now changed with the current study: In it, researchers from the University of Exeter prove that Bennett and Peter's calculations appear to be correct. They used a kind of giant barbecue grill with a small loudspeaker attached to a side strut. This produced sounds at different frequencies. The scientists then used a microphone to analyse how the sound waves propagated along the grating.
Theoretical prediction confirmed experimentally
The characteristic Rayleigh–Bloch behaviour was observed below a certain pitch: Directly around the struts, the volume was very high, whereas above and below the grating plane it decreased abruptly. As soon as the cut-off frequency was exceeded, the Rayleigh–Bloch waves disappeared. However, they were suddenly detectable again precisely in the frequency ranges predicted by Bennett and Peter's formulae. However, they no longer ‘nestled’ so closely to the struts, but also spread out somewhat vertically to the grating plane. ‘We interpret this as the influence of their imaginary part, which is very small at these frequencies, but still present,’ says Peter.
The results improve the understanding of these exotic waves and therefore also allow a better assessment of the conditions under which they can potentially become dangerous. However, they also make it possible to design antennas with which the Rayleigh–Bloch waves can be optimally transmitted. It may be possible in future to use the waves for communication purposes, for example, to transmit signals with low loss in a similar way to a fibre optic cable.
Professor Dr. Malte A. Peter
Applied Analysis
Tel:: +49 821 598 -5473
Mail: malte.peter@math.uni-augsburg.de
Chaplain, G.J., Hawkins, S.C., Peter, M.A. et al., Acoustic lattice resonances and generalised Rayleigh-Bloch waves. Commun Phys 8, 37 (2025): https://doi.org/10.1038/s42005-025-01950-4
Prof Dr Malte Peter with the grating that the researchers used to test their predictions.
Gregory Chaplain
University of Exeter
Schematic of the experiment. The black loudspeaker on the right of the frame generates sound waves w ...
Nature communications physics
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