Gregor Tanner

Sound radiation from vibrating plates – a classical phase space approach

Neekar M Mohammed, Stephen C Creagh, Martin Richter and Gregor Tanner

Paraphrasing Marc Kac’s famous quote, I’d like to address the question ‘Can you hear the structure of a classical phase space?’ The background to this question is the following: vibrating structures give off sound - and controlling or even simulating the radiated sound is something of importance for many engineering applications. Both, calculating the vibrational waves on complex structures - think aeroplanes or cars - and obtaining the acoustic radiation are non-trivial tasks for large structures. In particular, solving the linear elasticity (wave) equations directly can lead to very large models scaling unfavourably with frequency. Especially at high frequencies - now think noise - it is in general much easier to compute the mean vibrational energy distribution using phase space calculations based on the underlying ray dynamics. As a result, one obtains a phase space density on the whole structure, however, without phase information. Such phase information is vital for sound radiation computations based on, for example, boundary integral equations such as the Rayleigh integral. Can we still obtain the sound being radiated from these structures? To some extend yes - by making use of the ‘hidden’ phase information retained in the ‘momentum coordinate’ which can be recovered by applying an inverse Wigner transformation. I will dwell on these ideas using some simple examples.