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According to the classical Helmholtz picture, an organ pipe while generating its eigentone has two anti-nodes at the two open ends of a cylinder, the anti-nodes being taken as boundary condition for the corresponding sound. Since 1860 it is also known that according to the classical picture the pipe actually sounds lower, which is to say that the pipe so-to-speak sounds longer than it is, a long-standing enigma. As for the pipe's end, we have resolved this acoustic enigma by detailing the physics of the airflow at the pipe's open end and showing that the boundary condition is actually the pipe's acoustically resonating vortical sphere (PARVS). The PARVS geometry entails a sound-radiating hemisphere based on the pipe's open end and enclosing a vortex ring. In this way we obtain not only a physical explanation of sound radiation from the organ-pipe's open end, in particular, of its puzzling dependence upon the pipe's radius, but also an appreciation of it as realization of the sound of the flute, mankind's oldest musical instrument., Comment: 11 pages, 5 figures

Geometric perturbation theory is universal. A typical example is provided by the 3D wave equation, widely used in acoustics. We face vibrating eardrums as a binaural auditory input stemming from an external sound source. In the setup of internally coupled ears (ICE), which are present in more than half of the land-living vertebrates, the two tympana are coupled by an internal air-filled cavity, whose geometry determines the acoustic properties of the ICE system. The eardrums themselves are described by a 2-dimensional, damped, wave equation and are part of the spatial boundary conditions of the three-dimensional Laplacian belonging to the wave equation in the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic boundary-condition dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics the deviations from equilibrium are extremely small (nm). Perturbation theory is therefore natural and shown to be appropriate. In doing so, we use a time-dependent perturbation theory \`a la Dirac in the context of Duhamel's principle. The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order. Furthermore, both the initial and the quasi-stationary asymptotic state are derived and analyzed. Finally, we set the general stage for geometric perturbation theory where (d-1)-dimensional manifolds as subsets of the boundary of a d-dimensional domain are driven by their own dynamics with the domain pressure $p$ and an external source term as input, at the same time constituting time-dependent boundary conditions for $p$.

A systematic study of small, time-dependent, perturbations to geometric wave-equation domains is hardly existent. Acoustic enclosures are typical examples featuring locally reacting surfaces that respond to a pressure gradient or a pressure difference, alter the enclosure's volume and, hence, the boundary conditions, and do so locally through their vibrations. Accordingly, the Laplace-Beltrami operator in the acoustic wave equation lives in a temporally varying domain depending on the displacement of the locally reacting surface from equilibrium. The resulting partial differential equations feature nonlinearities and are coupled though the time-dependent boundary conditions. The solution to the afore-mentioned problem, as presented here, integrates techniques from differential geometry, functional analysis, and physics. The appropriate space is shown to be a (perturbation) fiber bundle. In the context of a systematic perturbation theory, the solution to the dynamical problem is obtained from a combination of semigroup techniques for operator evolution equations and metric perturbation theory as used in AdS/CFT. Duhamel's principle then yields a time-dependent perturbation theory, called geometric perturbation theory. It is analogous to, though different from, both Dirac's time-dependent perturbation theory and the Magnus expansion. Specifically, the formalism demonstrates that the stationary-domain approximation for vibrational acoustics only introduces a small error. Analytic simplifications methods are derived in the framework of the piston approximation. Globally reacting surfaces (so-called pistons) replace the formerly locally reacting surfaces and reduce the number of independent variables in the underlying partial differential equations. In this way, a straightforwardly applicable formalism is derived for scalar wave equations on time-varying domains., Comment: 121 pages and 16 figures