ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 (2002) 213 – 224 Four-dimensional Treatment of Linear Acoustic Fields and Radiation Pressure D. Stanzial, D. Bonsi Laboratorio di Acustica Musicale e Architettonica, FSSG-CNR, Fondazione G. Cini, Isola di San Giorgio Mag- giore, I-30124 Venezia, Italy G. Schiffrer Dipartimento di Fisica, Universit` a di Ferrara, Via Paradiso, 12, I-44100, Ferrara, Italy Summary A general expression of acoustic radiation pressure is here derived on the basis of the linear theory of classi- cal fields. Following this theory, the acoustic energy, the sound intensity and the sound momentum density are introduced, together with the wave-momentum flux density tensor, as components of a acoustic energy-momentum tensor in a unified space-time approach, formally similar to the relativistic formulation of electromagnetism. The related conservation laws are then expressed by the condition of vanishing 4-divergence of this tensor, showing in particular that the so-called radiation pressure is nothing but a consequence of the mo- mentum conservation law for the acoustic field. As an application, the radiation pressure is computed explicitly in two cases: a plane wave reflected on a flat wall and the field in the interior of an open organ pipe. In the latter case, indirect measurements of the radiation pressure have been also performed by an intensimetric technique, allowing to determine the complex reflection amplitude at the pipe’s end. Finally, as an appendix of the paper, the angular momentum conservation and the analogy between the acoustic and electromagnetic radiation pressure are analyzed to some extent. PACS no. 43.25.Qp 1. Introduction Following our previous works, mainly concerned with the rigorous definition of time averaged energetic properties of general linear acoustic fields (e.g. [1], [2], [3], [4]), the first aim of this paper is to cast a link between the energy and momentum density concepts from the point of view of their conservation laws formulated in the so-called acous- tic space-time, to be defined in subsection 2.2. The accom- plishment of this task has allowed us to give a contribution to the study of a classical research subject: the acoustic radiation pressure, a physical quantity that in our view is simply a consequence of the momentum conservation law for acoustic fields. Therefore, this quantity plays a role in all fields of acoustics, including the audible linear domain, being complementary and quite similar to that played by intensity for energy conservation. The second aim of this paper is to express the radia- tion pressure as a time-dependent quantity in terms of the solution of the wave equation with appropriate boundary conditions, without relying necessarily on the two histor- ical definitions of radiation pressure, due to Rayleigh and to Langevin, which often predominate in the acoustical lit- erature [5]. Furthermore, our approach can be extended in Received 22 October 2001, accepted 19 July 2002. principle to nonlinear acoustics, since it is based on the general physical principles of field theory. The theoretical definition of acoustic radiation pressure given here follows the development of the analogous quan- tity for the electromagnetic field, since it is based on the wave-momentum flux density tensor as a part of the energy-momentum tensor. In the electromagnetic case, the momentum flux density tensor was introduced by James C. Maxwell in his famous Treatise on Electricity and Magnetism, published in 1873 [6], and was there sub- divided in two parts, treated separately: the electric part, called the electrostatic stress, and the magnetic one, or electrokinetic stress (see Art.s 105-111 and 639-646 of the Treatise). In this way, Maxwell obtained an expression of the force which arises in the field itself due to the pres- ence of electromagnetic waves, the so-called “radiation pressure” (Art.s 792-793). According to this conception, radiation pressure turns out to be a non-linear effect, but just in the sense that it is a second-order quantity derived from a linear wave equation. In the linear acoustics context this approach allowed us to obtain a time-dependent form of the same quantity, which has been named acoustic radiation pressure thanks to the electromagnetic analogy. One interesting point is that the present treatment is formally in agreement with the one given by Beissner [7], which follows rigorously from the fluid mechanical theory based on the momentum c S. Hirzel Verlag EAA 213