Chapter Fourteen
Vibroacoustics (membranes, plates, and shells)
Abstract
After a brief introduction, the wave equation for a membrane in rectangular coordinates is derived. Next comes the solutions in rectangular and polar coordinates together with the Green's function. The membrane and free space wave equations are solved simultaneously to model a circular electrostatic loudspeaker radiating into free space. The solution of the plate wave equation in polar coordinates, together with the Green's function, follows and various edge boundary conditions are applied to a circular plate. In each case the eigenvalues are calculated and the eigenfunctions plotted. The shallow spherical shell is treated in a similar way to the plate initially but the boundary conditions are representative of an actual dome-shaped loudspeaker diaphragm with a compliant edge suspension and coil mass. The shell and free space wave equations are solved simultaneously to model sound radiation from a loudspeaker in an infinite baffle. The effect of varying the dome height is explored. The chapter concludes with a model of a novel induction loudspeaker in an infinite baffle. This exploits the phenomenon whereby a circular free plate excited by a point force at its center radiates has a flat on-axis response and radiates constant power at all frequencies.
Keywords
Bessel function; Eigenfunction; Eigenfrequency; Eigenvalue; Free space; Green's function; Homogeneous wave equation; Membrane wave equation; Plate wave equation; Shell wave equation; Electrostatic loudspeaker; Dome loudspeaker; Induction loudspeaker; Fluid-structure coupling
Part XXXVIII: Membranes
14.1. Introduction to vibroacoustics
Apart from a brief overview of loudspeaker diaphragm behavior in Section 6.14, until now we have treated all sound sources as if they were either perfectly rigid or perfectly flexible (resilient). Real diaphragms, of course, fall somewhere between these two extremes in their behavior. Until now, we have concentrated on airborne waves (or sound fields) radiating from sound sources into free space or confined within cavities or ducts. Vibroacoustics is the subject that deals with waves within structures such as membranes, plates, and shells. We shall begin with membranes, which are plates so thin that they can be treated as if they have zero-bending stiffness. Membranes are used as diaphragms in high-end electrostatic or planar-magnetic loudspeakers. However, unlike a resilient disk, which we treated in Section 13.8 and 13.9, a membrane is clamped at its perimeter and can therefore support standing waves. Plates are slightly more complicated because they have finite bending stiffness. We will examine how these can be used as loudspeaker diaphragms in Section 14.19, but for now we will simply calculate their modes with various boundary conditions.
The third kind of structure we shall consider is the spherical shell which is the most common kind of diaphragm and is found in devices ranging from microspeakers to hi-fi dome tweeters. These are more rigid than plates for the same thickness and therefore resonate at higher frequencies. We will examine the behavior of membranes, plates, and shells as loudspeaker diaphragms fully coupled into their surrounding air loads. Hence, we will introduce the concept of fluid–structure interaction.
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