Page 24 - Fall 2006

quency wave! I will show why this is so later, but first we will study the simpler structural wave types.
The simplest structural waves are those that deform an infinite material longitudinally and transversely. Longitudinal waves, sometimes called compressional waves, expand and contract structures in the same way acoustic waves deform fluids. The wave equation and sound speed for a longitudinal wave traveling in the x direction are
, and (1a) , (1b)
where w is the deformation (also in the x direction), B is the elastic bulk modulus and ρ is the mass density.
The bulk modulus relates the amount of volumetric con- traction (per unit volume) to an applied pressure:
. (2)
Low volumetric changes mean stiffer structures, and faster compressional waves.
So far, we have considered structural waves only in media large in all dimensions with respect to vibrational wavelengths. For audible frequencies, and for most practical structures, one or two geometric dimensions are small with respect to a wavelength. As a longitudinal wave expands or contracts a beam or plate in its direction of propagation, the walls of the structure contract and expand transversely due to the Poisson effect, as shown in Fig. 1. The Poisson’s ratio, which relates in and out of plane strain deformations accord- ing to:
(3)
determines the amount of the off-axis deformation, which
for incompressible materials like rubber approaches the
amount of the on-axis deformation (a Poisson’s ratio of
6
like beams and plates, since the free surfaces of the structur- al material are exposed to air or fluid. Since the stiffness of most fluids that might surround a beam or plate is smaller than that of the structural material, the free surfaces of the
structure act essentially as stress relievers, slowing down the compressional waves. The sound speeds of longitudinal waves in beams and plates are
, and (4a)
, (4b)
where cl is defined not by the Bulk Modulus, but by the Young’s Modulus E, which is related to the volumetric Bulk Modulus according to:
. (5)
For a typical Poisson’s ratio of 0.3, longitudinal wave speeds in plates and beams are 90% and 86% of those in infinite structural media, respectively.
As I mentioned earlier, the key difference between acoustic waves in structural materials and fluid media is a structure’s ability to resist shear deformation. This shear stiff- ness allows pure shear waves to propagate through a struc- ture, with the structure deforming in its transverse direction as the wave propagates in the axial direction (see Fig. 2 below). Shear wave behavior is governed by the same wave equation as longitudinal waves, and acoustic waves in fluid media:
. (6) However, shear waves, which travel at the speed
, (7)
are slower than longitudinal waves, since a structure’s shear modulus is smaller than its Bulk and Young’s Moduli. The shear modulus G is related to E and Poisson’s ratio according to:
. (8)
Bending waves in beams and plates
Most sound radiated by vibrating structures is caused by bending, or flexural waves traveling through beams, plates,
0.5).
Longitudinal waves are therefore slower in structures
Fig. 1. A longitudinal wave passing through a plate or beam (amplitudes highly exaggerated). As the material expands or contracts along the axis of the plate or beam, the Poisson effect contracts and expands the material in the transverse direc- tions.
22 Acoustics Today, October 2006
Fig. 2. A shear wave propagating through a plate or beam (amplitudes highly exag- gerated). The wave propagates along the plate or beam axis, while deforming the structure transversely.