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INTRODUCTION TO ACOUSTICS 323

between these two points after a force has been applied may be written as

(x,f) = 2S ij (x, t)dx i dx j (10.3)

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where S ij(x, t) is the second-order strain tensor defined by
± + 3u i + 9j±^-\ (]04)
Cy dXi dX{ dXj J
with the subscripts of i, j, and k being x, y, or z.
For rigid materials, the deformation gradient expressed in Equation (10.4) must be
kept small to avoid permanent damage to the structure; hence, the last term in the above
expression is assumed to be negligible, and so the expression for the strain-displacement
tensor is rewritten as

du t(x,t), , j ( x , t ) - ]
,
t
Sij(x, 0 = - — - -- h -4 - (10.5)
4 -
2 dxt J
10.4.2 Stress

When a body vibrates acoustically, elastic restoring forces, or stresses, develop between
neighbouring particles. For a body that is freely vibrating, these forces are the only ones
present. However, if the vibration is caused by the influence of external forces, two types
of excitation forces (body and surface forces) must be considered. Body forces affect the
particles in the interior of the body directly, whereas surface forces are applied to material
boundaries to generate acoustic vibration. In the latter case, the applied excitation does not
directly influence the particles within the body but it is rather transmitted to them through
elastic restoring forces, or stresses, acting between neighbouring particles. Stresses within
a vibrating medium are defined by taking the material particles to be volume elements,
with reference to some orthogonal coordinate system (Auld 1973a). In order to obtain a
clearer understanding of stress, we make the use of the following simple example. Let us
assume a small surface area AA on an arbitrary solid body with a unit normal n, which is
subjected to a surface force AF with uniform components AF i. The surface AA may be
expressed as a function of its surface components AA j and the unit normal components
as follows:
n j
A j = n j A A (10.6)
with the subscript j taking a value of 1, 2, or 3.
The stress tensor, T ij, is then related to the surface force and the surface area through

AF i

with the subscripts i and j taking a value of 1, 2, or 3.

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A tensor is a matrix in which the elements are vectors.