Introduction to Acoustics

28 Part A Propagation of Sound
Part A 3.2
the task of the present chapter is to explain in what
sense linear acoustics is an approximation. An extensive
discussion of the extent to which the linearization approximation
is valid is not attempted, but a discussion
is given of the manner in which the linear equations result
from nonlinear equations that are regarded as more
nearly exact.
The historical origins [3.1] of linear acoustics reach
back to antiquity, to Pythagoras and Aristotle [3.2], both
3.2 Equations of Continuum Mechanics
Sound can propagate through liquids, gases, and solids.
It can also propagate through composite media such as
suspensions, mixtures, and porous media. In any portion
of a medium that is of uniform composition, the general
equations that are assumed to apply are those that are
associated with the theory of continuum mechanics.
3.2.1 Mass, Momentum,
and Energy Equations
The primary equations governing sound are those that
account for the conservation of mass and energy, and for
changes in momentum. These may be written in the form
of either partial differential equations or integral equations.
The former is the customary starting point for the
derivation of approximate equations for linear acoustics.
Extensive discussions of the equations of continuum mechanics
can be found in texts by Thompson [3.5], Fung
[3.6], Shapiro [3.7], Batchelor [3.8], Truesdell [3.9]),
and Landau and Lifshitz [3.10], and in the encyclopedia
article by Truesdell and Toupin [3.11].
The conservation of mass is described by the partial
differential equation,
∂ρ
+∇·(ρv) = 0 , (3.2)
∂t
where ρ is the (possibly position- and time-dependent)
mass density (mass per unit volume of the material), and
v is the local and instantaneous particle velocity, defined
so that ρv·n is the net mass flowing per unit time per
unit area across an arbitrary stationary surface within the
material whose local unit outward normal vector is n.
The generalization of Newton’s second law to a continuum
is described by Cauchy’s equation of motion,
which is written in Cartesian coordinates as
ρ Dv
Dt
= �
ij
∂σij
ei + gρ. (3.3)
∂x j
of whom are associated with ancient Greece, and also
to persons associated with other ancient civilizations.
The mathematical theory began with Mersenne, Galileo,
and Newton, and developed into its more familiar form
during the time of Euler and Lagrange [3.3]. Prominent
contributors during the 19-th century include Poisson,
Laplace, Cauchy, Green, Stokes, Helmholtz, Kirchhoff,
and Rayleigh. The latter’s book [3.4], The Theory of
Sound, is still widely read and quoted today.
Here the Eulerian description is used, with each field
variable regarded as a function of actual spatial position
coordinates and time. (The alternate description is
the Lagrangian description, where the field variables are
regarded as functions of the coordinates that the material
being described would have in some reference
configuration.) The σij are the Cartesian components of
the stress tensor. The quantities ei are the unit vectors
in a Cartesian coordinate system. These stress tensor
components are such that
t = �
i, j
eiσijn j
(3.4)
is the traction vector, the surface force per unit area on
any given surface with local unit outward normal n.
The second term on the right of (3.3) is a body-force
term associated with gravity, with g representing the
vector acceleration due to gravity. In some instances, it
may be appropriate (as in the case of analyses of transduction)
to include body-force terms associated with
external electromagnetic fields, but such are excluded
from consideration in the present chapter.
The time derivative operator on the left side of (3.3)
is Stokes’ total time derivative operator [3.12],
D ∂
= + v·∇ , (3.5)
Dt
∂t
with the two terms corresponding to: (i) the time derivative
as would be seen by an observer at rest, and (ii) the
convective time derivative. This total time derivative applied
to the particle velocity field yields the particle
acceleration field, so the right side of (3.3) is the apparent
force per unit volume on an element of the continuum.
The stress tensor’s Cartesian component σij is, in accordance
with (3.4), the i-th component of the surface
force per unit area on a segment of the (internal and
hypothetical) surface of a small element of the continuum,
when the unit outward normal to the surface is in