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First we address the scalar case. For an ideal acoustic tube, we have
the following wave equation [27]:

(1) 
where denotes (scalar) pressure in the tube at the point
along the tube at time in seconds. If the length of the tube is
, then is taken to lie between and . We adopt the
convention that increases ``to the right'' so that waves traveling
in the direction of increasing are referred to as ``rightgoing.''
The constant is the speed of sound propagation in the tube, given
by
, where is the
``tension^{}'' of the gas in the tube, and
is the mass per unit volume of the tube. The dual
variable, volume velocity , also obeys (1) with
replaced by . The wave equation (1) also holds for an
ideal string, if represents the transverse displacement,
is the tension of the string, and is its
linear mass density.
The wave equation (1) follows from the more physically meaningful
equations [28, p. 243]:
Equation (2) follows immediately from Newton's second law of motion, while (3) follows
from conservation of mass and properties of an ideal gas.
The general travelingwave solution to (1), or
(2) and (3), was given by D'Alembert
as [27]

(4) 
where
are
the right and leftgoing wave components of pressure and velocity,
respectively, and are referred to as wave variables.
This solution form is interpreted as the sum of two
fixed waveshapes traveling in opposite directions along the tube. The
specific waveshapes are determined by the initial pressure
and velocity throughout the tube .
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