Next 
Prev 
Up 
Top

JOS Index 
JOS Pubs 
JOS Home 
Search
First we address the scalar case. For an ideal acoustic tube, we have the
following wave equation [60]:

(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
``spring constant'' or ``stiffness'' of the air in the
tube,^{2} 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
telegrapher's equations [24]:
Equation (2) follows immediately from Newton's second law of motion, while (3) follows
from conservation of mass and properties of an ideal gas [61].
The general travelingwave solution to (1), (2), and (3)
was given by D'Alembert [60] as

(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 uniform tube. The
specific waveshapes are determined by the initial pressure
and velocity throughout the tube for .
Next 
Prev 
Up 
Top

JOS Index 
JOS Pubs 
JOS Home 
Search
Download wgj.pdf