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### The Ideal Acoustic Tube

First we address the scalar case. For an ideal acoustic tube, we have the following wave equation : (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 right-going.'' 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 :   (2)   (3)

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 traveling-wave solution to (1), (2), and (3) was given by D'Alembert  as (4)

where are the right- and left-going 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 wave-shapes traveling in opposite directions along the uniform tube. The specific waveshapes are determined by the initial pressure and velocity throughout the tube for .

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