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Unformatted text preview: tive change in volume, which is assumed to
be small, is (324) In the limit , this becomes (325) The pressure perturbation
Equation (323), which yields associated with the volume perturbation follows from (326) or
(327) giving (328) where use has been made of Equation (325).
Consider a section of the gas column lying between
. The - directed force acting on its left boundary is directed force acting on its right boundary is
(i.e., - directed) acceleration of the section is and . The mass of this section is
, whereas the - . Finally, the average longitudinal
. Thus, the section's longitudinal equation of motion is written (329) In the limit , this equation reduces to (330) Finally, Equation (328) yields (331) where (332) is a constant with the dimensions of velocity, which turns out to be the sound speed in the gas. (See Section 7.1.) Figure 34: First three normal modes of an organ
As an example, suppose that a standing wave is excited in a uniform organ pipe of length
of the pipe lie at
where , and the open end at . Let the closed end . The standing wave satisfie...
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