A standing pressure wave can be established by sound waves within a pipe with both ends open as well as pipes with one closed end. When a single-standing pressure wave exists within a pipe, the frequency ( f ) of the sound emitted is regarded as being the first or ” fundamental ” sound frequency of the pipe. The mechanism by which the fundamental frequency ( f ) is established in pipes with one closed end vs. fully open pipes differs.
Consider an open pipe with an arbitrary length ( L ). Within the pipe, there is a springy and slightly compressible medium of gas. As a sound wave enters an open pipe, a region of high pressure travels the length of the pipe. As the pressure component of the wave leaves the pipe, it causes atoms in the gas immediately outside of the pipe’s opening to ” spring ” outward. This movement outward is followed by a collapse of these atoms back into the tube. Due to the momentum of the collapsing atoms, it is as if a pressure wave has been reflected back into the pipe by a closed-end. What differs in open pipe systems, however, is the location within the pipe where it is possible to form the first standing pressure wave.
Imagine a single pressure wave traveling the length ( L ) of an open pipe. As the pressure wave leaves the pipe, atoms immediately outside of the pipe’s open end expand, collapse, and send a pressure wave back through the pipe. Let’s now imagine a circumstance where a sound source sends a new sound wave down the pipe. At the same time, a compression wave that has just travelled the length of the pipe begins travelling back through the pipe. The two pressure waves will meet at the middle of the tube, and they act cooperatively to establish a standing pressure wave. This pressure wave will be established as long as the source of sound maintains the fundamental frequency ( f ). Thus, when a pressure wave is established at the center of the open pipe, and there will be two low-pressure regions established at each open end of the pipe.
When two sound waves meet at the center of the pipe, the collapsing wavefront has thus traveled through ( 3 / 4 ) of a complete cycle. Therefore, the incoming wave that it meets has traveled ( 1 / 4 ) of a complete cycle. For this reason, the standing pressure wave established at ( 1 / 2 ) the length ( L ) of an open pipe ( represented graphically with a node ) is the distance representing ( 1 / 4 ) of a complete revolution. Therefore, within open pipe systems, the length ( L ) of the open pipe = ( 1 / 2 ) of a full revolution, and ( L ) = ( 1 / 2 ) of a complete wavelength ( λ ) of the sound wave.
If ( L ) = ( 1 / 2 )( λ ), where ( λ ) = the wavelength in meters ( m ), then ( 2L ) = ( λ ). Recall that the velocity of sound is ( v ) = ( λ )( f ), where ( f ) = the frequency of a sound-emitting source. Therefore, the fundamental frequency ( f ) of a sound in an open sound system = ( v/λ ) = ( f ). Since ( λ ) = ( 2L), the fundamental frequency ( f ) = ( ( v ) / ( 2L ) ) in open pipe systems.
In order for two standing waves to be established within an open pipe system, the frequency of the incoming wave would have to be doubled. Under this circumstance, when one compression wave reaches the open end of the pipe, another compression wave has travelled to the center of the pipe. As a consequence, the collapsing pressure wave and the wave after it would meet at ( 3 / 4 )( L ) of the open pipe. As symmetry would have it, another standing wave is also established at ( 1 / 4 )( L ) within the pipe. The frequency of the sound emitted is a multiple of the fundamental frequency by ( 2 )( f ). In general, any multiple of the fundamental frequency is mathematically represented by ( n )( f1 ), where n = 1, 2, 3, etc. Therefore, the frequency ( f ) of a wave within an open pipe system = [ ( n )( v ) ] / ( 2L ), where ( v ) = the speed of sound ( 332 m / s ), ( L ) = the length of the pipe.
RENAISSANCE PHYSICS 