Before examining the physics of standing waves within pipes that are closed at one end, a review of systems within which both ends are open is due.
Periodic motion is used to model the behavior of systems that are harmonic or cyclical in nature. Within the midst of turbulent high-tide waves crashing against a shoreline, there is harmony. The water waves rhythmically rush up the shoreline against the weight of the earth’s atmosphere for a brief period of time. As the waves crests temporarily overcome the gravitational force of the earth, a vacuum is created behind them. The vacuum does not have the mass or energy needed to sustain the temporary displacement of water, and the water soon rushes back into the body from which it emerged. Within the midst of turbulent systems of particles, resonant states can and will occur under the right circumstances.
Resonance phenomena are more readily observed in less turbulent systems. When a child swings upon a swing, an initial input of energy propels them forward to some maximum displacement before swinging back to their origin. Upon reaching their origin of travel, very little energy was lost by the system. Thus, if an equal amount of energy from a push is put into the system, the swing will swing higher. In this circumstance, the correct timing ( period ) within which to most efficiently put energy into the system is determined by the length of the swing.
Let’s now reexamine the case of a source of sound-emitting pressure waves into a pipe with two open ends. Recall that sound waves, despite their representation in many pipe diagrams, have a tendency to spread outward. When traveling sound waves travel within the constraints of a pipe, the energy within the normally spherical wave gives rise to turbulence within the springy air molecules. As pressure builds within the pipe, the turbulent air system begins to push against the atmosphere from both open ends of the pipe. Upon pushing against the atmosphere, a pressure vacuum is created within the midst of the tube. As a consequence, the atmosphere pushes the turbulent air back into the tube. If a certain frequency of sound enters the system, the turbulent air system within the pipe takes on the characteristics of an elongated spring. The coordinated expansion of molecules against the atmosphere at both open ends becomes amplified with efficient energy inputs from sound waves entering at a fundamental frequency.
Eventually, the back-and-forth pushing of the energized air medium against the atmosphere reaches a state of equilibrium, and the vibrational equilibrium of the air columns is supported by a region of high pressure in the center of the pipe. Within this region of high pressure, the air is maximally confined and restricted from expanding. This region of air is represented by a node within open-pipe diagrams, and the turbulent air regions at the pipe’s open ends are represented with antinodes. When resonance within the system has become well established, a characteristic sound is produced, and the frequency giving rise to this sound is the fundamental frequency of the system ( f1 ).
It is now useful to treat the system as if single particles representing sound waves travel the length of the open pipe system. A complete cycle is defined by a sound wave moving the length of the pipe, and upon being reflected by the atmosphere, traveling back toward the source of sound at the other open end. When the sound particle reaches the end of the pipe, it has traveled ½ of a complete revolution. Upon being reflected by the atmosphere, let’s assume that the sound source sends out another sound particle. Each particle travels at the same speed, and they will meet and become compressed at the pipe’s center. Since the first particle has traveled ¾ of a complete revolution, the particle that meets it head-on must have traveled ¼ of a revolution.
Graphically, an antinode describes the circumstance at the pipe’s openings, and a node describes the pressure region at the center of the pipe. Thus, the distance between an antinode and a node within sound systems represents ¼ of a wavelength. This is the case in open AND closed pipe systems. Within the open pipe system, two ¼ wavelengths fit into the length of the pipe, and thus, the length ( L ) of the open pipe is ½ λ. When dealing with pipes with one closed-end, a region of restricted turbulence exists at the closed end, and a less restricted and turbulent region of pressure in equilibrium with the atmosphere is established at the open end. Thus, a single antinode and node exist within the pipe with one closed end. Since the distance between a node and antinode is ¼ λ, the length of a pipe with a closed-end is ¼ λ.
Recall that the velocity of a wave is defined by v = λf, where the velocity ( v ) is in units of meters per second ( m/s ), and generally speaking, the velocity of sound is ( 343 m/s ). The wavelength λ is measured in meters ( m ), and frequency ( f ) is in cycles per second ( 1/s or s^-1 ), and the unit of frequency is Hertz ( Hz ). We determine the frequency of open and closed-end pipes as follows:
Open End Pipes: If v = λf, then ( v/λ ) = f. Since λ = 2L in open pipe systems, f = ( v/2L ).
If the frequency of the sound source is doubled, a sound wave reflected by the atmosphere would be headed towards a wavefront traveling towards it from the halfway mark of the pipe, and the two wavefronts would meet ( and form a node ) at ¾ the pipe’s length. As symmetry would have it, another node would be established at ¼ the length of the tube. Therefore, when the second harmonic ( 2f1 = f2 ) establishes resonance within an open pipe, two nodes exist within the length of the pipe. Generally speaking, the nth harmonic of an open pipe will have n# nodes along the pipe’s length. Furthermore, the general frequency formula for open pipes is f = nv/2L, where n = 1,2,3,…etc.
Closed-End Pipes: If v = λf, then ( v/λ ) = f. Since λ = 4L in closed pipe systems, f = ( v/4L ). Within closed-end pipes, there must always be an antinode at the open end of the pipe, and there must be a node at the closed end of the pipe. This can only be established with odd multiples of the fundamental frequency, and thus, n = 1, 3, 5…etc. for closed-end pipes. As a consequence, the general frequency formula for closed-end pipes is f = nv/4L.
RENAISSANCE PHYSICS 