Open & Closed Pipe (Physics): Differences, Resonance & Equation

The physics of waves covers a diverse range of phenomena, from the everyday waves like water, to light, sound and even down at the subatomic level, where waves describe the behavior of particles like electrons. All of these waves exhibit similar properties and have the same key characteristics that describe their forms and behavior.

One of the most interesting properties of a wave is the ability to form a "standing wave." Learning about that concept in the familiar terms of sound waves helps you understand the operation of many musical instruments, as well as laying some important groundwork for when you learn about the orbits of electrons in quantum mechanics.

Sound is a longitudinal wave, which means the wave varies in the same direction as it travels. For sound, this variation comes in the form of a series of compressions (regions of increased density) and rarefactions (regions of decreased density) in the medium through which it travels, such as air or a solid object.

The fact that a sound wave is longitudinal means that the compressions and rarefactions hit your eardrum one after another, rather than multiple "wavelengths" hitting it at the same time. Light, by contrast, is a transverse wave, so the waveform is at right angles to the direction it travels.

Sound waves are created by oscillations, whether these are from your vocal cords, the vibrating string of a guitar (or other oscillating parts of musical instruments), a tuning fork or a pile of dishes crashing to the floor. All of these sources create compressions and corresponding rarefactions in the air surrounding them, and this travels as sound (depending on the intensity of the pressure waves).

These oscillations need to travel through some sort of medium because otherwise there would be nothing to create the compression and rarefaction regions, and so sound only travels at a finite speed. The speed of sound in air (at 20 degrees Celsius) is around 344 m/s, but it actually travels at a faster rate in liquids and solids, with a speed of 1,483 m/s in water (at 20 C) and 4,512 m/s in steel.

Vibrations and oscillations tend have what can be thought of as a natural frequency, or resonant frequency. In mechanical systems, resonance is the name for the reinforcement of sound or other vibrations that occurs when you apply a periodic force at the object's resonant frequency.

Essentially, by applying the force in time with the natural frequency at which an object vibrates or oscillates, you can amplify or prolong the motion – think about pushing a child on a swing and timing your pushes with the existing motion of the swing.

Resonant frequencies for sound are basically the same. A classic demonstration with tuning forks shows the concept clearly: Two identical tuning forks are attached to sound boxes (which essentially amplify the sound in the same way the sound box of an acoustic guitar does for the guitar string's oscillation), and one of them is struck with a rubber mallet. This starts the air around it vibrating, and you can hear the pitch produced by the natural frequency of the fork.

But if you stop the fork that you hit from vibrating, you will still hear the same sound, just coming from the other fork. Because the two forks have the same resonant frequencies, the motion of the air caused by the vibration of the air caused by the first fork actually made the second one vibrate too.

The specific resonant frequency for any given object depends on its properties – for example, for a string, it depends on its tension, mass and length.

A standing wave pattern is when a wave oscillates but doesn't appear to move. This is actually caused by the superposition of two or more waves, travelling in different directions but each having the same frequency.

Because the frequency is the same, the crests of the waves line up perfectly, and there is constructive interference – in other words, the two waves are added together and produce a larger disturbance than either would on its own. This constructive interference alternates with destructive interference – where the two waves cancel each other out – to produce the standing wave pattern.

If a sound of a certain frequency is created near a pipe filled with air, a standing sound wave can be created in the pipe. This produces resonance, which amplifies the sound produced by the original wave. This phenomenon underpins the workings of many musical instruments.

For an open pipe (that is, a pipe with open ends at each side), a standing wave can form if the wavelength of the sound allows there to be an antinode at either end. A node is a point on a standing wave where no motion takes place, so it remains in its resting position, while an antinode is a point where there is the most motion (the opposite of a node).

The lowest-frequency standing wave pattern will have an antinode at each open end of the pipe, with one node in the middle. The frequency where this happens is called the fundamental frequency or the first harmonic.

The wavelength associated with this fundamental frequency is 2L, where length, L, refers to the length of the pipe. Standing waves can be created at higher frequencies than the fundamental frequency, and each one adds an extra node to the motion. For example, the second harmonic is a standing wave with two nodes, the third harmonic has three nodes and so on.

Where the fundamental frequency is f₁, the frequency of the nth harmonic is given by f_n = nf₁, and its wavelength is 2L / n, where L again refers to the length of the pipe.

A closed pipe is one where one end is open and the other is closed, and like open pipes, these can form a standing wave with sound of an appropriate frequency. In this case, there can be a standing wave whenever the wavelength allows an antinode at the open end of the pipe and a node at the closed end.

For a closed pipe, the lowest-frequency standing wave pattern (the fundamental frequency or first harmonic) will have just one node and one antinode. For a closed pipe with length L, the fundamental standing wave is produced when the wavelength is 4L.

Again, there can be standing waves produced at higher frequencies than the fundamental frequency, and these are called harmonics. However, only odd harmonics are possible with a closed pipe, but each of them still produces an equal number of nodes and antinodes. The frequency of the nth harmonic is f_n = nf₁, where f₁ is the fundamental frequency and n can only be odd. The wavelength of the nth harmonic is 4L / n, again remembering that n must be an odd integer.

The most well-known applications of the concepts you've learned about are musical instruments, particularly woodwind instruments like the clarinet, flute and the saxophone. The flute is an example of an open pipe instrument, and so it produces standing waves and resonance when there is an antinode at both ends.

Clarinets and saxophones are examples of closed pipe instruments, which produce resonance when there is a node at the closed end (although it isn't completely closed because of the mouthpiece, sound waves still reflect as if it is) and an antinode at the open end.

Of course, the holes on the real-world instruments complicate matters slightly. However, to simplify the situation slightly, the "effective length" of the pipe can be calculated based on the position of the first open hole or key. Finally, the initial vibration that leads to the resonance is either produced by a vibrating reed or by the musician's lips against the mouthpiece.