Do one-dimensional systems really exist?

In physics we often speak of a one-dimensional system when referring to a real, and therefore three-dimensional, object that has one (or more) of the following characteristics:

1. its dimensions are much larger in one direction than in the other two. For example, a steel cable in a bridge.
2. the relevant physical phenomena take place essentially along one preferred direction of the system. For example, in an organ pipe (which can also be quite wide), the air column oscillates essentially along the length of the pipe.
3. we choose to describe the system approximately, focusing only on phenomena taking place along one of its dimensions.

If we adopt this point of view, we realise that in a great many musical instruments sound is in fact produced in one-dimensional systems such as

• strings, usually of metal or gut (in string instruments)
• air columns of various shapes (in wind instruments)

The main difference between the oscillations of the two media is that the oscillations of strings are mostly transverse, while those of air columns are only longitudinal

There are also essentially one-dimensional percussion instruments, such as the xylophone or the marimba.

Other percussion instruments, such as the snare drum, cymbals or the gong, can instead be treated as vibrating systems in two dimensions

Oscillations in one-dimensional systems

To describe the oscillations of these systems we can follow two equivalent routes:

1. the point of view of travelling waves
2. the point of view of normal modes

The point of view of travelling waves

• Consider an ideal vibrating string. From the superposition principle it follows that if two waves can propagate in a medium, so can their sum. D'Alembert used this idea to write the solution of the wave equation in terms of waves travelling forwards (progressive) and backwards (regressive) along the string. Any wave on the string can be broken down into progressive and regressive components.
• This point of view becomes particularly intuitive when there are obstacles or inhomogeneities on the string, so that it is useful to consider each wave in its incident, reflected or transmitted motion with respect to the obstacle (for applications of these concepts see, for example, refraction in one dimension (in Italian) and reflection in strings)
• Because the string has ends, waves on it cannot propagate indefinitely, but are reflected at the boundary. The precise way this reflection happens depends on the type of boundary conditions: on the page reflection in strings the various types of reflection are analysed in detail.

The point of view of normal modes

• Instead of studying the motion of a single pulse back and forth along the string (see reflection in strings and refraction in one dimension, in Italian), we can approach the problem in another way: what happens when we drive the string with a force applied from outside? In particular: what happens when the string is driven periodically?
• In this case we observe a feature common to all oscillating physical systems, called resonance: there are particular frequencies of the external drive at which the system responds by oscillating particularly strongly. These frequencies are called the natural frequencies of the system.
• When the system oscillates at one of its natural frequencies, its energy is not carried from one place to another along the system by travelling waves, but concentrates in certain regions. The system oscillates as a standing wave.
• If you know what normal modes are, you may already have made the connection: all the normal modes of oscillation of a system are in fact standing waves.
• Another equivalent way of looking at the question is to note that a physical system does not oppose the external force setting it into oscillation with the same resistance at all frequencies. There are always frequencies at which the system is particularly compliant. They correspond to the minima of the impedance of the system, and are its natural frequencies. Each natural frequency corresponds to one or more normal modes.

Links and further reading

• It is not obvious a priori that the description of standing waves from the point of view of travelling waves should coincide with the one in terms of normal modes, but this is indeed the case: every wave in the system can be broken down into travelling waves thanks to D'Alembert's solution, and every mode of oscillation of the system can be broken down into a combination of standing waves through the normal modes.
• To see in more detail how a wave is reflected at a fixed end, visit the section reflection in strings
• To see in more detail how a travelling wave interferes with its reflected wave, leading to the formation of antinodes and nodes, visit the section interference between reflected waves
• It is not always easy to calculate exactly the motion of an elastic string, such as the string of a musical instrument. However, it is fairly simple to find out its natural frequencies. In the section natural frequencies of a string you can find out how.
• The sound of a great many instruments depends on the properties of the vibrating string. The real world, however, almost always departs from the ideal case. See string instruments.
• Pages on impedance: impedance and the links on that page, impedance matching