Do two-dimensional systems really exist?

We call two-dimensional a physical system that has one (or more) of the following characteristics:

1. its dimensions are much larger in two directions than in the third. For example, a steel plate (as in cymbals) or a skin membrane (as in drums) has a thickness negligible compared with its diameter.
2. the relevant physical phenomena take place essentially in a surface of the system, not perpendicular to it.
3. we choose to describe the system approximately, focusing only on phenomena taking place in one of its planes, or in a particular surface.

Many percussion instruments use two-dimensional systems, such as metal plates (idiophones) or membranes of skin or other materials (membranophones), to produce sound.

Oscillations in two-dimensional systems

The point of view of travelling waves

While in a one-dimensional system such as a string waves can travel only forwards or backwards, in a two-dimensional system travelling waves are free to propagate in a plane (more generally in a curved surface), which means they have infinitely many directions of propagation available.

Let us examine the case of a plane. From geometry we know that any direction can be described in terms of two independent components x and y.

Imagine, by analogy with the one-dimensional case, a flat plate whose edge is clamped all around. If we think for a moment of wave propagation in terms of rays, then we can think of the wave as moving in the two-dimensional system like a ball on a billiard table. It is free to travel in any direction until it reaches an edge. There it is totally reflected (at least ideally), and changes the direction of its motion. After many reflections the whole plane will be in a state that is the superposition of all the incident and reflected waves at its edges.

Exactly as in the one-dimensional case, it can be seen that there are particular frequencies of the waves travelling in the plate for which this superposition gives a stationary result. That is, it produces a configuration in which the nodes and antinodes do not move.

The point of view of normal modes

From this point of view (see the page on normal modes), there is no particular difference from the normal modes in one-dimensional systems.

In the two-dimensional case too, a particular normal mode, or resonance of the system, coincides with an oscillation as a particular standing wave, that is, with a minimum of the impedance of the system at that frequency.

The boundary conditions determine which modes of vibration are possible or impossible. The particular oscillation frequency is determined by the frequency of the standing wave corresponding to the normal mode.

Further reading

• Percussion instruments do not just go "boom". Many percussion instruments have a precise tuning.
• Would a square drum sound like a round one?
• Are the normal modes of vibration of cymbals the same as those of a drum membrane, since both are two-dimensional and circular?
• Do we speak of harmonics for percussion instruments too? Are they different from those of string instruments?
• In the section devoted to percussion instruments you will find details about the instruments.
• The sections natural frequencies of a circular membrane, natural frequencies of circular plates and natural frequencies of a rectangular membrane give details on calculating the natural frequencies and normal modes of typical two-dimensional systems.