Illustration of the modes

• Consider a thin, perfectly elastic circular membrane fixed along its whole outer edge.
• This physical system describes well, for example, the membrane of the snare drum and of the timpani.

• Its normal modes are characterized by two integers ${\displaystyle \{n,m\}}$.
  • The first number n characterizes the tangential, or angular, oscillation (that is, along the concentric circles).
  • The second number m characterizes the radial oscillation (that is, in the direction from the centre to the edge).
• Given the radius of the membrane, the material it is made of, and its tension, the natural frequency of each mode is determined solely by these two numbers.
• The numbers are easy to interpret: the first gives the number of nodal diameters. The second gives the number of nodal circles (including the one at the edge, which is always present, since we are dealing with a membrane fixed along its whole edge).
• A diameter or a circle is nodal when it always stays at rest during the oscillation. In 1787 the physicist Chladni devised an ingenious method to reveal the nodal curves. It consists in sprinkling a fine powder on a plate (Chladni used metal or glass plates) and exciting it with a bow (in modern times, with a loudspeaker). The powder collects precisely along the nodal curves, showing them clearly (see the individual musical instruments for a discussion of the various nodal lines.)

• For example, {0,1} means that there are no nodal diameters, while there is one nodal circle.
• To identify the nodes in difficult cases, remember that, along a given "radial" or "tangential" direction, a node always lies between two antinodes, and vice versa.

Normal modes and harmonic sounds

• Normal modes are of fundamental importance in determining the timbre of the sound heard when the membrane is struck (or rubbed). In general, the sound coming from it is made up of many normal modes at the same time, in different proportions. Below we have isolated them so that their frequency can be identified more easily.

• In the following table, the areas coloured light blue indicate the areas of the membrane that vibrate in phase opposition with respect to the white areas. The black lines are nodal lines, and therefore correspond to the "Chladni figures" for this membrane. They are regions of the membrane that are always at rest in that particular normal mode. By clicking on the audio samples you can listen to the frequencies of the various modes, with the fundamental fixed at 110 Hz.

nodal lines and audio frequencies in the various normal modes

circ_memb_freq_0_1.mp3 mode ${\displaystyle \{0,1\}}$, ${\displaystyle f=f_{0}\!}$	circ_memb_freq_1_1.mp3 mode ${\displaystyle \{1,1\}}$, ${\displaystyle f=1.59f_{0}\!}$	circ_memb_freq_2_1.mp3 mode ${\displaystyle \{2,1\}}$, ${\displaystyle f=2.14f_{0}\!}$
circ_memb_freq_0_2.mp3 mode ${\displaystyle \{0,2\}}$, ${\displaystyle f=2.29f_{0}\!}$	circ_memb_freq_1_2.mp3 mode ${\displaystyle \{1,2\}}$, ${\displaystyle f=2.92f_{0}\!}$	circ_memb_freq_2_2.mp3 mode ${\displaystyle \{2,2\}}$, ${\displaystyle f=3.50f_{0}\!}$
circ_memb_freq_0_3.mp3 mode ${\displaystyle \{0,3\}}$, ${\displaystyle f=3.60f_{0}\!}$	circ_memb_freq_1_3.mp3 mode ${\displaystyle \{1,3\}}$, ${\displaystyle f=4.23f_{0}\!}$	circ_memb_freq_2_3.mp3 mode ${\displaystyle \{2,3\}}$, ${\displaystyle f=4.83f_{0}\!}$

Links and further reading

• You will have noticed that, unlike in the case of the string, not all the frequencies of the oscillations are in integer ratios with each other. See natural frequencies of the circular membrane for details on the values of the frequencies.
• What is shown on this page are only the normal modes of the membrane. In the section percussion you can find out how a membrane really moves when it is struck, and what relation there is between these normal modes and the sound heard coming from a drum.
• The study of normal modes and nodal lines is important for all other musical instruments too.
• See the section questions and answers for questions and curiosities about normal modes, musical instruments and more.