Illustration of the normal modes

• Consider a thin, perfectly elastic rectangular membrane fixed along its whole outer edge.
• This physical system is a two-dimensional generalization of the case of the string

• Its normal modes are characterized by two integers.
  • The first number refers to the oscillation in the x direction;
  • The second number refers to the oscillation in the y direction.
• Given the dimensions 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: for each direction, x or y, they give the maximum number of antinodes encountered when moving along any segment in that particular direction.
• Note: in this case the number of nodes along x is equal to the number of antinodes along x plus one. The same holds for y. Indeed, there is a node on each of the two edges of the cell.

• For example, {1,1} means that there is at most one antinode along x and one along y.
• We said "at most" because it could happen that we have chosen a particular segment that coincides with a nodal line. On that segment all the points are nodes, and there is never even a single antinode.

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

rect_memb_freq_1_1.mp3 mode ${\displaystyle \{1,1\}}$, ${\displaystyle f=f_0}$	rect_memb_freq_2_1.mp3 mode ${\displaystyle \{2,1\}}$, ${\displaystyle f=1.84f_0}$	rect_memb_freq_3_1.mp3 mode ${\displaystyle \{3,1\}}$, ${\displaystyle f=2.72f_0}$
rect_memb_freq_1_2.mp3 mode ${\displaystyle \{1,2\}}$, ${\displaystyle f=1.26f_0}$	rect_memb_freq_2_2.mp3 mode ${\displaystyle \{2,2\}}$, ${\displaystyle f=2.00f_0}$	rect_memb_freq_3_2.mp3 mode ${\displaystyle \{3,2\}}$, ${\displaystyle f=2.83f_0}$
rect_memb_freq_1_3.mp3 mode ${\displaystyle \{1,3\}}$, ${\displaystyle f=1.61f_0}$	rect_memb_freq_2_3.mp3 mode ${\displaystyle \{2,3\}}$, ${\displaystyle f=2.24f_0}$	rect_memb_freq_3_3.mp3 mode ${\displaystyle \{3,3\}}$, ${\displaystyle f=3.00f_0}$

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 the section on natural frequencies of the rectangular membrane for details on the values of the frequencies.
• See the section questions and answers for questions and curiosities about normal modes, musical instruments and more.