In Search of the Missing Fundamental: by Richard K. Jones
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Modes and Nodes

Mode: The vibrational mode of a vibrating circular membrane is the frequency at which the sections of the membrane are vibrating. This frequency is determined by counting the number of nodal lines and circles.

Node: In a vibrating circular membrane, a node is a place where the medium doesn’t move-as opposed to an anti-node, which has maximum movement. Nodal lines (diametric) and nodal circles (concentric) define the mode number. e.g. Mode 1,1 has one diametric node, and one concentric node.

Mode 1,1  the Principal Tone
(the one we hear as the pitch of the drum)
One diametric node and one concentric node

Unlike one-dimensional vibrating air columns and vibrating strings, vibrating circular membranes are two-dimensional. A two-dimensional circular membrane can have many modes of vibration occurring simultaneously; symmetrical (concentric modes), asymmetrical (diametric modes), and composite modes (which are combinations of the two). With respect to how timpani produce pitch, these modes are the shapes and patterns of vibrations of the head that move the air creating audible frequencies. By nature, these modes are not harmonic, that is the frequency of higher modes are not simple integers times the fundamental frequency as is found in the harmonic series.

Furthermore, since it is vibrating in two dimensions, the modes of a circular vibrating membrane create various combinations of diametric and concentric nodal points.  Nodes or nodal points (nodal lines and circles in the case of a circular membrane) are points which remain at rest while the other parts of the membrane are in a state of motion. Figure 2e is an animation of mode 4,1  which has four nodal diameters and one nodal circle. On the animation below, the nodal diameters and circles show up as white regions that don’t oscillate, while the red and blue regions indicate positive and negative displacements. Animations courtesy of  Dan Russell.

Mode 4,1
mode41Fig. 2e
Animation of a  vibrating membrane with one nodal circle
and four nodal diameters occurring in the vibration of mode 4,1.

Symmetrical and Asymmetrical  Motion

Mode 1,2Mode-1,2Figure 2e.1
The sixth mode of vibration mode 1,2. This is a composite mode, which vibrates with two
concentric modes of vibration and one diametric mode of vibration:
the nodal lines will encompass the entire circumference and diameter of the head

Mode 1,2 in motion
mode12

For a vibrating circular membrane, nodal lines and circles are points of minimal amplitude and the first nodal circle is always at the outer circumference (the outside edge) of the vibrating membrane. On a timpano, that is where the bearing edge of the bowl touches the head dictating the boundary conditions of the vibrating membrane. This is often referred to as the lip or rim of the bowl. In the customary mode designation, the first number gives the number of nodal diameters, and the second is the number of nodal circles. e.g. The indication (1,1) means that there is one nodal diameter (bisecting the circle) and one nodal circle (the outside edge). This (1,1) mode is the second mode of vibration of a circular membrane and is responsible for what we hear as the pitch of the timpani. It is called the principal tone and is not the fundamental with regard to a timpano’s sound spectrum.

Mode 1,1 of a vibrating circular membrane with
one diametric node and one concentric node

Figure 2f is a video of the superposition of 25 modes of vibration of circular membrane. Video courtesy of cptwell. The motion of a vibrating timpani head is very similar to the motion of the modes in these videos. With timpani however, certain modes are virtually dampened while others are not which helps give the instrument its sense of pitch.

Fig. 2f

Figure 2g is a short video by Dan Russell (Penn State) showing the motion and frequencies of the first five modes of a large air loaded circular membrane. Of particular importance to timpani are the (1,1) and (2,1) modes, which are the principal tone and the first partial (the fifth) of the set of preferred modes that produce the sense of pitch.

Notice that the frequency ratio between modes 1,1 (45Hz) and 2,1 (58Hz) is only 1.28 (439 cents), which is a sharp major third. If this were a timpano, the ratio for these two modes should be closer to 1.5 or a perfect fifth in order to be harmonic. The main contributing factors to the narrowness of the interval are the diameter, mass, stiffness, and tensioning of the rubber membrane.

Just as with strings, the diameter, mass, stiffness, and tensioning are important criteria in determining membrane inharmonicity/harmonicity.

Fig. 2g

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