5.13.2 Circular membrane
When a membrane that is stretched over a circular frame is struck, energy is supplied, which again causes the membrane to vibrate in a number of modes simultaneously.
The first six modes in which the circular membrane can vibrate are shown in Figure 20. The diagrams comprise circles that are concentric with the rim and lines that pass through the centre of the membrane.
These represent nodal circles and nodal lines. They are the two-dimensional equivalent of the nodes on a vibrating string. There is no upwards and downwards motion of the membrane at any position along a nodal circle or line. Note that since the membrane is fixed all around its circumference, there is a nodal circle here for all the modes.
The fundamental vibrational mode is shown in Figure 20(a). The only feature is the nodal circle around the outside of the membrane. The whole membrane falls and rises en masse.
The second mode of vibration is shown in Figure 20(b). In addition to the outer nodal circle, there is now a nodal line across the centre dividing the membrane into two semi-circular segments. The two segments vibrate with opposite phase – while one is rising, the other is falling. The angular position of the nodal line through the centre is determined by where the membrane is struck – it will be roughly at right angles to an imaginary line drawn between the point where the membrane is struck and the centre.
The third mode of vibration is shown in Figure 20(c). There are now two nodal lines and the outer nodal circle. In this mode, two diametrically opposite quarters of the membrane vibrate in phase while the other two segments vibrate with the opposite phase.
The fourth mode of vibration is shown in Figure 20(d). There are no nodal lines but there are now two nodal circles. As with the second mode, the two segments vibrate with opposite phase, the central disc rising while the outer ring falls and vice versa.
The fifth mode has six segments separated by three nodal lines, while the sixth mode combines one nodal line and two nodal circles. Higher modes of vibration (not shown) continue to become more complicated.
Run the Flash animation below. It shows three-dimensional representations of a circular membrane vibrating in each of its first six modes of vibration and should help you to visualise the motion of the membrane in each mode. Please note: to view this animation correctly, you will need to click on the ‘Launch in separate player’ link below.
It turns out that, just like we discovered for the rectangular bar, the natural frequencies of a circular membrane are not harmonically related. If the first natural frequency is denoted by f1, then the frequencies of the second, third, fourth, fifth and sixth modes of vibration are 1.59f1, 2.13f1, 2.29f1, 2.65f1 and 2.91f1 respectively. Again, you don't need to remember these values – just be aware that the natural frequencies do not form a harmonic series.