on, sofnis then-th harmonic. (Sometimes a different nomenclature is used:f 1 is called thefundamental
frequency,f 2 is called thefirst overtone,f 3 is thesecond overtone, and so on, sofnis the.n1/-th overtone.)
It turns out (as you can see from examining Fig. 10.1(b)) that this same condition (Eq. 10.2) also applies
to waves that arefreeat both ends: an integer number of half-wavelengths must fit into lengthL.
10.2 Fixed at One End and Free at the Other
A different situation occurs when the wave is fixed at one end and free at the other (Fig. 10.1(c)). From
examining the figure, you can see the pattern: an odd number of half-segments has to fit into distanceL.
Since each segment is a half wavelength, this means that an odd number of quarter-wavelengths must fit into
lengthL:
LDn
4.nD1;3;5;7;:::/ (10.3)Again using the relationvDf and solving forf, we find the condition for standing waves in this case is
fnDnv
4L.nD1;3;5;7;:::/ (10.4)Although we’ve been talking about string waves, this analysis refers to both transverse and longitudinal
waves (sound waves, for example). As we’ll see later, musical instruments work by creating standing sound
waves which satisfy these same conditions.
10.3 Vibrations of Rods and Plates
A rod may be set vibrating (longitudinally) by holding or clamping it at some point and stroking it with rosin.
There will be a node at the point where the rod is clamped, and antinodes at each end. For example, clamping
the rod at its center point will create standing waves free at both ends (where there are antinodes) and fixed
in the center (where the rod is clamped), resulting in annD 1 standing wave, as shown in Figure 10.1(b),
nD 1. Clamping the rod at^1 / 4 its length from one end again creates a node at the clamped point and antinodes
at the two ends, resulting in annD 2 standing wave (Figure 10.1(b),nD 2 ).
Standing waves can also be created in two-dimensional plates or membranes. Figure 10.2 shows the
standing wave modes of a circular membrane such as a drum head. Notice in this case that the frequencies of
the standing wave modes arenotinteger multiples of the fundamental frequencies, so they arenotharmonics.
Figure 10.2: Modes of vibration of a circular membrane, showing nodal lines. (Figure from D. Livelybrooks,
Univ. of Oregon.)