Suppose that the initial displacement of the string at is
It is readily demonstrated that
Let us multiply Equation (271) by , and integrate over from 0 to . We obtain
As an example, suppose that the string is initially at rest, so that
When a stringed instrument, such as a guitar, is played, a characteristic pattern of normal mode oscillations is excited on the plucked string. These oscillations excite sound waves of the same frequency, which propagate through the air and are audible to a listener. The normal mode (of appreciable amplitude) with the lowest oscillation frequency is called the fundamental harmonic, and determines the pitch of the musical note that is heard by the listener. For instance, a fundamental harmonic that oscillates at Hz corresponds to ``middle C''. Those normal modes (of appreciable amplitude) with higher oscillation frequencies than the fundamental harmonic are called overtone harmonics, because their frequencies are integer multiples of the fundamental frequency. In general, the amplitudes of the overtone harmonics are much smaller than the amplitude of the fundamental. Nevertheless, when a stringed instrument is played, the particular mix of overtone harmonics that accompanies the fundamental determines the timbre of the musical note heard by the listener. For instance, when middle C is played on a piano and a harpsichord the same frequency fundamental harmonic is excited in both cases. However, the mix of excited overtone harmonics is quite different. This accounts for the fact that middle C played on a piano can be easily distinguished from middle C played on a harpsichord.
Figure 27 shows the ratio for the first ten overtone harmonics of a uniform guitar string plucked at its midpoint: that is, the ratio for odd- modes, with , calculated from Equation (292). It can be seen that the amplitudes of the overtone harmonics are all small compared to the amplitude of the fundamental. Moreover, the amplitudes decrease rapidly in magnitude with increasing mode number, .
In principle, we must include all of the normal modes in the sum on the right-hand side of Equation (289). In practice, given that the amplitudes of the normal modes decrease rapidly in magnitude as increases, we can truncate the sum, by neglecting high- normal modes, without introducing significant error into our calculation. Figure 28 illustrates the effect of such a truncation. In fact, this figure shows the reconstruction of , obtained by setting in Equation (289), made with various different numbers of normal modes. The long-dashed line shows a reconstruction made with only the largest amplitude normal mode, the short dashed-line shows a reconstruction made with the four largest amplitude normal modes, and the solid line shows a reconstruction made with the sixteen largest amplitude normal modes. It can be seen that sixteen normal modes is sufficient to very accurately reconstruct the triangular initial displacement pattern. Indeed, a reconstruction made with only four normal modes is surprisingly accurate. On the other hand, a reconstruction made with only one normal mode is fairly inaccurate.
Figure 29 shows the time evolution of a uniform guitar string plucked at its mid-point. This evolution is reconstructed from Equation (289) using the sixteen largest amplitude normal modes of the string. The upper solid, upper short-dashed, upper long-dashed, upper dot-short-dashed, dot-long-dashed, lower dot-short-dashed, lower long-dashed, lower short-dashed, and lower solid curves correspond to , , , , , , , , and , respectively. It can be seen that the string oscillates in a rather peculiar fashion. The initial kink in the string at splits into two equal kinks that propagate in opposite directions along the string, at the velocity . The string remains straight and parallel to the -axis between the kinks, and straight and inclined to the -axis between each kink and the closest wall. When the two kinks reach the wall, the string is instantaneously found in its undisturbed position. The kinks then reflect off the two walls, with a phase change of radians. When the two kinks meet again at the string is instantaneously found in a state that is an inverted form of its initial state. The kinks subsequently pass through one another, reflect off the walls, with another phase change of radians, and meet for a second time at . At this instant, the string is again found in its initial position. The pattern of motion then repeats itself ad infinitum. The period of the oscillation is the time required for a kink to propagate two string lengths, which is . This is also the oscillation period of the normal mode.