THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

4.7 Standing Waves 77Figure 4.3. Different modes of vibrations for a string for the fundamental and the firsttwo harmonics.The cases for which m = 0 and m = n correspond to the boundary conditions atthe fixed points at the two ends of the string. However, there are additional (n − 1)locations, called nodal points or nodes, where the displacement produced by thenth harmonic mode of vibration remains at zero, as illustrated in Figure 4.3. Thissituation may be viewed as one in which the harmonic wave moving in the positivex-direction cancels precisely at the nodal points at all times t the harmonic wavemoving in the opposite direction. Because the points of zero displacement remainfixed, the resultant wave pattern constitutes what is known as standing waves. Thedistance between nodal points for the nth harmonic mode of operation is L/n,andthe points of maximum vibrational amplitudes are referred to as antinodes or loops.Let us take a “snapshot” of the vibrating string at a particular time for the sixthharmonic mode (Figure 4.4). The displacement of the string frozen in time occursas a sinusoidal function of x.This function repeats itself every length 2L/n of thex-coordinate, which, in turn, is equal to the wavelength λ n of the harmonic waves.In this case of the sixth harmonic mode, the repetition occurs every L/3 of thestring length. The wavelength is related to the velocity of propagation byλ n = c/f n