Page 15 - Fall 2008

Fig. 5. (a) The modes of vibration for a string with zero scatterers. The zero frequency equilibrium position of the string is shown as mode “a.” The vertical position of the dashed line for the different modes is proportional to the frequency of the mode; the harmonic sequence of frequencies is evident. Each frequency has two “degenerate” modes, corresponding to cos(2πx/λ) and sin(2πx/λ). (b) The result of placing point mass scatterers on the string, dividing the length shown into four equal parts, with the mass of the scatterers being infinite. Some of the modes are “suppressed” down to the equilibrium state “a” from Fig. 5 (a). Some of the modes from Fig. 5 (a) are unchanged, because the infinite masses wind up at nodal points, as for modes “i” and “q.” Other modes have their modes “pulled” so as to become equivalent to “i,” “q,” etc.
with the string free of scatterers) form a harmonic sequence, given by f = νc/2l., ν= 0, 1, 2,···. The zero frequency given by ν=0 corresponds to the equilibrium position of the string, with no transverse displacement. In our treatment, this zero frequency equilibrium position will actually be counted as a mode.
With the notation and conditions established above, the first seventeen relevant modes of vibration of a string that is free of scatterers is shown in Fig. 5a. The letters labeling the different modes will be used for later reference. The zero fre- quency equilibrium state of the string with no transverse dis- placement is shown as mode “a.” For each mode, the dashed line indicates the equilibrium position of the string. The positions where the string crosses the dashed line are called “nodal points,” that for standing waves never move. The fig- ure is drawn so that the vertical position of the dashed line of each mode is proportional to the frequency of the mode. For the plain string, the frequencies are harmonic, so the modes are drawn in the figure with equal vertical spacing. At each vertical level (or frequency) there are two modes given by
cos(2πx/λ) and sin(2πx/λ) that have the same frequency. Modes that have the same frequency are said to be “degener- ate.” In aid of understanding what happens when scatterers are added, one should take particular note of modes “i” and “q” in Fig. 5a.
Infinite masses
The next step in understanding the effect of periodic scatterers is to jump from the extreme case of the string with no scatterers (m=0), to the opposite extreme case with the string having point mass scatterers, where the masses are infinite, that is m = ∞. With reference to Fig. 5b, infinite masses will be placed on the string with a lattice constant a = l / 4, beginning with the leftmost end of the section of string. In our example, five infinite masses will be placed on the string, dividing it into four equal segments as shown in Fig. 5b. Because an infinite mass cannot be moved, any mode of the string that “tries” to move at the site of an infinite mass will be completely suppressed. Thus, with reference to the labeling in Fig. 5a, modes “b” through “h” are suppressed
14 Acoustics Today, October 2008