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

80 4. Vibrating StringsTherefore, all even modes n = 0, 2, 4,...haveA 2 = A 4 = A 6 =···=0and the odd modes result in non-zero A n coefficients:A 1 = 8dπ , A 2 3 =− 8d9π , A 2 5 = 8d ,25π2etc. (4.19)The amplitudes of the various harmonic modes are given by the numerical valuesofA n . In general, it may be observed that no harmonics are generated having a nodeat the point of the string initially plucked. As the nodal number n increases, theassociated amplitudes decrease from the value of the fundamental amplitude, i.e.,the fundamental A 1 is 9 times larger than A 3 and 25 times larger than A 5 , and so on.Example 3: Sharp Blow Applied to String. If the string is struck a sharp blow (asopposed to being plucked, as described above), v 0 (x,0) has nonzero values but noinitial displacement exists. Then all the coefficients A n are zero and the coefficientsB n are given by Equation (4.17b). A common example of a struck string is theimpact of a piano hammer striking a string. It is interesting to note that pianos aredesigned in such a way that the impact point of the hammer is one-seventh of thedistance from one end of the string, thus eliminating the seventh harmonic (whichwould have produced a discordant sound).4.9 Energy of Vibrating StringIn any nondissipative system the total energy content remains constant, equal to thevalue of the maximum kinetic energy. For the nth mode of vibration the maximumvalue of the kinetic energy of a segment of length dx isdE n = ω nδ (A22 n + Bn2 )sin 2 k n xdx (4.20)which is established by simply applying the relation dE = (mv)dv/2 in conjunctionwith Equation (4.6) and the fact that the mass of the string element is givenbym = δ dxwhere δ is the linear density. With integration over the variable x from 0 to L, themaximum kinetic energy of the string isE n = ω2 n δ (A22 n + Bn) 2 L2 = m (4 ω2 n A2n + Bn)2Here m is the total mass of the string and (A 2 n + B2 n ) is the square of the maximumdisplacement of the nth harmonic. In a conservative system (which describes thedissipationless vibrating string) the maximum potential energy is also equal to themaximum kinetic energy of the system. From Equation (4.20) the energy of the