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

84 4. Vibrating Stringsy(0, L) = 0. The displacement at x = 0, however, has an amplitudey 0 = (F tan kL)/kT (4.30)Singularities of Equation (4.30) occur when cos kL = 0, i.e.,orandkL = ωLc(2n − 1)π= , n = 1, 2, 3,...2ω n =(2n − 1)πc2L(2n − 1)f n = c4LThese singularities connote infinite amplitudes which, of course, do not occurin real strings, because dissipative forces neglected in the foregoing analysis doactually exist. But the amplitudes do achieve maximum values at these frequencies.In a similar fashion we can ascertain the minimum amplitudes from the conditionkL =±1, i.e.,andkL = ω n L/c n = 1, 2, 3,....ω n = nπcLorf n = nc2LThe minimum amplitudes decrease progressively with increasing frequencies.In fact, on comparing with Equation (4.10) it is noted that the frequencies ofminimum amplitudes are identical to those of the free-string vibration, and theterm antiresonance has been applied to describe those frequencies. Differentiationof Equation (4.29) with respect to time t yields the complex velocity v of thestring,v = FeiωtT/c· sin k(L − x)cos kLThe input mechanical impedance then becomesZ s = feiωtv= T iccos kLsin kL=−iδc cot kLwhich exists as a pure reactance, with no power absorbed by the string. The amplitudeof vibration is a maximum at cot kL = 0, which occurs at the frequencygiven by f n = nc/2L. For extremely low frequencies the input impedance has thelimitsZ s =− iδckL =−iT ωL