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

4.11 Strings of Finite Lengths: Forced Vibrations 83is analogous to the characteristic electrical impedance of an infinite transmissionline.The average power input to the string is found from the average value of theinstantaneous power W = fv evaluated at x = 0, orW = F 22δc = δcV 022where V 0 is the velocity amplitude at x = 0.4.11 Strings of Finite Lengths: Forced VibrationsIn the case of a finite string, the reflections from the far end generate frequenciesthat cause the input impedance to change greatly with the frequency of the drivingforce. If the support at finite x = L is fully rigid and no dissipative forces occurin the string, the input impedance becomes a pure reactance, and no power isconsumed in the string.Because the complex expression for transverse waves on a finite string now needsa term descriptive of the reflected wave, Equation (4.21) needs to be rewritten asy = Ae i(ωt−kx) + Be i(ωt+kx) (4.25)for all times t. The boundary condition at x = 0is( ) ∂yFe iωt =−T∂x x=0for all values of t. Inserting Equation (4.25) into Equation (4.26) yields(4.26)F =−T (−ikA + ikB) (4.27)Applying y(L, t) = 0 for the rigid clamp at x = L, Equation (4.25) becomes0 = Ae −ikL + Be ikL (4.28)Solving the Equations (4.27) and (4.28) for A and B results inA =FikT ·e ikLe ikL + e −ikL =Fe ikL2ikT cos kLandB =− FikT · e −ikL=−Fe−ikLe ikL + e−ikL 2ikT cos kLSubstituting the above constants into Equation (4.26) yieldsy = FeiωtikT · eik(L−x) − e ik(L−x)= Feiωt2 cos kL kT· sin k(L − x)cos kL(4.29)The real portion of Equation (4.29) graphs a pattern of standing waves on thestring with nodes occurring at those points where sin(L − x) = 0, in addition to