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

82 4. Vibrating Stringswhere A is a complex constant of which magnitude equals the displacement amplitudeof the wave motion and whose phase angle renders the difference in phasebetween the motion of the string and the driving force.In complex format the harmonic driving force can be written asf = Fe iωt (4.22)In Figure 4.5, the driving force is shown being applied to the string at an angle θthe string makes with the horizontal. This angle is given by( ) ∂ytan θ =∂xThe force exerted in the horizontal direction at the support at the end of the stringis −T cos θ. Because the displacements are assumed small, cos θ ≈ 1, and themagnitude of this force in the horizontal direction is essentially equal to tensionT in the string. From similar considerations, the transverse force exerted by thesupport on the string is −T sin θ, approximated by( ) ∂yf =−T sin θ =−T(4.23)∂x x=0Equation (4.23) indicates that for any applied transverse force the shape of thestring at x = 0 will vary. Inserting f and y from Equations (4.21) and (4.22) intoEquation (4.23) yields for this boundary conditionorx=0Fe iωt =−T (−ik)Ae i[ωt−k(0)]A =F(4.24)ikTThe term F/kT represents the magnitude of this complex amplitude A. InsertingEquation (4.24) into Equation (4.21) and then differentiating with respect to timeresults in the complex velocity vv = F( cT)e i(ωt−kx)The mechanical (or wave) impedance Z s of the string is defined as the ratio of thedriving force to the transverse velocity of the string at x = 0:Z s = T c = √ T δ = δcIt turns out that Z s is a real quantity, with no imaginary load. The mechanicalload presented by the string to the driving force is purely resistance. The inputimpedance exists as a function of the linear density δ and the tension appliedto the string, and it does not depend on the applied driving force; this means itis a property characteristic of the string, not the wave propagation in the string.The input, often termed characteristic or mechanical impedance (or resistance)