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

References 854.12 Real Strings: Free VibrationReal strings manifest some degree of stiffness, causing the observed frequenciesto be higher than the theoretical values for idealized strings. This results from thepresence of elastic boundary forces augmenting the action of tensile forces consideredpreviously, with the net effect of increased restoring forces. The presenceof stiffness exerts a greater influence with increasing frequency, and the overtonesof a stiff string no longer constitute an exact harmonic series.We must also be attentive to the fact that clamping at the ends of the string maynot be exactly rigid and that yielding can occur at these points. The wave impedanceat the ends will constitute the transverse mechanical impedance of the supports.Consider a case where the left end of a finite string at x = 0 is attached to apivot representing the slightly loose clamp. The transverse force f 0 exerted by thestring on the hinge is( ) ∂yf 0 = T sin θ ≈ T∂x x=0Here y is the complex expression for the transverse wave on the string, as givenby Equation (4.25). Because the motion of the swivel must match that of the endof the string, the velocity is( ) ∂yv 0 = = iωy 0∂t x=0Let z 0 denote the transverse mechanical impedance of the hinge at x = 0( ) ∂yz 0 = f 0= Tv 0 iω · ∂x x=0and hence the boundary condition at x = 0 becomesy 0y 0 = T (∂y/∂x) x=LiωZ Lwhere Z L is the mechanical impedance of the swivel located at x = L. Ifthesupports were truly rigid, Z 0 = Z 1 = 4, and the boundary conditions reduce toy 0 = y L = 0.ReferencesFletcher, Neville H. and Rossing, Thomas D. 1998. The Physics of Musical Instruments,2nd ed. New York: Springer-Verlag, Chapter 2. (This text provides an excellent expositionon the physical principles of musical instruments.)Jean, Sir James. 1968. Science and Music. New York: Dover Publications, Chapter 3.(Although rather skimpy on the mathematical details, the exposition provides a goodinsight into the physics of vibrating string to the reader.)