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

4Vibrating Strings4.1 IntroductionIn dealing with vibrating systems it is commonly assumed that the entire mass ofthe system is concentrated at a single point and the motion of the system can bedescribed by giving the displacement as a function of time. This rather simplifiedapproach yields approximations rather than accurate closed-form solutions. Aspring, for example, certainly does not concentrate its mass at one end, nor cana loudspeaker be accurately depicted as being a massless piston engaged in anoscillating motion. The loudspeaker diaphragm consists of a considerable portionof its mass spread out over its surface, and each part of the diaphragm can vibratewith a motion that is different from those of other segments.The vibrational modes of a loudspeaker constitute a complex affair, so it wouldbehoove us to study simpler modes of vibration, say, those of a vibrating string orbar, so that we can readily visualize the transverse vibrations. Even in the simplestof cases, certain simplifying assumptions have to be made which cannot be fullyjustified in the real physical world.4.2 The Vibrating String: Basic AssumptionsConsider a long, heavy string stretched to a moderate tension between two rigidsupports. A momentary force is applied to the string that becomes displaced fromits equilibrium position. The displacement does not remain in the initial position;it breaks up into two separate disturbances that propagate along the string apartfrom each other as shown in Figure 4.1. The propagation velocity of all smalldisplacements depends only on the mass and tension of the string, not on the shapeand amplitude of the initial displacement. The wave generated by such a transverseperturbation is generally known as a transverse wave.4.3 Derivation of the Transverse Wave EquationIn Figure 4.2, a portion of a string under tension T and rigidly clamped at itsends is shown. The string has negligible stiffness and a uniform linear density δ.71