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

5Vibrating Bars5.1 IntroductionThe theory underlying the physical operation of vibrating bars is of great interest toacousticians, because a number of acoustic devices employ longitudinal vibrationsin bars and frequency standards are established by producing sounds of specificpitches in circular rods of different lengths. The analysis of vibrating bars facilitatesour understanding of acoustic waves through fluids, for the mathematicalexpressions governing the transmission of acoustic plane waves through fluid mediaare similar to those describing the travel of compression waves through a bar.Moreover, if the fluid is confined inside a rigid pipe, the boundary conditions beara close correlation to those of a vibrating bar. An example of devices falling intothe category of vibrating bars include piezoelectric crystals which are cut so thatthe frequency of the longitudinal vibration in the direction of the major axis of thecrystal may be used to monitor the frequency of an oscillating electric current orto drive an electroacoustic transducer.The principal mode of sound transmission in bars is through the propagationof longitudinal waves. Here the displacement of the solid particles in the baroccurs parallel to the axis of the bar. The lateral dimensions of a bar are smallcompared with the length, so the cross-sectional plane can be pictured as movingas a unit. In reality, because of the Poisson effect that generally occurs in solidmaterials, the longitudinal expansion of the bar results in a lesser degree of lateralshrinkage and expansion; but this lateral motion can be disregarded in very thinbars.5.2 Derivation of the Longitudinal Wave Equation for a BarIn Figure 5.1, a bar of length L and uniform cross-sectional area  is subjectedto longitudinal forces which produce a longitudinal displacement ξ of each ofthe molecules in the bar. This displacement in long, thin bars will be the same ateach point in any specific cross section. If the applied longitudinal forces vary ina wavelike perturbative manner, the displacement ξ is a function of both x and t89