Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur
LECTURE-13 WAVE PROPAGATION IN SOLIDS
Longitudinal Vibrations In Thin Plates Unlike 3-D solids, thin plates have surfaces which may be free from constraints, thereby developing Poisson strains. Thus in plates, longitudinal strains will produce lateral strains due to Poisson contraction. As a consequence, pure longitudinal waves can not exist in thin plates and thus waves in one direction will generate disturbances in other directions as well. Hence, waves in thin isotropic plates are termed as quasilongitudinal.
For thin plates, deriving the wave equation for quasilongitudinal disturbances involves the same steps as that needed for longitudinal waves in a 3-D solid, except the fact that stress-strain relationship for plates is different. It is known from elasticity theory that longitudinal stress and longitudinal strain in a thin plate are related as: Eq. 13.1
Thus the quasi-longitudinal wave equation for a thin plate is: Eq. 13.2 Here c l is the wave propagation velocity in an isotropic homogenous plate and is expressed as: Eq. 13.3
Longitudinal Vibrations In 1-D Bars For long 1-D homogenous isotropic bars, the wave propagation equation is very similar to that for a thin plate, except for the fact that its development requires a different stress-strain relationship, which is: Using this relationship, we get 1-D wave equation for bar as: Eq. 13.4 where, Eq. 13.5
Transverse (Shear) Waves In Solids Unlike fluids, solids can resist shear deformation as well. In case if fluids, shear stress are associated with flow gradients, and then gradients are driven by viscous dissipative effects. Thus, shear waves generated in fluids dissipate rapidly and hence are not of much importance. However, in solids, it is the shear modular G, which couples shear stress to shear strain. Thus shear waves in solids do not dissipate as rapidly as in case of fluids.
Consider a small accelerating material element subjected differential shear stress as shown in Fig. 13.1.
Here the element experiences shear stress τ xy on its left face, and on its right face. Because of the imbalance of their shear stresses, the material element accelerates in y direction and this motion is governed by Eq. 13.6. or Eq. 13.6 Further, the application of shear stress on left and right faces causes the material element to distort by angle r, and the governing equation for it is:
or Eq.13.7 Combining Eqs. 13.6 and 13.7 we get: or Eq.13.8 Eq.13.9 where, Eq.13.10 Thus, shear waves travel in 2-D solids with a speed of (G/ρ) and this speed is smaller than that of quasi-longitudinal waves.
Torsional Waves In Bars Bars when subjected to torsional forces, exhibit torsional waves. Such torsional waves are essentially shear waves. The governing equation for such waves is: where, Eq. 13.11 θ=torsional twist of bar at a given cross-section. c t =Speed of torsional shear wave I p =Polar moment of inertia per unit length of bars. GJ=Torsional stiffness.
Bending Waves In Bars Bending waves in bars (and plates) are of great practical significance from an acoustical stand point. This is so because bending waves in solids can generate significant transverse displacements (w.r.t. propagation direction of bending waves) and these transverse displacements of significant amplitude can very effectively disturb adjacent fluid to generate considerable sound levels. Further, bending waves are associated with transverse impedance which may be similar magnitude as that of sound wave in adjacent fluid. This implies that the energy exchange levels in solid and fluid media associated with bending waves may be significant.
Bending Waves In Bars The bending wave equation for a bar can be written as: Eq. 13.12 where, E=Young s modulus of material I=Moment of inertia of bar at cross-section of interest m=mass per unit length of bar ɳ=Transverse displacement of bars at position x at its neutral axis. Unlike longitudinal waves, shear wave, and torsional wave in solids, where the governing equation is a 2 nd order PDE in x and t, governing equation for bending waves for bars, Eq. 13.12 is a 4 th order PDE in x, and 2 nd order PDE in t.
It is because of this reason, that the propagation velocity of bending waves, c b, is determined to be: Eq. 13.13 Thus, unlike other waves discussed earlier, bending waves propagation velocity depends on angular frequency of excitation. Hence if a bar is excited in bending at several frequencies, then all of these waves will disperse as they propagate through the media. Thus, bending waves are dispersive in nature. Finally, the solution for bending waves excited by a simple harmonic source, contains 4 terms. This is in contrast with the solution for longitudinal shear or torsional waves.
Out of these four terms, two represent waves travelling in +ve and ve directions at a speed of. The other two terms represent non-propagating fields, and their amplitudes decay with distance. The velocity of the decaying fields is imaginary. Further these decaying fields do not transport energy and hence do not qualify as waves. However, in literature, they are referred as evanescent waves or alternatively as near fields.
References Acoustics, Beranek Leo L., Acoustical Society of America, 1993. Introduction to Acoustics, Finch Robert D., Pearson Prentice Hall, 2005. Fundamentals of Acoustics, Kinsler Lawrence E., et al, 4 th ed., John Wiley & Sons, 2005. Sound and Structural Vibration, Fahy Frank, et al, 2 nd ed., Academic Press 2007.