The Velocity of Compressional Waves in Rods of Rectangular Cross Section

Abstract

Solutions for compressional waves in an isotropic rod of rectangular cross section are formulated from the general elastic equations of motion. Two general modes of propagation are constructed which satisfy the boundary conditions for the free vibrations of a long rod for which the width 2d is considerably greater than the thickness 2a. The thickness mode is shown to have an infinite phase velocity (cut-off frequency) at that frequency for which the half wave-length in an infinitely wide rod would be equal to 2d. The phase velocity of both of these propagation modes is shown to approach that of Rayleigh surface waves when the wave-lengths are much smaller than the lateral dimensions. The calculated thickness mode cut-off frequencies are compared with measurements reported earlier and the agreement is shown to be good. A comparison between the predicted and measured phase velocity dispersion is made for various values of aid. The agreement is found to be satisfactory as long as d is several times greater than a.

Mechanical and thermal properties of ceramics :

“…This point of view has been developed by LeCorre [91] and by Joel Further discussion of the Laval-Raman approach can be found in Huntington's review paper [7]. The propagation of elastic compressional waves in rectangular bars [103], in impulsively loaded cylindrical rods [104], and in explosively loaded plates [105] has also been studied. Truesdell [106] has discussed the propagation of waves in an isotropic elastic material subject to a finite static strain.…”

“…The apparatuses described so far can be used when there is Probably the largest uncertainty in comparative methods arises from failure to meet the basic underlying requirements of eqs (106) and (107) By analogy with the development of eqs (98) to (100) (103) and (104) + errors due to failure to meet requirements of equation (112) where the terms are analogous to those following eq (105 [71] and Moore, McElroy, and Graves [72].…”

“…[51] in which dead weights If it is intended to determine, as closely as possible, the "true" tensile strength of a brittle material, the work of Bortz and Wade [7] merits careful study. On the other hand, the designer concerned with a particular piece of hardware would be wise to follow Boland and Walton's [44] Ternary systems, 28,32 Test data (see also Methods of testing), [14][15]243 Test procedures (see also Methods of testing), 14-15, 133, 243 Thermal activation, 203 Thermal conductivity, 3, measurement methods, [96][97][98][99][100][101][102][103][104][105][106][107][108] Thermal diffusivity, 64 Thermal expansion, 41,94 Thermal expansion anisotropy, stresses due to, 94,230 Thermal history, effect on thermal conductivity, [94][95] Thermal properties, IV, …”

Parallel strip waveguide for ultrasonic flow measurement in harsh environments

“…Therefore other techniques must be used to more accurately model the propagation of ultrasound in such finite width strips. several techniques have been used previously to study the propagation of ultrasonic waves in such narrow strips including semi-analytical finite element modeling [14] and various analytical models [15], [16]. These earlier works have shown that such strips can support lamb-like modes.…”

On Coupled Longitudinal and Lateral Vibrations of Elastic Rods

“…Speci"cally, Chree [3] in 1889 discussed an approximate dispersion relation for long wavelength modes in an in"nitely long body with an arbitrary cross-section. Later, Morse [4,5] and Kynch and Green [6] examined this problem for rectangular cross-sections. Most of their results pertain to large wavelength vibrations.…”