For convenience of post-processing of modal results eigen vector [[phi].sub.m] is split by separating nodal degrees of freedom (dof) with respect to displacements [u.sub.n] on [GAMMA], that correspond to the longitudinal wave
in the fluid (denoted by L), displacements [u.sub.[tau]] on [GAMMA], that would excite shear wave (denoted by S), displacements [u.sub.z] along OZ axis on entire cross-section (denoted by z) and introduced displacements [u.sub.C] in the core of cross-section (denoted by C) (Figure 2(b)).
One of transducers acts as the transmitter sending the bulk wave (mainly, longitudinal wave
) across the specimen, and the other transducer plays the role as the receiver based on the principle of electromagnetic induction.
Before soaking in the solutions, the specimens were dried in an electric oven at 105[degrees]C for 24 h, and then, the initial dry weight, longitudinal wave
velocity, and compressive strength were recorded.
Relations (15), (18), and (19) display how a longitudinal wave
changes the acoustic index of a medium.
[c.sub.3] in (19) is the velocity of the longitudinal wave
and it can be found using the following equation:
Compared with those at room temperature, the longitudinal wave
velocity, density, and dynamic tensile strength were less to different extents after heat treatments, while the physical mechanical geometrical factors dropped linearly with increasing heating rate.
[b.sub.17] = 2[[beta].sub.13] + [[beta].sub.33]/[c.sub.33] (17) means that the longitudinal wave
is generated due to thermoelastic effect.
According to the experimental measurement result, the spread speed of 64 K ultrasound in drilling mud with a density of 1.1 g/[cm.sup.3] is 1380 m/s, where as that of the ultrasound longitudinal wave
in the steel material is approximately 5850 m/s, and the f transverse wave is 3230 m/s.
Larger changes in ultrasonic longitudinal wave
velocity were detected for unmodified POM while for POM/HAp nanocomposites with 5% of HAp longitudinal wave
velocity was almost constant even after [10.sup.6] cycles, proving an enhancement of the mechanical properties stability by HAp nanoparticles.
where [V.sub.pj], [p.sub.pj], and [E.sub.pj] denote the elastic longitudinal wave
velocity, density, and elastic modulus of the jth pile (including fictitious soil pile) segment, respectively.
The above estimates indicate that identify in vitro particular longitudinal wave
current distribution in the metal conductors can only mode transmission there through of DC, AC or pulse current [i.sub.0](t) which provide a flow in cross sections [S.sub.0] of conductive material a relatively large current densities [[delta].sub.0m].
The [u.sup.v] displacement wave equations can be expressed as a longitudinal wave
equation for the dilatation [epsilon] and a transverse wave equation for the rotation tensor [[omega].sup.[mu]v] .