Besides, according to the radial and axial
wave number and penetration number, the samples can be further divided into three categories, namely, (1) Group 1: radial thick-axial medium samples, whose radial penetration number is [N.sup.r.sub.p] > 3 and the axial
wave numbers and penetration numbers are [N.sup.a.sub.w] > 0.1 and [N.sup.a.sub.p] < 3; (2) Group 2: radial medium-axial medium samples, whose
wave numbers and penetration numbers in radial and axial directions are all [N.sub.w] > 0.1 and [N.sub.p] < 3; (3) Group 3: radial medium-axial thick samples, whose radial penetration numbers are [N.sup.r.sub.w] > 0.1 and [N.sup.r.sub.p] < 3, and axial penetration numbers are [N.sup.a.sub.p] > 3.
It is worth to note that, due to the limitations of the standard finite element method in solving Helmholtz equations with high
wave numbers, to solve the Helmholtz problem (9) by FEM, the time step r should be carefully chosen, in order to ensure the
wave number in (9) is not high, and for high dimension problem, dense grids need to be generated to ensure accuracy, resulting in increased computational time.
These plots show, as expected, that the distribution [rho](k) is strongly peaked at the
wave number [k.sub.0], corresponding to the energy of the initially bound state and is negligible for nontunneling components.
This patch consists total (m + n)/2 ripples having the same
wave number [gamma] and amplitude a.
For numerical solving the wave dispersion relations, it is practically to normalize all variables and we do that normalizing the wave length, [lambda] = 2[pi]/[k.sub.z], to the tube radius, a, that implies a dimensionless
wave number [k.sub.z]a.
The complex frequency [omega]* is solved as function of the
wave number k* and the parameter set ([LAMBDA], [S.sub.T], M, and [phi]).
[??] is defined by [mathematical expression not reproducible], and k is the electromagnetic
wave number. [F.sub.pq] is the polarization factor.
In each case, the Raman scattered signal got shifted to 455, 424, and 380 nm; but the shift in their
wave number approximately corresponded to 3060 [cm.sup.-1], confirming the Raman effect.
where [A.sub.SH] and [B.sub.SH] are the amplitudes of the ascending and descending SH waves in the soil layer, respectively, k = [omega] cos [[psi].sup.L.sub.SH]/[c.sup.L*.sub.s] is the
wave number along the x-direction, and t = -i[square root of (1 - 1/[cos.sup.2][[psi].sup.L.sub.SH])] and [[psi].sup.L.sub.SH] are the included angles of the ascending or descending SH waves in the soil layer in the horizontal direction