if
where T(t) and Y(x, t) are the periodic and quasi-periodic functions in time, respectively; and function Y(x, t) is small compared to function X(x)T(t) in some energy norm. The last condition can be verified both a priori and a posteriori.
1.7. Short-wave (high-frequency) asymptotics. Possible generalizations of DEEM
DEEM can be considered a special case of short-wave (high-frequency) asymptotics. The corresponding algorithms are known as the method of geometric optics, the ray method, the semi-classical approximation, the WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) approach, the method of edge waves, the Keller–Rubinow method, etc. (Keller and Rubinow 1960; Maslov and Fedoryuk 1981; Babich et al. 1985; Babich and Buldyrev 1991; Chen et al. 1991, 1992; Chen and Zhou 1993; Bauer et al. 2015). They were independently developed in various fields of mathematics, mechanics and physics.
Note an interesting fact: Ufimtsev proposed the asymptotic method of edge waves (Ufimtsev 1962, 2003, 2014). According to Rich and Janos (1994) and Mitzner (2003), this theory played a critical role in the design of American stealth aircrafts F-117 and B-2. It is a fascinating example of the direct application of asymptotic formulas in engineering practice!
The key to short-wave asymptotics is the ansatz φ(x)exp(iε−1S (x)), in the nonlinear case – φ(x)Φ(iε−1S (x)), where
Let us show the generalization of DEEM based on the WKBJ approach using the toy problem – natural oscillation of a beam of variable cross-section (Bauer et al. 2015):
[1.55]
with clamped edges.
In dimensionless variables, we obtain
where
Functions φ1,φ2 are supposed to be smooth enough to avoid turning points.
The solution to equation [1.56] is sought in the form:
Substituting ansatz [1.58] into equation [1.56] and applying ε-splitting, we obtain a recurrent system of equations:
The eikonal equation [1.59] has the following solutions:
From the transport equation [1.60], we obtain
In the first approximation, the general solution of equation [1.56] can be written as follows:
In expression [1.61], function u0(ξ) is “frozen” at either end of the interval (i.e. we can change u0(ξ) to u0(0) or u0 (ξ) to u0 (1)) for rapidly decaying components).
Using solution [1.61] and boundary conditions [1.57], we obtain the frequency of oscillations:
Formula [1.62] at φ1 = φ2 = 1 coincides with Bolotin’s formula [1.18]. Thus, the WKBJ method generalizes the DEE method to the problems with variable coefficients.
As it is mentioned in Chen et al. (1991, 1992), the short-wave (high-frequency) asymptotics gives the same results as the DEE approach for domains of simple geometry. At the same time, DEE does not cover the cases of different geometries (circular, elliptical, etc.) or non-self-adjoint problems. The short-wave asymptotics in the form of the Keller–Rubinow approach (Keller and Rubinow 1960) in Chen et al. (1991) allows more ready extension to other geometries and is more aptly generalizable to dissipative boundary conditions. In other words, it gives the possibility to overcome degeneration of the DEE case.
1.8. Conclusion: DEEM, highly recommended
The importance and usefulness of a particular calculation method is determined by its wide application when studying practically important systems and phenomena. From this point of view, the importance of DEEM is not in doubt.
From the very beginning, DEEM was originated by Bolotin for the analysis of overhead power lines (Bolotin et al. 1958).
DEEM was used to obtain estimates for the density of natural frequencies of shallow shell vibrations (Gavrilov 1961b; Bolotin 1963; Stearn 1970; Zhinzher and Khromatov 1971, 1972a, 1972b; Moskalenko 1972, 1975). This is very important during the study of random vibrations of elastic structures (Bolotin 1966; Birger and Panovko 1968; Moskalenko 1968; Bolotin 1984; Elishakoff et al. 1994).
The