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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 , 0 < ε ≪ 1. In the DEE method, ε = 1/λ, where λ is the nondimensional frequency. As a result, the construction of the asymptotics can be reduced to solving the eikonal and transport equations (Birger and Panovko 1968). However, short-wave (high-frequency) asymptotics can be treated as a multiscale approach (Maslov and Fedoryuk 1981).

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

[1.56]

[1.57]

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:

[1.58]

Substituting ansatz [1.58] into equation [1.56] and applying ε-splitting, we obtain a recurrent system of equations:

[1.59]

[1.60]

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:

[1.61]

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:

[1.62]

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.