The Malyuzhinets—Popov diffraction problem revisited

It is natural to characterize the position of an observation point $\mathcal{M}$ near point $\mathcal{O}$ by coordinates $(s,n)$, where $n\geq 0$ is the length of the perpendicular from $\mathcal{M}$ to the contour $\mathcal{C}$ and $s$ is the length of the arc between $\mathcal{O}$ and the foot of the perpendicular, as shown in Fig. 2. These coordinates are orthogonal; however, in contrast to the classical case [7, 9], the mapping $(x,y)\mapsto(s,n)$ is not smooth.

Furthermore, $s$ and $n$ and, accordingly, $x$ and $y$ will be small in comparison with the curvature radius $h^{-1}$. We use the following elementary relation which results from (4) by a simple calculation:

We also use the expression for the Laplacian [9]:

We start with the observation (traceable to Leontovich and Fock) that, as in the case of the tangential incidence of a plane wave at a smooth obstacle, in the vicinity of point $\mathcal{O}$

where $V$ oscillates slower than the exponentials. This inspired us to follow the classical research strategy [7, 8, 9] based on separating out a rapidly oscillating factor and introducing stretched coordinates, leading to a problem for the so-called parabolic equation.

We introduce dimensionless stretched coordinates by

with the same powers of $k$ as in the original definition of Leontovich and Fock [7]. The dimensionless large parameter

of our problem is in agreement with that of Fock. We seek the outgoing wavefield in a form similar to that of the Leontovich–Fock ansatz [7]:

where the attenuation factor $U$ oscillates slower than $e^{iks}$ and the dots stand for smaller-order terms with respect to the large parameter $k/h$.

Formulas connecting coordinates $(\sigma,\nu)$ with the Cartesian coordinates follow from (9):

These formulas entail useful relations involving polar coordinates $r=\sqrt{x^{2}+y^{2}}$ and $\varphi=\arctan y/x$

With the help of (15a), we rewrite the factor $V$ in (11) as

and, following Fock, require the smallness of the remainder terms. This implies the restrictions on distance from the singular point $\mathcal{O}$ which can be summarized as

Further research refers to the area characterized by the conditions (19).¹¹1Limitations for the corresponding area in the Fock problem, presented in [9], were different because the speed of propagation was assumed nonconstant.

Rewriting (10) in stretched coordinates (12), expanding the coefficients in powers of $(h/k)^{\frac{2}{3}}$ and substituting (18) and (14) into the Helmholtz equation (1) and boundary condition (3), we immediately obtain the problem for the main term $U_{0}$ of the outgoing wave:

We note that in a remarkable paper by A. Popov [1], which we followed up until now, the exponential on the right-hand side of (20b) was missing, and thus, his further results require refinement.

We seek the main term of the attenuation factor $U_{0}$ in the form of a Fourier-type integral as follows:

Using the representation for the Heaviside function

where $(\xi-i0)^{-1}:=\lim\limits_{\varepsilon\to+0}(\xi-i\varepsilon)^{-1}$, and the convolution theorem [30], we arrive at

Here,

is an inhomogeneous Airy function (A), and ^′ denotes the differentiation with respect to $\xi$. The function $I(\xi)$ vanishes in the lower half of the complex plain $\mathbb{C}^{-}$ as $|\xi|\to\infty$.

Assume that $\widehat{U}_{0}$ is analytic with respect to $\xi$ and decreases to zero in $\mathbb{C}^{-}$ as $|\xi|\to\infty$.²²2This assumption is a form of causality condition and is equivalent to the condition $U_{0}(\sigma)=0$ for $\sigma<0$. The integral in (23a) can be evaluated with the help of Jordan’s lemma and residue theorem, and problem (23) takes the form

The desired solution of (25) is

where $w_{1}$ is the Airy function in Fock’s definition (A). We choose this particular solution of the Airy equation (25a), guided by the same argument as in Fock [7] and Babich and Kirpichnikova [9], as it corresponds to the outgoing wave for the harmonic time-dependence $e^{-ikt}$, which we omit. Therefore,

The zeroes of $w_{1}^{\prime}$ are located at the upper half-plane; thus, the integrand is analytic with respect to $\xi$ in $\mathbb{C}^{-}$ and decreases there as $|\xi|\gg 1$ (A). Hence, $U_{0}\equiv 0$ as $\sigma\leq 0$, and hereinafter we assume $\sigma>0$.

The main term of the attenuation factor for the total wavefield is, according to (6), the sum of the main terms of (14) and (18):

This can be equivalently rewritten in the form

convenient for studying the wavefield in the shadow zone.

The expression (27) is obtained under the assumption that the inequalities (19) are satisfied. When neither $\sigma$ nor $\nu$ is large (the area indicated in Fig. 3 by $\mathcal{D}_{1}$), the integral does not allow for a high-frequency asymptotic simplification. Henceforth, we investigate (27) in areas $\mathcal{D}_{2}$–$\mathcal{D}_{6}$ where at least one of $\sigma$ and $\nu$ is large,

The condition (30) guarantees that all integrals that will be encountered below will contain a large parameter and can be subjected to an asymptotic analysis.

The foregoing analysis relies on a well-developed asymptotic evaluation technique of the integrals of rapidly oscillating functions [31]. For different parts of the integration interval, all or some special functions in (27) can be replaced by their approximations. We will transform expression (27) using the approach of Fock [7], Brown [8] and Babich and Kirpichnikova [9], implying the simplification of integrands in small vicinities of critical points of respective phase functions. We matched the asymptotics of (27) derived in the well-illuminated area $\mathcal{D}_{2}$ with the cylindrical wave (7), which yields an expression for the diffraction coefficient. The approximation of (29), which we constructed inside the deep shadow area $\mathcal{D}_{6}$, allows matching with the Friedlander–Keller formulas.

Neighboring areas defined by the corresponding inequalities intersect in the vicinity of their common boundaries, and the respective asymptotic formulas that we construct match in these overlapping zones.

On the half-line $-\xi\gg 1$, all special functions in (27) can be replaced by their approximations (A). The integrand takes the form

with the phases

The phase $\Psi_{1}$ has a unique critical point $\xi_{1}$ where $\Psi_{1}^{\prime}(\xi_{1})=0$:

Under the condition (31), $\xi_{1}\gg 1$, whence the first term on the right-hand side of (33) has no critical point on the half-line under consideration and its contribution is negligible.

Now, we address the second term. The equation for the critical point $\xi_{2}$ of $\Psi_{2}$ is

whence $(-\xi)^{\frac{1}{2}}=(\nu-\sigma^{2})/(2\sigma)$. Because the inequality (31) guarantees that the right-hand side is positive and large, the phase has a critical point

on the half-line under consideration.

On the interval $|\xi|<const$, only the function $w_{1}(\xi-\nu)$ in (27) can be replaced by its approximation (102), which gives

with $\Psi_{1}$ introduced in (34). Under condition (31) its critical point is far outside the interval and the respective contribution can thus be ignored.

We split the half-line $\xi\gg 1$ into three pieces: $1\ll\xi\ll\nu$, $|\xi-\nu|<const$ and $\xi\gg\nu$.

Where $1\ll\xi\ll\nu$, we can substitute into (27) asymptotic expressions for $w_{1}(\xi-\nu)$, $I^{\prime}(\xi)$ and $w_{1}^{\prime}(\xi)$ ((102) and (108)), and the integrand takes the form

with

On the interval $|\xi-\nu|<const$, functions $I^{\prime}(\xi)$ and $w_{1}^{\prime}(\xi)$ can be replaced by their approximations ((102) and (108)), but $w_{1}(\xi-\nu)$ cannot. Hence, we have

where

Finally, as $\xi\gg\nu$, all special functions can be asymptotically approximated ((102) and (108)), and the integrand becomes

with

From formulas (40)–(45) we conclude that the contribution of semi-axis $\xi\gg 1$ to (27) is exponentially small.

The analysis above shows that only the critical point (38) contributes to (27). Then, the standard stationary phase method [31] yields

where

Relations (16) and (17) allow us to match the expression (46) with a cylindrical wave (7) in the area where $\sigma\gg 1$, but the remainder terms in (16) are small, which can be written as follows:

The matching easily gives the following formula for the diffraction coefficient $A(\varphi;k)$:

This formula is in agreement with the result of A. Popov [1]. Similar to the case of non-tangential incidence (see, e.g., [26, 28, 29]), the diffraction coefficient is linear in the jump of the curvature $h$, but the singularity when approaching to the limit ray is different.

We establish that in the area described by $kr\gg 1$ and (32) the total wavefield is the sum of incident and diffracted waves.

As $\nu\gg 1$, replacing the function $w_{1}$ by its asymptotic expansion (102) gives

where the phase $\Psi_{1}$ introduced in (34) has the critical point $\xi_{1}$ (36). When $\xi_{1}$ is negative and not small, it contributes to the integral; however, when $\xi_{1}$ is positive and not small, it does not. These cases correspond to the location of the observation point not too close to the limit ray in the shadow zone and illuminated region, respectively. Accordingly, these areas are characterized by the inequality

which is ((16), (17)) equivalent to

If the critical point of phase $\xi_{1}$ and the endpoint $0$ merge, then $\xi_{1}$ contributes to the integral whenever it is negative or positive. The standard stationary phase method is not applicable; however, the expression (56) can be rewritten in terms of the Fresnel integral

as follows. Consider a very narrow neighborhood of the limit ray where the inequalities

hold true. In polar coordinates the second one has the form

Under the condition (60), $|\xi_{1}|\ll\sqrt{\nu}$, and the main contribution to the integral (56) is given by the interval $0\leq-\xi\ll\sqrt{\nu}$. Expanding the integrand of (56) in powers of $1/\nu$ yields

On the interval under consideration, the cubic term in $\xi$ and higher-order terms are small. Taking a quadratic approximation of the phase and main-term approximation of amplitude and extending integration to the half-line $(-\infty-i\varepsilon,0]$, $\varepsilon>0$, we derive the following expression for $F$:

This can be immediately rewritten in terms of the Fresnel integral (59)

with

Under the condition (60), the remainder terms in (64) are small.

In the area described by (60), the phase of the exponential and the argument of the Fresnel integral allow for the following geometrical interpretation (see (15)–(17)):

Variable $\mathrm{Z}$ is typical of wavefields description in areas where waves of different natures merge [3]. In a part of the area characterized by (60) (namely, where (57) holds), $\mathrm{Z}$ can be large, and the Fresnel integral allows an asymptotic expansion.

We start with the analysis of the function $D$ (54).

In the part of the integration interval where $const<\xi\leq 0$, the function $w_{1}(\xi-\nu)$ can be replaced solely by its asymptotic expansion (102). As established in Section 4, the derivative of the phase of the integrand vanishes at the point $\xi_{1}$ (36). Even if the critical point $\xi_{1}$ lies inside the interval under consideration, it gives no stationary-phase contribution because the integrand does not rapidly oscillate there.

On the half-line $-\xi\gg 1$, all special functions can be replaced by their approximations ((102) and (107)), which gives

Here, $\Psi_{2}$ is introduced in (35). In Section 4, we show that under the condition (31), phase $\Psi_{2}$ has one critical point $\xi_{2}$ (38), of which the contribution to $U_{0}$ has the form (46).

Now, we consider the vicinity of the limit ray, where the inequalities

hold true. In polar coordinates, they read

The area characterized by (68) is wider than that described by (60).

The inequality (68) implies that $-\xi_{2}\ll\nu^{\frac{1}{4}}$, whence the main contribution to $D$ is given by the segment $0\leq-\xi\ll\nu^{\frac{1}{4}}$. We replace $w_{1}(\xi-\nu)$ by its approximation (102):

Here, $\Psi_{1}$ is introduced in (34). Then, we expand the integrand in powers of $1/\nu$. Phase $\Psi_{1}$ is rewritten as (62). As $|\xi|\ll\nu^{\frac{1}{4}}$, the quadratic term in $\xi$ and higher-order terms in (62) are small and can be discarded. Therefore, we rewrite the integrand as follows:

On the half-line $-\xi\gg 1$, the functions $w_{1}$ and $H$ can be replaced by their approximations (102) and (107), respectively, and we arrive at an integrand with the phase $\widetilde{\Psi}_{2}=-\frac{2}{3}(-\xi)^{\frac{3}{2}}+(\sigma-\sqrt{\nu})\xi$. A straightforward calculation (similar to that concerning the phase $\Psi_{2}$; see (35)–(38)) shows that $\widetilde{\Psi}_{2}$ has exactly one critical point

provided that $\sqrt{\nu}-\sigma\gg 1$, cf. (31). The condition (68) implies that $-\widetilde{\xi}_{2}\ll\nu^{\frac{1}{4}}$, whence the integral of (71) over the half-line $-\xi\geq const\,\nu^{\frac{1}{4}}$ is negligible. Therefore, under the conditions (68), function $D$ has the following asymptotic representation:

A similar manipulation with $G$ yields