REFLECTION AND REFRACTION OF SPHERICAL WAVES

L. M. Brekhovskikh

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Reflection and refraction of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I. Reflection of Spherical Waves

1. Reflection of a spherical wave in the geometrical-optics approximation . . . . . . . . . . 6
2. Expansion of a spherical wave into plane waves . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Field of the reflected wave in the wave zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4. Weyl–van der Pol formula. The case of a strip located near the saddle point . . . . . . 17
5. Visual interpretation of the results. Limits of applicability of geometrical optics . . . . 21
6. Field of a raised radiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7. Lateral waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8. Field in a region close to the angle of total internal reflection . . . . . . . . . . . . . . . . 31

II. Refraction of Spherical Waves

1. Field of the refracted wave in the geometrical-optics approximation . . . . . . . . . . . 33
2. Corrections to geometrical optics for refracted waves . . . . . . . . . . . . . . . . . . . . . 35
3. The case in which one of the media has appreciable absorption . . . . . . . . . . . . . . 39
4. Sound field in water from a radiator located in air . . . . . . . . . . . . . . . . . . . . . . . 40

1. Introduction

The theory of reflection and refraction of plane waves was developed already by Fresnel. In contrast to it, a complete theory of the reflection and refraction of spherical waves has been created only in recent years. This gap in time is explained chiefly by the fact that in optical phenomena spherical waves may always be regarded as practically plane, since the corresponding radii of curvature are large in comparison with the wavelength. Only in the fields of radio and acoustics do we encounter spherical waves as such.

L. M. BREKHOVSKIKH

As has been shown by works of recent years (A. A. Andronov\(^1\), Ott\({}^{2,3}\), Krüger\(^4\)), the problem of reflection and refraction of spherical waves can be solved by means of methods analogous to those known in wave optics. An exposition of the principal achievements in this direction constitutes the aim of the present article. It turns out, moreover, that all particular cases in the problem of the propagation of radio waves in the presence of an interface between two homogeneous media (Sommerfeld’s problem), which have usually been solved by different methods (for example, the case of high conductivity of the ground, considered by V. A. Fock\(^5\) and M. A. Leontovich\(^6\); the case of moderate conductivity, considered by Weyl\(^7\); the case of a raised dipole, etc.), can be treated from a unified point of view. This point of view, which also covers the case of refracted waves, has many advantages and is based on a mathematical method adequate to the phenomena occurring in the propagation of short waves, whose role in practice has recently increased greatly.

We shall consider in parallel the cases of electromagnetic and sound waves, since the mathematical apparatus for these cases is completely identical. In addition, in practice there is an almost complete parallelism in all problems concerning the propagation of sound and radio waves. Among them the problem of reflection and refraction of spherical waves is the simplest.

Among the more complicated practically important problems one may mention, for example, the problem of the propagation of spherical waves in plane-layered media. It is therefore very essential to establish a common language between these two branches of science. This would also be useful for such questions as modeling electromagnetic phenomena by acoustic ones and conversely.

2. Reflection and refraction of plane waves

The basis of the method for considering reflection and refraction of spherical waves will be their expansion into plane waves. Here the class of plane waves is understood more broadly than usual: it includes the so-called inhomogeneous waves, or waves with complex direction cosines. Therefore we must first consider the reflection and refraction of plane waves, taking this generalization into account as well. In the case of electromagnetic waves we shall confine ourselves to the consideration of waves for which the electric vector lies in the plane of incidence. We shall characterize the electromagnetic field by the Hertz vector\(^8\), which in the case under consideration will have only one component, normal to the interface. Let the \(z\)-axis be perpendicular to the interface, and let the \(xz\)-plane coincide with the plane of incidence. Denoting by \(\Phi(x,z)\) the only nonzero component of the Hertz vector along the \(z\)-axis and assuming that all quantities (written in complex form) depend on time through

multiplying factor \(e^{-i\omega t}\), we shall have for the components of the electromagnetic field*):

\[ E_x=\frac{\partial^2\psi}{\partial x\,\partial z},\qquad E_y=0,\qquad E_z=-\frac{\partial^2\psi}{\partial x^2}; \tag{1} \]

\[ H_x=0,\qquad H_y=-\frac{i\omega\varepsilon}{c}\frac{\partial\psi}{\partial x},\qquad H_z=0. \tag{2} \]

We shall describe the field of a sound wave by the sound potential. Since in the subsequent formulas it will enter in exactly the same way as the vertical component of the Hertz vector in the formulas for the electromagnetic field, we shall likewise denote it by \(\psi(x,z)\).

The sound pressure \(p(x,z)\) and the oscillatory velocity \(\mathbf v(x,z)\) are expressed in terms of the sound potential as follows:

\[ p(x,z)=i\omega\rho\psi(x,z),\qquad \mathbf v=\operatorname{grad}\psi, \tag{3} \]

where \(\rho\) is the density of the medium.

Let a plane electromagnetic or sound wave be incident on the interface \(z=0\) at an arbitrary angle. The vertical component of the Hertz vector or the sound potential of such a wave will be given by the formula

\[ e^{ik(x\cos\alpha-z\sin\alpha)}, \tag{4} \]

where \(k\) is the wave number \(\left(k=\dfrac{2\pi}{\lambda}\right)\), and \(\alpha\) is the grazing angle of the wave, i.e. the angle formed by the normal to the wave front with the interface (Fig. 1).

Owing to the presence of the interface, in the upper medium we shall have a reflected wave:

\[ V e^{ik(x\cos\alpha+z\sin\alpha)}, \tag{4'} \]

where \(V\) is the reflection coefficient, and in the lower medium a refracted wave:

\[ W e^{ik_1(x\cos\alpha_1-z\sin\alpha_1)}, \tag{4''} \]

where \(k_1\) is the wave number for the lower medium, and \(\alpha_1\) is the grazing angle of the refracted wave. The coefficients \(V\) and \(W\) are determined from the boundary conditions. In the electromagnetic case these conditions are the continuity of \(E_x\) and \(H_y\) upon passage through the boundary. If quantities pertaining to the lower medium are denoted by the index 1, then

Fig. 1. Reflection and refraction of a plane wave.

*) In the general case, when the Hertz vector has all three components different from zero, the electric and magnetic fields in a medium with dielectric constant \(\varepsilon\) are determined by the formulas

\[ \mathbf E=\frac{\omega^2\varepsilon}{c^2}\psi+\operatorname{grad}\operatorname{div}\psi,\qquad \mathbf H=-\frac{i\omega\varepsilon}{c}\operatorname{rot}\psi. \]

We shall everywhere take the magnetic permeability \(\mu\) equal to unity.

while quantities in the upper medium will be left without an index, then, according to (1) and (2), the boundary conditions can be written as:

\[ z=0,\quad \frac{\partial \psi}{\partial z}=\frac{\partial \psi_1}{\partial z},\quad \psi=n^2\psi_1, \tag{5} \]

where \(n=\sqrt{\dfrac{\varepsilon_1}{\varepsilon}}\) denotes the refractive index.

In the acoustic case, the boundary conditions require continuity of the normal component of velocity and of the sound pressure. According to (3), these conditions can be written as:

\[ z=0,\quad \frac{\partial \psi}{\partial z}=\frac{\partial \psi_1}{\partial z},\quad \psi=\frac{\rho_1}{\rho}\psi_1. \tag{5'} \]

The conditions (5) and (5′) can be written in a single form, namely:

\[ z=0,\quad \frac{\partial \psi}{\partial z}=\frac{\partial \psi_1}{\partial z},\quad \psi=m\psi_1, \tag{6} \]

where \(m=n^2=\dfrac{\varepsilon_1}{\varepsilon}\)—in electrodynamics, and \(m=\dfrac{\rho_1}{\rho}\)—in acoustics.

In what follows, using this notation, we shall write the formulas in electrodynamics and in acoustics in a unified form.

Thus, for the field in the upper medium we have

\[ \psi=e^{ik(x\cos\alpha-z\sin\alpha)}+Ve^{ik(x\cos\alpha+z\sin\alpha)} \]

and for the field in the lower medium:

\[ \psi_1=We^{ik_1(x\cos\alpha_1-z\sin\alpha_1)}. \]

Substituting these expressions into (6) and then putting \(z=0\), we obtain, in the well-known way,

\[ n\cos\alpha_1=\cos\alpha \tag{7} \]

and, further,

\[ V=\frac{m\sin\alpha-\sqrt{\,n^2-\cos^2\alpha\,}}{m\sin\alpha+\sqrt{\,n^2-\cos^2\alpha\,}}, \tag{8} \]

\[ W=\frac{1}{m}(1+V). \tag{9} \]

These formulas are the fundamental ones in the theory of reflection and refraction of plane waves.

Let us also note the following entirely obvious circumstance, which we shall use below. If the field of the incident wave at the point \(x_0,z_0\) is equal to \(\psi_0\), then the field of the reflected wave at the point \(x,z\) will be:

\[ \psi_0Ve^{ik[(x-x_0)\cos\alpha+(z+z_0)\sin\alpha]}, \tag{10} \]

where in the exponent there is the total phase increment of the wave. Similarly, for the field of the refracted wave we obtain:

\[ \psi_1=\psi_0 W'e^{ik[(x-x_0)\cos\alpha+z_0\sin\alpha-zn\sin\alpha_1]}, \tag{11} \]

where \(z<0\).

Let us now pass to a generalization of the concept of a plane wave. Expression (4) is a solution of the wave equation

\[ \frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial z^2}+k^2\psi=0. \]

It is not difficult to verify that it remains a solution if by \(\alpha\) one understands an arbitrary complex quantity

\[ \alpha=\alpha'+i\alpha'', \]

and then we shall have:

\[ \left. \begin{aligned} \cos\alpha&=\cos\alpha'\,\operatorname{ch}\alpha''-i\sin\alpha'\,\operatorname{sh}\alpha'',\\ \sin\alpha&=\sin\alpha'\,\operatorname{ch}\alpha''+i\cos\alpha'\,\operatorname{sh}\alpha''. \end{aligned} \right\} \tag{12} \]

As a result, expression (4) is written in the form

\[ e^{-k(x\sin\alpha'+z\cos\alpha')\operatorname{sh}\alpha'' +ik(x\cos\alpha'-z\sin\alpha')\operatorname{ch}\alpha''}. \tag{13} \]

The limits of variation of \(\alpha'\) may be restricted to the interval \((0,2\pi)\), while the quantity \(\alpha''\) may take all values from \(-\infty\) to \(+\infty\). The last expression describes a wave with variable amplitude, the planes of constant amplitude \(x\sin\alpha'+z\cos\alpha'=\mathrm{const.}\) being, as is easy to see, normal to the planes of constant phase \(x\cos\alpha'-z\sin\alpha'=\mathrm{const.}\), i.e. to the wave fronts. Such waves are called inhomogeneous plane waves. An inhomogeneous plane wave propagates in a direction making an arbitrary angle \(\alpha'\) with the \(x\)-axis, and has a wavelength \(\lambda'\) smaller than the usual wavelength \(\lambda=\dfrac{2\pi}{k}\). Indeed, from (13) it follows that \(k_x=k\cos\alpha'\operatorname{ch}\alpha''\) and \(k_z=-k\sin\alpha'\operatorname{ch}\alpha''\), so that

\[ \left(\frac{2\pi}{\lambda'}\right)^2=k_x^2+k_z^2=k^2\operatorname{ch}^2\alpha''>k^2, \]

whence it follows that \(\lambda'<\lambda\).

It also follows from this that the velocity of propagation of an inhomogeneous wave is less than the velocity of an ordinary wave and depends on \(\alpha''\). The amplitude of an inhomogeneous wave decreases or increases in the direction perpendicular to the propagation.

In particular, for purely imaginary \(\alpha\) (\(\alpha'=0\)) the inhomogeneous wave, as is seen from (13), propagates along the \(x\)-axis, while its amplitude increases or decreases along the \(z\)-axis.

Since in deriving formulas (7)—(9) we nowhere used the condition that \(\alpha\) and \(\alpha_1\) be real, these formulas are also valid for complex \(\alpha\) and \(\alpha_1\), i.e. for inhomogeneous waves. For ordinary plane waves the reflection coefficient (8) is a real quantity if \(m\) and \(n\) are real. Thus, the phase of the reflected

of the wave coincides with the phase of the incident wave (if \(V>0\)) or is opposite to it (if \(V<0\)). In the case of inhomogeneous waves, however, the reflection coefficient is complex, and the phase change may be arbitrary.

Let us see how the character of an inhomogeneous wave changes upon its reflection and refraction. For simplicity, take an inhomogeneous wave propagating along the \(x\)-axis. Setting in (13) \(\alpha'=0\), we obtain for it the expression

\[ e^{-kz\,\operatorname{sh}\alpha''+ikx\,\operatorname{ch}\alpha''}. \tag{14} \]

The process of reflection, as is seen from Fig. 1 and also from comparison of expressions (4) and (4′), may be regarded as a change in the sign of \(\alpha\). Therefore, if (14) is treated as the incident wave, then the reflected wave will be:

\[ Ve^{kz\,\operatorname{sh}\alpha''+ikx\,\operatorname{ch}\alpha''}. \]

Thus, if the incident wave had an amplitude decreasing with increasing \(z\), then the reflected wave will have an amplitude increasing with increasing \(z\), and conversely.

Essential changes in the character of an inhomogeneous wave may also occur upon its refraction. There may be cases when an inhomogeneous wave, upon refraction, is transformed into an ordinary plane wave, and conversely. Indeed, let us again consider, for simplicity, a wave with purely imaginary \(\alpha\). Here, according to (12), \(\cos\alpha=\operatorname{ch}\alpha''\). Therefore, from (7), for the grazing angle of the refracted wave we obtain:

\[ \cos\alpha_1=\frac{\operatorname{ch}\alpha''}{n}. \]

For \(n>\operatorname{ch}\alpha''\), \(\alpha_1\) will be a real quantity. Thus, an inhomogeneous wave upon refraction is transformed into an ordinary plane wave.

Let, further, \(\alpha\) be real; then from (7)

\[ \cos\alpha_1=\frac{\cos\alpha}{n}; \]

for \(n<\cos\alpha\) we shall have \(\cos\alpha_1>1\), i.e. \(\alpha_1\) is imaginary. Here an ordinary plane wave has been transformed into an inhomogeneous one. The latter case is nothing other than the well-known phenomenon of total internal reflection.

I. REFLECTION OF SPHERICAL WAVES

3. Reflection of a spherical wave in the geometrical-optics approximation

Let there be a radiator \(O\) (Fig. 2) in the form of a vertical dipole in the electromagnetic case and of a pulsating sphere of small radius in acoustics. The field created by such a radiator in the surrounding space, as before, can be characterized by the Hertz vector

(possessing only one vertical component) in the first case, and the acoustic potential in the second. We shall denote by $\psi_0$ the vertical component of the Hertz vector and the acoustic potential of the field produced by the radiator in the absence of the boundary of separation. To within a constant factor depending on the power of the radiator, we shall have:

\[ \psi_0=\frac{e^{ikR}}{R}, \]

where $R$ is the distance from the radiator to the point of observation. We shall call this expression a spherical wave.

The presence of the boundary separating the media will be manifested in the fact that, in order to obtain the total field $\psi$ in the upper medium, one must add to $\psi_0$ the reflected wave $\psi_r$:

\[ \psi=\psi_0+\psi_r. \tag{15} \]

In the lower medium the field will be described by the refracted wave, which we shall denote by $\psi_1$. The determination of $\psi_r$ and $\psi_1$ will constitute our problem.

The simplest method, suitable for sufficiently high frequencies, is the application of geometrical (ray) optics. It consists in neglecting the curvature of the wave upon reflection and assuming that it is reflected as a plane wave. The sphericity of the wave is then taken into account only by a factor characterizing the decrease of the wave amplitude due to the spreading of its front. As a result, for the reflected wave we obtain:

Fig. 2. Ray diagram for reflection of a spherical wave.

\[ \psi_r=V(\chi)\frac{e^{ikR_1}}{R_1}, \tag{16} \]

where the factor $e^{ikR_1}$ gives the phase gain of the wave along the ray. Here $R_1$ is the length of the ray $OAP$ (Fig. 2), connecting the radiator with the point of observation and reflected from the boundary according to the law of geometrical optics; $\chi$ is the grazing angle of this ray and, finally, $V(\chi)$ is the reflection coefficient of a plane wave, determined by expression (8).

When expression (16) is used to compute the components of the electric and magnetic fields, one obtains the so-called reflection formulas. Below, in § 7, we shall determine under what conditions the expression obtained from the exact theory, which takes into account the curvature of the wave front, passes into (16).

Thus the limits of applicability of the reflection formulas will be determined.

4. Expansion of a Spherical Wave into Plane Waves

For a more exact investigation of the reflection and refraction of spherical waves, it proves convenient to represent them in the form of a superposition of plane waves, since the formulas for the reflection and refraction of plane waves are known. Here there is a very close analogy with the problem of wave reflection in diffraction problems (see also the article by V. A. Fock\(^{9}\)). The usual method applied in diffraction problems is the expansion of the incident wave into waves having the same symmetry as the body on which the diffraction occurs. Analogously, in the problem of reflection, the spherical wave is expanded into plane waves, since reflection from a plane boundary is being considered. In general, our problem of the reflection of a spherical wave may be regarded as the problem of its diffraction by a plane.

Fig. 3. \(\vec n\) is the normal to the front of an arbitrary plane wave. The angles \(\alpha\) and \(\varphi\) characterize its direction in space.

Let us consider in more detail the expansion of a spherical wave into plane waves. We shall characterize the direction of the normal to the front of each of the plane waves by the angles \(\alpha\) and \(\varphi\) (Fig. 3). Any one of the plane waves will be given by the expression

\[ e^{i(k_x x+k_y y+k_z z)}, \tag{17} \]

where

\[ k_x=k\cos\alpha\cos\varphi,\qquad k_y=k\cos\alpha\sin\varphi,\qquad k_z=k\sin\alpha \tag{18} \]

are the components of the wave vector along the coordinate axes.

The formula for the expansion of a spherical wave into plane waves (17) has the form\(^*\)

\[ \frac{e^{ikR}}{R} = -\frac{ik}{2\pi} \int_{-\frac{\pi}{2}}^{i\infty} \int_{0}^{2\pi} e^{ik(x\cos\alpha\cos\varphi+y\cos\alpha\sin\varphi+z\sin\alpha)} \cos\alpha\,d\alpha\,d\varphi , \tag{19} \]

where it is temporarily assumed that the radiator is located at the origin of coordinates and, consequently, \(R=\sqrt{x^2+y^2+z^2}\). In the exponent without—

\(^*\) See \(^{10}\), p. 138. Expression (19) differs from those found in the literature only in that we use the grazing angles \(\alpha\) instead of the angles of incidence \(\gamma\). In this case \(\alpha=\dfrac{\pi}{2}-\gamma\).

REFLECTION AND REFRACTION OF SPHERICAL WAVES

the “plus” sign is taken if \(z>0\), and the “minus” sign if \(z<0\), which is necessary for convergence of the integral.

Integration with respect to \(\varphi\) is carried out from \(0\) to \(2\pi\), and with respect to \(\alpha\)—first over real values from \(-\dfrac{\pi}{2}\) to \(0\), and then over imaginary values from \(0\) to \(i\infty\) (Fig. 4). Thus, in addition to ordinary plane waves propagating in all directions, the expansion contains inhomogeneous waves corresponding to imaginary \(\alpha\), propagating with a shortened wavelength in the \(xy\)-plane and exponentially decaying (or increasing) in amplitude in the direction of the \(z\)-axis (see the preceding paragraph). The necessity of introducing waves of this kind in the expansion of a spherical wave follows from the fact that by superposition of ordinary plane waves alone one cannot obtain a field that would have the required singularity at \(R=0\) and remain finite at all other points. We shall show how such a singularity is obtained when inhomogeneous waves are included.

Assuming \(\alpha=i\alpha''\), we obtain, according to (18) and (12), for inhomogeneous waves:

\[ \begin{gathered} k_x=k\,\operatorname{ch}\alpha''\cos\varphi,\qquad k_y=k\,\operatorname{ch}\alpha''\sin\varphi,\\ k_z=ik\,\operatorname{sh}\alpha''. \end{gathered} \tag{20} \]

Fig. 4. Path of integration in the complex plane of angles in the expansion of a spherical wave into plane waves.

For \(\alpha''\to\infty\), from (20) we obtain \(k_x\to\infty,\ k_y\to\infty,\ k_z\to i\infty\). This means that we have waves propagating in the horizontal plane with a wavelength tending to zero and at the same time decaying in the vertical direction with an attenuation coefficient tending to infinity.

At \(x=y=z=0\), the superposition of an infinite number of these waves [integral (19)] gives an infinite value of the Hertz vector or of the acoustic potential. On moving away from this point, finite values are obtained either because of attenuation (for \(z\ne0\)), or because of dephasing of the waves (for \(x\ne0\) or \(y\ne0\), since all waves have the same phase only when \(x=y=0\)).

Here it is appropriate to make the following remark. The expression for the primary wave

\[ \frac{e^{ikR}}{R} \]

does not change under any rotation of the coordinate system, since the origin is assumed to coincide with the emitter. However, for different positions of the coordinate axes, different inhomogeneous waves are taken in the expansion. Namely, each time those inhomogeneous waves are taken which decay along the chosen direction of the \(z\)-axis. Hence it follows that one and the same field can be represented by a superposition of plane-

L. M. BREKHOVSKIKH

waves whose direction cosines are chosen according to different laws. This remark is of fundamental importance for us. The choice of the most convenient law, which is equivalent to finding a suitable path of integration in the complex plane, can substantially simplify the problem (and the possibilities available here are not, generally speaking, reducible to mere rotations of the coordinate system). This idea underlies the method set forth in the following paragraphs.

In the presence of an interface between media, each of the plane waves gives rise to reflected and refracted waves. As a result, the Hertz vector (or the acoustic potential) in the upper medium will be given by an integral of the form (19), where under the integral sign there will stand not only the direct wave but also the reflected wave. In the lower medium we shall have an analogous integral of the refracted plane waves.

For what follows it is convenient to assume that the origin of the rectangular coordinate system \((x, y, z)\) lies on the interface, and that the radiator is placed at the point \((0,0,z_0)\) (Fig. 2). In this case, in expression (19) \(z\) must be replaced by \(z+z_0\). Moreover, let us assume that the observation point lies in the plane \(xz\). Then \(y=0\).

Taking now into account the phase increment for each of the reflected plane waves, given by formula (10), we obtain for the reflected spherical wave

\[ \psi_r=-\frac{ik}{2\pi} \int_{\frac{\pi}{2}}^{i\infty}\int_0^{2\pi} e^{ik[x\cos\alpha\cos\varphi+(z+z_0)\sin\alpha]} V(\alpha)\cos\alpha\,d\alpha\,d\varphi \tag{21} \]

and, analogously, taking also (9) into account, for the refracted wave*)

\[ \psi_1=-\frac{ik}{2\pi m} \int_{\frac{\pi}{2}}^{i\infty}\int_0^{2\pi} e^{ik(x\cos\alpha\cos\varphi+z_0\sin\alpha)-ik_1z\sin\alpha_1} [1+V(\alpha)]\cos\alpha\,d\alpha\,d\varphi. \tag{21'} \]

Here, according to the law of refraction (7),

\[ k_1\sin\alpha_1=k\sqrt{\,n^2-\cos^2\alpha\,}. \]

*) The method used here has a very wide range of applicability. Thus, it is not difficult to see that all our arguments leading to formula (21′) would remain valid if the medium for \(z<0\) had parameters that vary arbitrarily with \(z\). In this case \(V(\alpha)\) would denote the reflection coefficient of plane waves from such a medium. This makes it possible to approach, from a new point of view, a broad and important class of problems, on which we shall dwell in more detail elsewhere.

The sign of the root must be chosen so that its imaginary part is positive, i.e.

\[ \operatorname{Im}\sqrt{\,n^{2}-\cos^{2}\alpha\,}>0, \tag{22} \]

which is necessary in order that, as \(z\to -\infty\), the amplitude of the refracted wave should tend to zero.

5. The field of the reflected wave in the wave zone

The analysis of integral (21) proves to be considerably more complicated than its derivation. We shall carry out this analysis for the wave zone, i.e. for distances from the emitter large in comparison with the wavelength. In this case the integral can be represented in such a way that the principal role in it will be played only by those plane waves whose direction almost coincides with the direction of the ray \(OAP\) (Fig. 2), constructed according to the laws of geometrical optics.

A convenient mathematical device here is the so-called saddle-point method. It serves to estimate values of integrals of the form

\[ I=\int\limits_{C} e^{\rho f(\zeta)}F(\zeta)\,d\zeta \tag{23} \]

for large values of \(\rho\). Here \(C\) denotes some path in the complex plane \(\zeta=\zeta'+i\zeta''\) along which the integration is performed. In a particular case the path \(C\) may include only real values of \(\zeta\).

The reader will find the general theory of the saddle-point method in the literature \(^{11,12}\). In its main features this method reduces to the following. The path of integration in the complex plane may be deformed without thereby changing the value of the integral. Making use of this, we shall try to draw the path of integration so that that part of the path which determines essentially the whole value of the integral is as short as possible. Then, as we shall see, the integrand can be replaced by another, simpler function (usually by a power series), agreeing sufficiently accurately with the integrand on this essential segment of the path of integration. In doing so we need not concern ourselves with what it will be on the other, inessential segments.

Without restricting the generality, \(\rho\) in (23) may be regarded as real and positive. Separate in \(f(\zeta)\) the real and imaginary parts:

\[ f(\zeta)=f_{1}+if_{2}. \]

Then the exponential under the integral in (23) will be written as

\[ e^{i\rho f_{2}+\rho f_{1}}. \tag{24} \]

It is evident from this that the path of integration must be chosen in such a way that on it the function \(f_{1}\) has, at some point, a maximum

and decrease as rapidly as possible when moving away from this point. But the imaginary and real parts of an analytic function—in our case \(f_1\) and \(f_2\)—have the property that, in the plane \(\zeta=\zeta' + i\zeta''\), the lines of steepest descent of one of them are lines of constant value for the other\(^*\). Since the path of integration must follow the line of steepest descent of \(f_1\), it follows from this that it must coincide with the line \(f_2=\mathrm{const.}\)—the line of constant phase. The point situated on this path at which \(f_1\) takes its maximum value is called the saddle point. At it, the derivative of \(f_1\), taken in the direction of the path of integration, must be equal to zero. Since along this path \(f_2=\mathrm{const.}\), it follows that the derivative of \(f_2\) is also equal to zero. Thus, upon differentiating in the direction of the path of integration,

\[ \frac{df}{d\zeta}=0. \tag{25} \]

But if the derivative of an analytic function at some point is equal to zero in at least one direction, then it is equal to zero in all the others as well, since, as is known, it is the same in all directions. Thus the saddle point must be a root of equation (25).

Thus, the most advantageous path of integration must pass through the saddle point, determined by equation (25), and depart from it along the line of steepest descent of the function \(f_1\), coinciding with the line \(f_2=\mathrm{const.}\) For brevity, we shall call such a path the “saddle path of integration.” If \(\rho\) is large, then the exponential (24), upon moving away from the saddle point, will decrease rapidly, so that only a small part of the path of integration, containing the saddle point, will play an essential role.

Suppose that, solving equation (25), we have found for the saddle point \(\zeta=\zeta_0\). Then on the saddle path of integration one must have

\[ f(\zeta)=f(\zeta_0)-s^2, \tag{26} \]

where \(s\) runs through all real values from \(-\infty\) to \(+\infty\), and at the saddle point itself \(s=0\). Indeed, in this case, according to (26), the imaginary part of the function \(f(\zeta)\) remains all the time equal to \(f_2(\zeta_0)\), while the real part decreases on both sides of the saddle point. We now make in (23) the substitution of the variable \(\zeta\) by the new variable \(s\). In the plane of the complex variable \(s\), the saddle path of integration will be represented by the real axis\(^ {**}\).

\(^*\) This is a consequence of the well-known Cauchy–Riemann relations. See, for example, \({}^{13}\), p. 254.

\(^ {**}\) When transforming the original path of integration into the saddle path, one must take into account the singular points of the integrand, if any are present.

As a result, denoting

\[ F(\zeta)\,\frac{d\zeta}{ds}=\Phi(s), \tag{27} \]

we write integral (23) in the form

\[ I=e^{\rho f(\zeta_0)}\int_{-\infty}^{+\infty} e^{-\rho s^2}\Phi(s)\,ds . \tag{28} \]

Since \(\rho\) is assumed to be large, only small values of \(s\) will be essential under the integral sign. Consequently, it is expedient to represent the function \(\Phi(s)\) as a power series in \(s\)*:

\[ \Phi(s)=\Phi(0)+\Phi'(0)s+\frac{1}{2}\Phi''(0)s^2+\cdots . \]

Substituting this series under the integral sign, after substituting the values of known integrals we obtain

\[ I=e^{\rho f(\zeta_0)}\sqrt{\frac{\pi}{\rho}}\left[\Phi(0)+\frac{1}{4\rho}\Phi''(0)+\cdots\right]. \tag{29} \]

Thus, the method of steepest descent makes it possible to represent the value of the integral in the form of a series in inverse powers of the large parameter \(\rho\). If the function \(\Phi(s)\) varies sufficiently slowly in comparison with the exponential \(e^{-\rho s^2}\), i.e., if its derivatives are sufficiently small, then in (29) one may restrict oneself to one or several of the first terms.

Let us apply these considerations to the calculation of the integral with respect to \(\varphi\) in (21). It may be written in the form**)

\[ \int_0^{2\pi} e^{i\rho\cos\varphi}\,d\varphi, \tag{30} \]

where \(\rho=kx\cos\alpha\). We shall assume that \(\rho\gg1\), and first consider the case when \(\alpha\) is real. Comparing with (23), we obtain \(f(\varphi)=i\cos\varphi,\ F(\varphi)=1\). In applying the method of steepest descent to this integral it is necessary to imagine the complex plane \(\varphi=\varphi'+i\varphi''\), the initial path of integration in which is the segment of the real axis \((0,2\pi)\). The saddle points according to (25) are found from the equation

\[ \frac{d}{d\varphi}\cos\varphi=0 \]

*) The function \(\Phi(s)\), defined by (27), usually has to be sought directly in the form of a power series in \(s\), since finding its explicit expression in closed form is difficult (examples see below).

**) Integral (30) is well known. It is equal to \(2\pi J_0(\rho)\), where \(J_0(\rho)\) is the Bessel function. The expression (32) obtained below is an asymptotic representation of this function. We carry out the investigation of the integral not for its own sake, but in order to illustrate, with a simple example, the method constantly used below.

and will be

\[ \varphi=0,\pi,2\pi. \]

According to (26) we introduce a new variable \(s\), connected with \(\varphi\) by the equation

\[ \cos\varphi=\pm 1+i s^2, \tag{31} \]

and draw in the \(\varphi\)-plane, through the saddle points, paths of integration such that along them \(s\) changes from \(-\infty\) to \(+\infty\). Equating, in the last expression, the real parts on both sides of the equality, we obtain for these paths:

\[ \cos\varphi'\operatorname{ch}\varphi''=\pm 1. \]

Here the upper and lower signs before unity, here and in (31), refer to the paths passing respectively through the points \(\varphi=0\) (or \(2\pi\)) and \(\varphi=\pi\). These paths of integration are shown in Fig. 5 by dotted lines.

Fig. 5. Replacement of the original path of integration (heavy line) by saddle paths (dotted lines).

Instead of integration along the real axis from \(0\) to \(2\pi\), we may integrate along these paths in the direction indicated by the arrows, which is equivalent to varying \(s\) within real values. According to (27), we have: \(\Phi(s)=\dfrac{d\varphi}{ds}\). Using (31), we obtain, accurately through \(s^2\) inclusive,

\[ \frac{d\varphi}{ds}=\sqrt{2}\,e^{\mp i\pi/4}\left(1\mp i\frac{s^2}{4}\right). \]

The choice of the upper or lower signs is made in the same way as in (31). As a result, we obtain integrals of the form (28), where the function \(\Phi(s)\) is already expanded in a series in \(s\). Substituting the values of the corresponding tabulated integrals, we shall finally have:

\[ \int_{0}^{2\pi} e^{i\rho\cos\varphi}\,d\varphi = \sqrt{\frac{2\pi}{\rho}} \left[ e^{i\rho-\frac{i\pi}{4}} \left(1+\frac{1}{8i\rho}\right) + e^{-i\rho+\frac{i\pi}{4}} \left(1-\frac{1}{8i\rho}\right) \right]. \tag{32} \]

Here the first term in the square brackets is obtained from the saddle points \(0, 2\pi\), and the second from the point \(\pi\)*). It can be shown that expression (32) is also valid for complex \(\rho\).

* The saddle points \(\varphi=0,2\pi\) correspond to waves propagating from the radiator to the receiver, while the saddle point \(\pi\) corresponds to waves propagating from the receiver to the radiator. The expansion of a spherical wave into plane waves includes both directions of propagation. It will be seen further that as the distance increases the role of the “reverse” plane waves decreases.

Substituting (32) into (21), we obtain:

\[ \psi_r=-\frac{ik}{\sqrt{2\pi x}} \left[ \int\limits_{\frac{\pi}{2}}^{i\infty} e^{i\rho-\frac{i\pi}{4}} \left(1+\frac{1}{8i\rho}\right) e^{ik(z+z_0)\sin\alpha}V(\alpha)\cos\alpha\,d\alpha + \int\limits_{\frac{\pi}{2}}^{i\infty} e^{-i\rho+\frac{i\pi}{4}} \left(1-\frac{1}{8i\rho}\right) e^{ik(z+z_0)\sin\alpha}V(\alpha)\cos\alpha\,d\alpha \right]. \tag{33} \]

Here it is convenient to transform the second integral by introducing a new variable \(\beta=\pi-\alpha\), along which the path of integration will go from \(\pi/2\) to \(\pi-i\infty\). In this case \(\sin\alpha=\sin\beta\), \(\cos\alpha=-\cos\beta\), \(\sqrt{\cos\alpha}=-i\sqrt{\cos\beta}\) (the minus sign in the last equality follows from an analysis of the right- and left-hand sides of the equality on remote sections of the corresponding paths of integration) and, according to (8), \(V(\alpha)=V(\beta)\). As a result, the integrand in the second integral in (33) is written in the same way as in the first, but its limits of integration will be \(\pi-i\infty\) and \(\pi/2\). Consequently, these two integrals can be combined into one with a path of integration going from \(\pi-i\infty\) through the point \(\pi/2\) and going to \(i\infty\) (the path \(\Gamma\) in Fig. 6). If, in addition, in the exponent one substitutes \(\rho=kx\cos\alpha=kR_1\cos\chi\cos\alpha\) and \(z+z_0=R_1\sin\chi\) (see Fig. 2), then expression (33) is written as

\[ \psi_r=-\sqrt{\frac{k}{2\pi x}}\,e^{\frac{i\pi}{4}} \int\limits_{\pi-i\infty}^{i\infty} e^{ikR_1\cos(\alpha-\chi)} \left(1+\frac{1}{8kx\cos\alpha}\right) V(\alpha)\sqrt{\cos\alpha}\,d\alpha. \tag{34} \]

The analysis of this integral can also be carried out by the method of steepest descent, since the quantity \(kR_1\) is assumed large.

Comparing with (23), we set \(\rho=kR_1,\ f(\alpha)=i\cos(\chi-\alpha)\) and

\[ F(\alpha)=\left(1+\frac{1}{8kx\cos\alpha}\right)V(\alpha)\sqrt{\cos\alpha}. \]

The saddle point, determined by the equation \(\dfrac{df}{d\alpha}=0\), will be \(\alpha=\chi\). The transition to the variable \(s\)—see (26)—will be made according to the relation

\[ \cos(\alpha-\chi)=1+is^2. \tag{35} \]

We replace the integration along the path \(\Gamma_1\) in the \(\alpha\)-plane by integration along a new (saddle) path of integration, which would correspond to real values of \(s\). Taking the real parts of both sides of equality (35) and taking (12) into account, we obtain the equation of this path in the \(\alpha\)-plane:

\[ \cos(\alpha'-\chi)\operatorname{ch}\alpha''=1. \]

The path will intersect the real axis at the point \(\alpha=\chi\) at an angle of \(45^\circ\) and will go off, on one side, to \(-\dfrac{\pi}{2}+\chi+i\infty\), and on the other, to \(\dfrac{\pi}{2}+\chi-i\infty\) (the path \(\Gamma\) in Fig. 6). Thus, integration over complex values of \(\alpha\) is replaced by integration over real values of \(s\) from \(-\infty\) to \(+\infty\). The function \(\Phi(s)\), determined by expression (27), can without difficulty be found in the form of a series in powers of \(s\), after which, according to the general formula (29), we obtain:

\[ \psi_r=\frac{e^{ikR_1}}{R_1}\left[V(\chi)-\frac{iN}{kR_1}\right], \tag{36} \]

where

\[ N=\frac{1}{2}V'''(\chi)\cos^2\chi - V''(\chi)\sin\chi. \tag{37} \]

If we introduce the notation \(q_0=\sqrt{\,n^2-\cos^2\chi\,}\) \((\operatorname{Im} q_0>0)\), \(\sin\chi=\gamma\), and restrict ourselves to the case of small \(\gamma\), then the calculation of the derivatives gives:

\[ N=-\frac{m\left[2m\sqrt{n^2-1}+\gamma(2n^2+1)\right]}{(m\gamma+q_0)^3}. \tag{38} \]

It is easy to give a physical interpretation of the mathematical operations performed in computing the integral (34). First of all, the deformation of the original integration path \(\Gamma_1\) into the path \(\Gamma\) corresponds to the possibility, noted in the preceding paragraph, of representing one and the same field by the superposition of plane waves chosen in different ways. The choice of the path \(\Gamma\) corresponds to the fact that the field is composed of those plane waves which have at the observation point the same phase \((\operatorname{Re}\cos(\alpha-\chi)=1)\), namely the same as the wave reflected at the grazing angle \(\chi\). The path on which the phase is constant, according to the general properties of analytic functions

is also the path on which (when the condition \(\operatorname{Im}\cos(\alpha-\chi)>0\) is satisfied in the present case) the quantity of the exponent under the integral decreases most rapidly. This is why, in analyzing the integral, only small \(s\), i.e. \(\alpha\) close to \(\chi\), turned out to be essential. This means that the field at the point of observation is, in the main, composed of plane waves reflected from the boundary at angles close to \(\chi\)—the angle of reflection of the ray constructed according to the laws of geometrical optics.

In accordance with this, the principal term in (35) is the first term in brackets, which gives the reflected wave in the approximation of geometrical optics. The second term may be regarded as a correction which, however, in a number of cases plays a very substantial role; in particular, as we shall show below, when the radiator and receiver are at distances from the boundary that are small compared with the wavelength.

Expression (38) is in one respect not complete. It does not include the so-called lateral wave, obtained when the analysis of the deformation of the path of integration in the complex plane is carried out more carefully. However, in the electromagnetic case, when the lower medium is the earth, this wave plays no role, since it is rapidly attenuated with distance. In the reflection of a spherical wave from a lossless dielectric, and also in acoustics, it may play a role, but we shall discuss this in more detail in § 9.

6. The Weyl–Van der Pol Formula. The Case of a Pole Located Near the Saddle Point

Let us make one refinement of expression (36) and at the same time determine the limits of its applicability.

The main steps in deriving formula (36) were the substitution of (35) and the expansion of the entire integrand in (34), except for the exponent, in a series in powers of \(s\). It is necessary, of course, that this series converge. However, the radius of convergence of the integrand is limited by the fact that \(V(\alpha)\) becomes infinite (has a pole of first order) at the point \(\alpha=\alpha_p\), where \(\alpha_p\) is found, in the case of electrodynamics, from the equation

\[ n^2 \sin \alpha_p + \sqrt{\,n^2-\cos^2\alpha_p\,}=0, \tag{39} \]

which gives:

\[ \cos\alpha_p=\pm\frac{n}{\sqrt{n^2+1}}. \tag{40} \]

If the location of the pole in the \(s\)-plane is denoted by \(s_0\), then, according to (35),

\[ \cos(\alpha_p-\chi)=1+is_0^2, \tag{41} \]

the series in powers of \(s\) will converge inside a circle of radius \(s_0\), on whose boundary there lies a pole. The smaller \(s_0\) is (i.e., the larger \(|n|\) is), the more the circle of convergence contracts to the point of the saddle \(s=0\). For the applicability of the saddle-point method used above it is necessary that the entire region of significant values of \(s\) under the integral

\[ \int_{-\infty}^{+\infty} e^{-kR_1s^2}\,ds \]

lie inside the circle of convergence.

Let us denote the upper boundary of the significant values of \(s\) by \(s_1\). It is of order of magnitude \(s_1 \sim \dfrac{1}{\sqrt{kR_1}}\). If, in expanding into a series, one is to be limited to the term in \(s^2\), then it is necessary that the zone of significant values of \(s\) be considerably smaller than \(s_0\), which may be written in the form

\[ \left(\frac{s_1}{s_0}\right)^2 \ll 1 \]

or

\[ kR_1s_0^2 \gg 1. \tag{42} \]

The quantity \(w^2 = kR_1s_0^2\) is customarily called the numerical distance. Consequently, for the application of the formulas (36) and (37) obtained above, it is necessary that the numerical distances be large.

Using expressions (40) and (41), we obtain*):

\[ w^2 = ikR_1\left(1-\frac{n\cos\chi-\sin\chi}{\sqrt{\,n^2+1\,}}\right). \tag{43} \]

If \(\chi\) is small and \(|n|\) is large, so that the quantities \(\chi^2\) and \(\dfrac{1}{n^2}\) may be neglected in comparison with unity, we shall have:

\[ w^2=\frac{ikR_1}{2n^2}(1+2n\chi). \tag{44} \]

For large \(|n|\), i.e., for high soil conductivity and long waves, condition (42) may fail to be satisfied. However, one can obtain an expression whose applicability is limited not by condition (42), but by the usual condition for the saddle-point method \(kR_1 \gg 1\), if the saddle-point method is somewhat modified, as was done by V. A. Fock\(^{14}\) and Ott\(^{3}\), by different methods (see also \(^{15}\)). We shall first set forth this question in general form.

Suppose that we have an integral of the form

\[ I=\int_{-\infty}^{+\infty} e^{-kRs^2}\Phi(s)\,ds, \tag{45} \]

*) In (40) we take the plus sign, since the pole lying near the origin of coordinates (\(\cos\varphi_p \cong 1\)) is significant for us. From (41) we also have

\[ \sin\varphi_p=-\frac{1}{\sqrt{\,n^2+1\,}}. \]

The plus sign here is inadmissible because then from (39) it would follow that \(\operatorname{Im}\sqrt{\,n^2-\cos^2\varphi_p\,}<0\), which contradicts condition (22).

where \(kR\) is large. Suppose that the function \(\Phi(s)\) has a simple pole at the point \(s=s_0\), where \(s_0\) is, in general, complex. We multiply and divide the integrand in (45) by \(s^2-s_0^2\):

\[ I=\int_{-\infty}^{+\infty} e^{-kRs^2}\Phi(s)\left(s^2-s_0^2\right)\frac{ds}{s^2-s_0^2}; \]

the function \(\Phi(s)\left(s^2-s_0^2\right)\) no longer has a singularity at \(s=s_0\). Expand it in powers of \(s\):

\[ \Phi(s)\left(s^2-s_0^2\right)=\sum_{m=0}^{\infty}\frac{c_m}{m!}\,s^m . \tag{46} \]

It can be shown\(^3\) that

\[ \int_{-\infty}^{+\infty}\frac{e^{-kRs^2}s^{2m}\,ds}{s^2-s_0^2} = \]

\[ =\pm \frac{1\cdot3\cdot5\cdots(2m-1)\sqrt{\pi}\,w^{2m-1}} {2^{m-1}(kR)^{m-\frac12}} \,e^{-w^2} \int_{w}^{i\infty}\frac{e^{u^2}}{u^{2m}}\,du, \]

where the “plus” sign must be taken if \(s_0\) lies in the upper half-plane, and the “minus” sign if it lies in the lower half-plane (here \(w=\sqrt{kR}\,s_0\)).

As a result, for the integral (45) we obtain:

\[ I=\sum_{m=0}^{\infty} \frac{2\sqrt{\pi}\,c_{2m}w^{2m}e^{-w^2}} {4^m m!(kR)^m(\pm s_0)} \int_{w}^{i\infty}\frac{e^{u^2}\,du}{u^{2m}} . \tag{47} \]

Thus the problem is solved for any position of the pole, including arbitrarily close to the saddle point. The integrals in (47) admit numerical integration by ordinary methods.

In contrast to the usual saddle-point method, the final result (47) is not a series in powers of \(\frac{1}{kR}\), since \(kR\) also enters into \(w\). However, if the numerical distance is large, then, representing the integral in (47) by integration by parts in the form of a semiconvergent series in powers of \(\frac{1}{w}\), we obtain at once, as is not difficult to show, the same series in powers of \(\frac{1}{kR}\) that is obtained if the usual saddle-point method is applied to (45) (i.e., if \(\Phi(s)\) is expanded in a series in powers of \(s\) and termwise integration is carried out).

Let us now apply the method set forth above to our problem. The Hertz vector in the upper medium is given by the sum of expressions (19) and (21), which we shall write in the form

\[ \psi=\frac{e^{ikR}}{R}-\frac{ik}{2\pi}\int\limits_{\frac{\pi}{2}}^{i\infty}\int\limits_{0}^{2\pi} e^{ikx\cos\alpha\cos\varphi}\times \]

\[ \times\left[e^{ik(z+z_0)\sin\alpha}+\bigl(V(\alpha)-1\bigr)e^{ik(z+z_0)\sin\alpha}\right]\cos\alpha\,d\alpha\,d\varphi . \]

Here the integral of the first term in the square brackets, by analogy with (19) (in the latter one must put \(y=0\)), gives \(\dfrac{e^{ikR_1}}{R_1}\), i.e., a wave as if issuing from the fictitious radiator \(O'\) (see Fig. 2). The integral of the second term is reduced to the form (34), with the difference that instead of \(V(\alpha)\) there will stand \(V(\alpha)-1\). As a result we obtain (neglecting \(\dfrac{1}{8ikx\cos\alpha}\) in comparison with 1):

\[ \psi=\frac{e^{ikR}}{R}+\frac{e^{ikR_1}}{R_1}+ \]

\[ +\sqrt{\frac{k}{2\pi x}}\,e^{\frac{i\pi}{4}} \int\limits_{\pi-i\infty}^{i\infty} e^{ikR_1\cos(\alpha-\chi)}[1-V(\alpha)]\sqrt{\cos\chi}\,d\alpha, \tag{48} \]

where, according to (8), where \(m=n^2\),

\[ 1-V(\alpha)=\frac{2q(\alpha)}{M(\alpha)}, \]

where

\[ q(\alpha)=\sqrt{\,n^2-\cos^2\alpha\,},\qquad M(\alpha)=n^2\sin\alpha+q . \]

Making the substitution \(\cos(\alpha-\chi)=1+is^2\) and changing the path of integration so that the integration is performed over real values of \(s\) (the path of steepest descent), with

\[ d\alpha=\sqrt{2}\,e^{\frac{3\pi i}{4}}\, \frac{ds}{\sqrt{1+\dfrac{is^2}{2}}}, \tag{49} \]

we bring the integral in (48) to the form (45), after which, according to (47), we shall have

\[ \psi=\frac{e^{ikR}}{R}+\frac{e^{ikR_1}}{R_1}+ \]

\[ +\sqrt{\frac{2k}{x}}\,e^{ikR_1+\frac{i\pi}{4}} \sum_{m=0}^{\infty} \frac{c_{2m}w^{2m}e^{-w^2}}{4^m m!(kR_1)^m s_0} \int\limits_{w}^{i\infty}\frac{e^{u^2}}{u^{2m}}\,du . \tag{50} \]

Here \(c_{2m}\) are the expansion coefficients:

\[ \frac{2q(\alpha)}{M(\alpha)}(s^2-s_0^2)\frac{d\alpha}{ds} =\sum_{m=0}^{\infty}\frac{c_m}{m!}\,s^m . \tag{51} \]

From (51) we obtain*) for \(|n|\gg 1,\ \chi\ll 1\):

\[ c_0=\frac{2s_0}{n},\qquad c_2=\frac{i s_0}{2n}. \tag{52} \]

Substituting this into (49), we note that for large \(kR_1\) one may restrict oneself to the first term of the sum, and as a result we obtain:

\[ \psi=\frac{e^{ikR}}{R} +\frac{e^{ikR_1}}{R_1} \left( 1+\frac{1}{n}\sqrt{8kR_1}\,e^{\frac{i\pi}{4}-w^2} \int_w^{i\infty} e^{u^2}\,du \right). \tag{53} \]

In the special case when the radiator is located at the boundary of separation \((R=R_1)\), we obtain the well-known formula of Weyl–Van der Pol (see\(^{10}\), p. 105 or\(^{5}\), p. 954):

\[ \psi=\frac{2}{R}e^{ikR} \left( 1+\frac{1}{n}\sqrt{2kR}\,e^{-\frac{i\pi}{4}-w^2} \int_w^{i\infty} e^{u^2}\,du \right) \quad **). \]

7. Visual interpretation of the results. Limits of applicability of geometrical optics

In the preceding paragraphs we have seen that the field at an arbitrary point \(P\) is composed mainly of plane waves whose grazing angle differs from \(\chi\)—the grazing angle of the ray constructed according to the laws of geometrical optics—only by small quantities.

This is in full agreement with the known result\(^{16,17,18}\), according to which, in the reflection of a spherical wave from an infinite reflecting plane, an essential role is played not by the entire plane, but only by a certain “effective zone” on it. The latter embraces a definite number of Fresnel zones constructed about the ray reflected according to the laws of geometrical optics. In this case

*) Since \(M(s_0)=0\), then

\[ M(s)=(s-s_0)M'(s_0)+\frac{1}{2}(s-s_0)^2M''(s_0)+\cdots \]

and, consequently,

\[ \frac{s^2-s_0^2}{M(s)} = \frac{s+s_0}{M'(s_0)+\frac{1}{2}(s-s_0)M''(s_0)}. \]

Here

\[ M'(s_0)=\frac{dM(z)}{d(\alpha)}\,\frac{d\alpha}{ds}, \]

etc.

**) Van der Pol’s work dates to 1931. Shuleikin, already in 1923, obtained a formula giving the same numerical results\(^{25}\).

It is often assumed\(^{17,18}\) that the first Fresnel zone may be taken as the effective zone and that the remaining zones play no essential role. It can be shown that this is permissible only in the case when the order of magnitude of the field at the observation point \(P\) is being considered.

According to what was said above, the interval of angles essential in the integration of plane waves is, in order of magnitude, equal to

\[ \frac{1}{\sqrt{kR}} . \]

It can be shown that, when the transmitter and receiver are located sufficiently high above the interface, the angle in the \(xz\) plane at which the first Fresnel zone is seen from the receiver is, in order of magnitude, also equal to

\[ \frac{1}{\sqrt{kR_1}} . \]

However, as the transmitter and receiver approach the interface, the angle at which the first Fresnel zone is seen increases continuously, whereas the interval of essential angles in the saddle-point method remains unchanged. This is where the difference appears between the method of Fresnel zones, in which spherical waves from secondary radiators on the boundary are summed, and the method of integration of plane waves according to Weyl. Considerations connected with Fresnel zones were used by E. L. Feinberg\(^{19}\) in his work on the effective path of radio waves.

The ordinary saddle-point method is applicable in those cases when the reflection coefficient changes little as the angle varies within the indicated limits. In electrodynamics, in the case of large conductivities of the lower medium and small grazing angles, this condition is not fulfilled, which necessitates the use of a modified saddle-point method. In fact, for \(\alpha = 0\), according to (8), \(V = -1\) for any \(m\) and \(n\). However, if \(m\) (which in the electrodynamic case is equal to \(n^2\)) is large, then already for sufficiently small, but nonzero, \(\alpha\), \(V(\alpha) \simeq +1\). The greater \(n^2\) is, the shorter is the interval of \(\alpha\) within which \(V(\alpha)\) changes from \(-1\) practically to \(+1\).

From formula (8) we obtain, putting \(m = n^2\),

\[ \left(\frac{dV}{d\alpha}\right)_{\alpha=0} = -\frac{2n^2}{\sqrt{n^2 - 1}} . \tag{54} \]

As \(|n|\) increases, the derivative also increases, which means that for large \(|n|\) and sufficiently small \(\alpha\) the reflection coefficient is a very rapidly varying function. This is also natural, since in the preceding paragraph it was shown that in this case, in the complex plane of angles, a pole lies near the origin. The idea of the modified saddle-point method consists precisely in passing to a function which has no pole near the saddle point and therefore is slowly varying.

Let us note that \(\left(\dfrac{dV}{d\alpha}\right)_{\alpha=0}\), according to (54), is also large when \(n\) is close to \(1\). In this case, from the value \(V = -1\) at \(\alpha = 0\), the coeffi-

coefficient of reflection rapidly decreases to small values when \(\alpha\) is finite. Here, too, the usual method of reverberation is inapplicable*).

Let us now consider the question of when one may use the formulae of geometrical optics obtained in the first approximation of the reverberation method.

The second term in (36), according to (37), vanishes identically if \(V=\mathrm{const.}\), i.e. if the coefficient of reflection does not depend on the angle of incidence. This occurs, in particular, for absolutely reflecting surfaces, where \(V=\pm 1\)**). The applicability of geometrical optics in this case is well known; moreover, here it is convenient to represent the reflected wave as if it were emanating from an imaginary (“virtual”) radiator \(O'\) in Fig. 2. In acoustics, the reflection coefficient is independent of the angle, apart from cases of total reflection, also when \(c=c_1\) (\(n=1\)). Indeed, putting \(n=1\) in (8) and taking into account that \(m=\dfrac{\rho_1}{\rho}\), we obtain:

\[ V=\frac{\rho_1-\rho}{\rho_1+\rho}. \]

In this case geometrical acoustics is also strictly valid. An analogous situation also occurs for certain cases of anisotropic media.

In many practical cases the correction term in (30), although not identically zero, gives so small an addition to the total field of the direct and reflected waves that it may be neglected. Let us derive a criterion showing when this can be done.

Let us first turn to the case when the radiator or receiver (for definiteness we shall assume the former) is located at the interface. Then \(R=R_1\), and the condition for the smallness of the correction term in (36), in comparison with the sum of the direct wave \(\dfrac{e^{ikR}}{R}\) and the reflected wave \(V(\chi)\dfrac{e^{ikR}}{R}\), is written:

\[ kR\, |1+V(\chi)| \gg |N|. \]

Hence, using expression (8) for \(V(\chi)\) and expression (38) for \(N\), we obtain, for small \(\gamma=\sin\chi\):

\[ kz \gg \left|\frac{m\sqrt{\,n^2-1\,}}{\left(m\sin\chi+\sqrt{\,n^2-1\,}\right)^2}\right|. \tag{55} \]

*) The case of \(n\) very close to 1 has been considered in detail by the author elsewhere\(^3\).

**) The case \(V=-1\) is realized, for example, in the reflection of a sound wave traveling from water at the water–air interface.

If \(\chi\) is so small that \(m\sin\chi \ll \sqrt{n^2-1}\), (55) is written as:

\[ kz \gg \left|\frac{m}{\sqrt{n^2-1}}\right|. \tag{56} \]

Thus, for geometrical optics to be applicable, it is necessary that the elevation of the receiver (emitter) above the interface be sufficiently large in comparison with the wavelength.

For \(|m|\to\infty\) we have a transition to an absolutely reflecting boundary, since, according to (8), in this case \(V\to 1\). Correspondingly, the right-hand side of (55) tends to zero, whence it follows that geometrical optics (acoustics) is valid in this case for any values of \(z\).

If neither the emitter nor the receiver is located on the interface, then the criteria obtained remain approximately valid, but instead of \(z\) there will enter the sum \(z+z_0\), and

\[ \sin\chi=\frac{z+z_0}{R_1}. \]

8. Field of a Raised Emitter

In practical calculations of the intensity of the electromagnetic field one often uses a formula first obtained, apparently, by Gershelman\(^{20}\), and later, by another method, by Niessen\(^{21}\). This formula refers to the case of a raised emitter and, when this emitter is a vertical dipole, has the form

\[ \psi(z_0,x,z)=\frac{e^{ikR}}{R}-\frac{e^{ikR_1}}{R_1}+\psi(0,x,z+z_0). \tag{57} \]

Here we use the following notation: in the first argument of \(\psi\) stands the height of the emitter above the interface, while in the second and third places stand the coordinates of the point of observation. Thus, in order to obtain, for an arbitrary point \((x,z)\), the vertical component of the Hertz vector of a dipole raised to the height \(z_0\), one must take the direct wave and add to it: a) the wave coming from the imaginary emitter, taken with a minus sign; b) the vertical component of the Hertz vector of a dipole situated on the interface, calculated not for the point \((x,z)\), but for the auxiliary point \(P'(x,z+z_0)\). This prescription for calculating the field of a raised dipole is very useful, but appears somewhat “mystical.” Moreover, both derivations of formula (57) are quite cumbersome. In substance, formula (57) and the prescription following from it can be obtained from very transparent considerations\(*\).

\(*\) These considerations arose in a discussion of the question with B. V. Malanov.

The vertical component of the Hertz vector at an arbitrary point consists of the direct and reflected waves

\[ \psi(z_0,x,z)=\frac{e^{ikR}}{R}+\psi_r(z_0,x,z). \tag{58} \]

But, as is seen from the original expression (21) for the reflected wave, the quantities \(z\) and \(z_0\) enter it only in the form of their sum and, consequently, it will not change if the height of the emitter is decreased and the height of the receiver increased by one and the same amount. Consequently,

\[ \psi_r(z_0,x,z)=\psi_r(0,x,z+z_0). \tag{59} \]

This also follows from visual considerations, since under such a change in the heights of the emitter and receiver neither the total phase accumulation nor the angle of incidence on the boundary of any of the plane waves of which the field of the reflected wave is composed changes. In Fig. 7 this is shown for a wave incident on the boundary at the grazing angle \(\chi\). From this property of the reflected wave formula (57) also follows at once. Indeed, the reflected wave for an emitter located on the boundary, i.e. \(\psi_r(0,x,z+z_0)\), can be expressed through the total field \(\psi(0,x,z+z_0)\) for this case, since the total field is the sum of the direct and reflected waves:

\[ \psi(0,x,z+z_0)=\frac{e^{ikR_1}}{R_1}+\psi_r(0,x,z+z_0). \tag{60} \]

Fig. 7, showing that the length of the reflected ray \(R_1\) and the angle at which it is reflected \(\chi\) do not change if the emitter is lowered and the receiver raised by one and the same amount.

Here the first term represents the direct wave. The distance from the emitter to the receiver is here \(CP'=R_1\) (Fig. 7). Expressing from this \(\psi_r(0,x,z+z_0)\) and substituting in (59), we obtain \(\psi_r(z_0,x,z)\). Substitution, in turn, of this expression in (58) gives formula (57).

The derivation presented shows that formula (57) is also valid for the acoustic potential in the case of an emitter raised above the interface.

9. Lateral waves

In order to obtain a complete expression for the field of the reflected wave, one must add to expression (36) one more term, corresponding to the so-called lateral wave. We shall first give the for—

mathematical foundations for this and obtain an explicit expression for this term, and then clarify its physical meaning.

In § 4 the integration along the original path \(\Gamma_1\) was replaced by integration along the path \(\Gamma\) (Fig. 6). As we have already seen, the choice of different paths of integration physically means nothing other than the representation of one and the same field in the form of a superposition of plane waves, whose direction cosines are chosen according to different laws. Let us introduce into this transformation of the path of integration certain corrections.

Fig. 8. Picture in the complex plane in the presence of a lateral wave. The lateral wave is given by the integral over the path \(\Gamma_2\), running along the edges of the cut.

The integrand in (34) is double-valued, since \(V(\alpha)\) contains the radical
\[ \sqrt{\,n^2-\cos^2\alpha\,} \]
[see (8)], whose sign can be chosen in two ways. For convenience of reasoning in such cases one imagines two equivalent copies of the \(\alpha\)-plane (two sheets of the Riemann surface), on each of which the integrand will take different values, corresponding to one or the other choice of the sign of the root. Let on one sheet (we shall call it the “upper” one) the sign of the root be chosen so that
\[ \operatorname{Im}\sqrt{\,n^2-\cos^2\alpha\,}>0, \]
and on the other (the “lower” one)
\[ \operatorname{Im}\sqrt{\,n^2-\cos^2\alpha\,}<0. \]
These sheets are joined along the lines
\[ \operatorname{Im}\sqrt{\,n^2-\cos^2\alpha\,}=0; \]
one of them, lying in the region of the complex \(\alpha\)-plane that interests us, is shown in Fig. 8 as a dotted line. It begins at the point \(A\), where \(\alpha=\arccos n\) (the branch point), and goes off to negative imaginary infinity, approaching the straight line
\[ \alpha'=\frac{\pi}{2}. \]
Here it is assumed that \(n\) has a small imaginary part and \(\operatorname{Re} n<1\).

The original path of integration, according to condition (22), lies on the upper sheet. Its deformation into the saddle path of integration is possible only when the beginning and the end of the saddle path also lie on the upper sheet.

This will not be the case for the position of the branch point shown in Fig. 8. Here the saddle path of integration crosses the dotted line and, consequently, passes from one sheet to the other, since the imaginary part of the root \(\sqrt{\,n^2-\cos^2\alpha\,}\) then passes through zero and changes sign. To avoid this difficulty, we construct the path of integration as indicated in Fig. 8. From the point
\[ \frac{\pi}{2}+\chi-i\infty \]
it goes to the point
\[ \frac{\pi}{2}-i\infty \]
and then goes around

dotted line. After this it crosses the dotted line and, already on the lower sheet, goes to the point \(\dfrac{\pi}{2}+\chi-i\infty\), and from there along the normal path \(\Gamma\), the initial part of which, indicated by dots, lies on the lower sheet. Now the beginning and the end of the path of integration lie on the upper sheet, but to the path there is added the integral along both sides of the dotted line (the integral along the edges of the cut).

Thus, the complete expression for the reflected wave will consist of two parts:

\[ \psi_r=\psi'_r+\psi''_r, \tag{61} \]

where the first part is given by the integral along the saddle path and was investigated above [see (36)], while the second is given by the expression

\[ \psi''_2=-\sqrt{\frac{k}{2\pi x}}\,e^{i\frac{\pi}{4}} \left[ \int_{\frac{\pi}{2}-i\infty}^{\chi_0} e^{ikR_1\cos(\chi-\alpha)}V(\alpha)\sqrt{\cos\alpha}\,d\alpha + \int_{\chi}^{\frac{\pi}{2}-i\infty} e^{ikR_1\cos(\chi-\alpha)}V^{+}(\alpha)\sqrt{\cos\alpha}\,d\alpha \right] \tag{62} \]

(compare with (34); we neglect \(\dfrac{1}{kx}\) in comparison with unity). Here the notation \(\chi_0=\arccos n\) has been introduced. \(V^{+}(\alpha)\) differs from \(V(\alpha)\) in that in it the sign of the root \(\sqrt{\,n^2-\cos^2\alpha\,}\) has been changed to the opposite*). Interchanging the limits of integration in the first integral in (62), we can reduce both integrals to one, taken from \(\chi_0\) to \(\alpha=\dfrac{\pi}{2}-i\infty\), with the difference \(V^{+}-V\) standing under the integral. According to (8) we have:

\[ V^{+}-V= \frac{4m\sin\alpha\sqrt{\,n^2-\cos^2\alpha\,}} {(m\sin\alpha)^2-n^2+\cos^2\alpha}. \]

It is convenient to carry out all further calculations by the method of steepest descent, analogous to the saddle-point method. For this purpose we deform the path of integration so that from the branch point \(\alpha=\chi_0\) it goes along the line on which the phase of the expression \(e^{ikR_1\cos(\alpha-\chi)}\) remains constant, i.e. \(\operatorname{Re}\cos(\alpha-\chi)=\text{const.}\) (the path \(\Gamma_2\) in Fig. 8). Since it is assumed that \(kR_1\gg1\), the exponential will have an appreciable value only in a small region near

\[ \text{*) In going around the branch point, the square root changes sign (see, for example, § 92).} \]

from the beginning of the path of integration, whose width is of order \(\dfrac{1}{\sqrt{kR_1}}\) (see above, § 6). Expanding, as in the saddle-point method, the remaining part of the integrand in a series in powers of the deviation from the branch point (where the zero term here vanishes), after integration we obtain:

\[ \psi_r''= \frac{2 i n e^{ik\left[nr+\sqrt{1-n^2}(z+z_0)\right]}} {mkx^2(1-n^2)\left(1-\operatorname{ctg}\chi_0\,\operatorname{tg}\chi\right)^{3/2}} . \tag{63} \]

The results given apply to the case in which the branch point lies to the right of the saddle path of integration \(\Gamma\) (Fig. 8), i.e. to the case \(\chi<\chi_0\). A ray constructed according to the laws of geometrical optics would then undergo total internal reflection. If the branch point lies to the left of the path \(\Gamma\), then the latter will cross the cut twice, so that its beginning and end will lie on the upper sheet of the Riemann surface. As a result, the need for a contour in the integral encircling the cut disappears, and the entire solution for the reflected wave will be given by the integral along the saddle path \(\Gamma\) studied in the preceding paragraphs. Thus, the lateral wave is a phenomenon of the same character as total internal reflection in the case of plane waves, with the difference that it occurs not only for \(n<1\), but also for \(n>1\) (see below).

Fig. 9. Arrangement of the front of the lateral wave relative to the fronts of the direct and reflected spherical waves for the case when the speed in the lower medium exceeds the speed in the upper one.

As is seen from formula (63), in the \(r,z\) plane the front of the lateral wave is rectilinear (in space it will be conical) and normal to the straight line issuing from the imaginary source at the angle \(\chi_0\), as shown in Fig. 9. The lower edge of the front of the lateral wave coincides with the edge of the front of the wave propagating in the lower medium with velocity \(c_1\) (the case \(c_1>c\) is considered). Its upper front merges with the front of the reflected wave. Formula (63) shows that the amplitude of the lateral wave decreases with distance as \(\dfrac{1}{x^2}\), and increases on moving away from the boundary (as \(\chi\) increases) owing to the increase of the factor \(\left(1-\operatorname{ctg}\chi_0\,\operatorname{tg}\chi\right)^{-3/2}\). For \(\chi=\chi_0\), formula (63) is invalid. This case will be considered in the following paragraph.

The lateral wave, just like the second term in (36), is a correction to the field computed in the geometrical-optics approximation [the first term in (36)]. This is evident, for example, from the fact that as \(\lambda \to 0\) \(\left(k=\frac{2\pi}{\lambda}\to\infty\right)\), which corresponds to the transition to geometrical optics, the amplitude of the lateral wave in (63) tends to zero. Nevertheless, the lateral wave admits a vivid interpretation with the aid of notions analogous to those of geometrical optics. For this one must imagine that, in addition to the ordinary ray reflected from the boundary at the grazing angle \(\chi\) (the ray \(OQP\) in Fig. 10), there also arrives at the point \(P\) from the point \(O\) a ray which falls on the boundary at the angle of total internal reflection \(\chi_0\), then propagates in the lower medium along the interface, and finally emerges again into the upper medium, again at the angle of total internal reflection (the ray \(OCDP\)). For this it is necessary, of course, in agreement with what we had above, that \(\chi<\chi_0\). The ray \(OCDP\) has this in common with the ordinary ray \(OQP\): both correspond to a minimum of the length of the optical path (Fermat’s principle).

Fig. 10. The ray \(OQP\) corresponds to the reflected spherical wave; the ray \(OCDP\), to the lateral wave.

Indeed, it can be shown that the phase lag in the propagation of the wave from \(O\) to \(P\) along the ray \(OCDP\) will be smaller than for all other possible rays whose paths lie partly in the lower medium*). This phase lag will also be smaller than for the ray \(OQP\).

From the wave point of view, the origin of the lateral wave must be imagined as follows.

The expansion of the spherical wave contains plane waves which, upon incidence on the boundary, produce in the lower medium inhomogeneous waves propagating along the boundary. Among such plane waves are, for example, ordinary plane waves having a grazing angle \(\alpha<\chi_0\). All the inhomogeneous waves in the lower medium, adding together, at large distances give a wave process propagating along the boundary with velocity \(c_1\) and producing at the boundary a disturbance with spatial period \(\lambda_1=\frac{2\pi}{k_1}\). This disturbance causes the appearance of a new wave in the upper medium. However, since the wavelength \(\lambda\) in the upper medium is smaller than \(\lambda_1\) \((\lambda=\lambda_1\cos\chi_0)\), in order to have periodicity along the boundary with period \(\lambda_1\), the wave must be inclined so that the normal to its front forms the angle \(\chi_0\) with the boundary. It is precisely in this way that the front of the lateral wave is arranged.

*) See\({}^{5}\), Ch. XII and, in particular, p. 528.

A lateral wave for this case is analogous to the well-known “head wave” of Mach, and the angle \(\chi_0\) is the Mach angle.

Although the lateral wave is only a correction to geometrical optics, in a number of cases it plays the principal role. Certain methods in seismometry, in particular, are based on its observation (the Mintrop wave).

Under laboratory conditions Schmidt\(^{22}\) observed the lateral wave in the reflection of sound waves from the boundary between two liquids. In Fig. 11 one of his numerous photographs, obtained by the Toepler method, is reproduced.

Fig. 11. Image of the reflection of a spherical wave, obtained experimentally. The rectilinear front of the lateral wave is clearly visible.

In the photograph the front of the lateral wave is clearly visible. An increase in the amplitude of this wave along the front as it recedes from the boundary of separation is also noticeable.

In an analogous way the integral along the edges of the section is analyzed for \(n > 1\) (the velocity in the lower medium is smaller). The resulting analytic expression for the lateral wave will coincide with (63). Assuming in (63) \(n > 1\), we obtain a wave with a front normal to the interface, propagating along it with velocity \(c_1\) and exponentially attenuating in the direction \(z\).

The occurrence of the lateral wave in this case is explained by the same considerations as above. The only difference is that in this case, to the wave propagating in the lower medium along the boundary with velocity \(c_1\), there corresponds, according to the law of refraction (7), in the upper medium not an ordinary wave with an inclined front, but an inhomogeneous wave. This also explains the occurrence of the exponential factor giving attenuation in the direction \(z\).

The lateral wave plays the main role in the case when the radiator and the receiver are in a strongly absorbing medium near the boundary.

with a weakly absorbing medium. This occurs, for example, when the radiator and receiver of radio waves are located in the earth, near its surface, at a large distance from one another. In this case the direct and reflected waves will propagate in the earth and therefore will be attenuated. As a result there remains a lateral wave, which penetrates from the earth into the air, propagates there, and then again enters the earth. By its propagation to the side of the main path lying in the earth, this wave justifies its name, lateral.

If by \(n\) we understand the refractive index of the absorbing medium relative to the nonabsorbing one, then the formula for the vertical component of the Hertz vector is obtained from (63) by replacing \(n\) by \(\dfrac{1}{n}\) and \(k\) by \(k_1\), since the upper and lower media have now exchanged roles. In doing this it must be taken into account that \(m=n^2\).

As a result, for the vertical component of the Hertz vector in this case we obtain

\[ \psi=\frac{2in^2}{kx^2}\, \frac{e^{ik\left[x+\sqrt{n^2-1}\,(z+z_0)\right]}} {(n^2-1)\left(1-\operatorname{ctg}\chi_0\,\operatorname{tg}\chi\right)^{3/2}}, \tag{64} \]

where, as above, \(\chi_0=\arccos n\); however, now this is a complex angle, since \(n\) is complex.

As we see, with distance the field decreases proportionally to \(\dfrac{1}{x^2}\), while with the deepening of the radiator and receiver into the absorbing medium it decreases as \(e^{-\varkappa(z+z_0)}\), where \(\varkappa=k\,\operatorname{Im}\sqrt{n^2-1}\).

This result was first obtained, by an entirely different method, by V. V. Vladimirskii and M. A. Leontovich.

In acoustics an analogous situation arises when the radiator and receiver of high-frequency sound are located in a layer of water containing air bubbles in sufficiently large concentration. Here the waves propagating in the layer itself also attenuate rapidly. The expression for the sound potential in this case is obtained from (64) by replacing the factor \(2in^2\) by \(2i\dfrac{\rho_1}{\rho}\).

10. Field in a region close to the angle of total internal reflection

The results obtained in §§ 5, 9 become invalid when the angle of sliding lies sufficiently close to the angle \(\chi_0=\arccos n\). Indeed, for \(\chi=\chi_0\) the amplitude of the lateral wave (63) tends to infinity, which is devoid of any physical meaning. The quantity \(N\) in the correction term in (36) behaves in the same way, as is not difficult to show; [from (38) this is not evident, since in it it has already been assumed that \(\chi\ll\chi_0\)]. These peculiarities are explained by the fact that, for \(\chi\) close to \(\chi_0\), use of the saddle-point method becomes

illegal, since the integrands in the corresponding integrals [for example, in (21)] in this case are not slowly varying.

The calculation of the lateral wave over the whole interval of angles where it exists, and also of the spherical reflected wave for angles \(\chi\) close to \(\chi_0\), was carried out by the author\(^1\). Using the value of one integral calculated by V. A. Fock, the expressions for both waves could be found in terms of Weber functions (functions of a parabolic cylinder). In this case the usual condition \(kR_1 \gg 1\) must be satisfied. The expressions obtained in this way pass over into the expressions (36) and (63) obtained above if the condition

\[ kR_1(\chi-\chi_0)^2 \gg 1, \tag{65} \]

is satisfied, i.e. if \(\chi\) is sufficiently far from the angle of total internal reflection.

Without giving the general, rather cumbersome expressions, we shall indicate only that at the point \(\chi=\chi_0\) itself the expression for the total field of the reflected wave \(\psi_r=\psi'_r+\psi''_r\), including also the lateral wave, whose front, according to Fig. 9, here merges with the front of the reflected spherical wave, will be:

\[ \psi_r=\frac{e^{ikR_1}}{R_1} - \frac{2.32ne^{\,i\left(kR_1+\frac{\pi}{8}\right)}}{ mR_1 (kR_1)^{\frac14}\sqrt{\sin 2\chi_0} } \, . \tag{66} \]

For sufficiently large \(kR_1\), the second term may be neglected. The remaining first term, as in (36), is the field obtained in the approximation of geometrical optics (acoustics), since according to (8), \(V(\chi_0)=1\). However, the correction to geometrical optics given by the second term decreases with distance considerably more slowly than in (36), namely as

\[ \frac{1}{(kR)^{\frac14}}, \]

and not as

\[ \frac{1}{kR_1}. \]

The lateral wave \(\psi''_r\) at \(\chi=\chi_0\) has a finite value equal to half of the second term in (66). The complete formulas, which we do not give here, show that, when the angle \(\chi\) is increased starting from \(\chi=\chi_0\), the reflected wave has a single spherical front, as is seen in Fig. 9; when \(\chi\) is decreased starting from \(\chi=\chi_0\), it is divided into two parts, one of which preserves the spherical front, while the other part, which we call the lateral wave, has a conical front, which again is seen in Fig. 9. As \(\chi\) recedes from \(\chi_0\), the amplitude of the lateral wave continuously decreases. For sufficiently large \(\chi-\chi_0\), when condition (65) is satisfied, the lateral wave is given by expression (63). If the radiator is located at the boundary of separation, then the minimum grazing angle is \(\chi=0\). Comparing the value then obtained from formula (63) with half of the second

term in (66), we see that the amplitude of the lateral wave in this case at the boundary is

\[ 0.58\,(kR_1)^{3/4}\frac{1-n^2}{n^2\sqrt{\sin^2\chi_0}} \tag{67} \]

(or, in order of magnitude, \((kR)^{3/4}\) times smaller than for \(\chi=\chi_0^*\)).

II. REFRACTION OF SPHERICAL WAVES

The field of refracted spherical electromagnetic and sound waves is of practical interest in a number of cases, for example, in calculating the field of radio waves beneath the earth from a radiating antenna located in the air, in calculating the sound field under water produced by an aerial radiator, etc. The field of refracted waves was investigated in the works of Krüger\(^4\), Ott\(^2\), Džerdžoj\(^{{21}}\) and the author\(^{{15}}\). Below, the results of these works are set out in a compact form permitting a clear physical interpretation of the question.

11. The field of the refracted wave in the approximation of geometrical optics

Through any point \(S\) in the lower medium there passes one of the rays issuing from the radiator \(O\) and refracted at the boundary according to the law of refraction of geometrical optics. In Fig. 12 this is the ray \(OTS\). It is shown for the case \(n<1\), i.e. \(c_1>c\). The grazing angles \(\chi\) and \(\chi_1\), as we know [see (7)], are related by

\[ n\cos\chi_1=\cos\chi,\qquad n=\frac{k_1}{k}. \tag{68} \]

The amplitude of the wave at the point \(S\) in the approximation of geometrical optics can be calculated from the condition of conservation of energy inside the ray tube, and the phase from the optical length of the ray. Let us first calculate the phase. It is equal to

\[ 2\pi\left(\frac{OT}{\lambda}+\frac{TS}{\lambda_1}\right)=k\cdot OT+k_1\cdot TS, \]

where \(k\) and \(k_1\) are wave numbers, and \(OT\) and \(TS\) are the lengths of the segments of which the ray consists.

Fig. 12. Ray diagram for refraction of a spherical wave.

For these lengths, according to Fig. 12, we have the expressions

\[ OT=\frac{z_0}{\sin\chi},\qquad TS=\frac{D}{\sin\chi_1}, \]

where \(D=-z\) is the distance of the observation point \(S\) from the interface,

\[ \text{*) Here we do not dwell on the case of } n \text{ close to } 1,\text{ when } \chi_0\cong \sqrt{1-n^2} \text{ is close to zero. On this see elsewhere.}^{23} \]

Taking into account the law of refraction (68), we obtain for the phase advance:

\[ k\left(\frac{z_0}{\sin\chi}+\frac{n^2D}{\sqrt{\,n^2-\cos^2\chi\,}}\right). \tag{69} \]

The angle \(\chi\), for a given position of the source and receiver, is found by eliminating \(\chi_1\) from equation (68) and the equation

\[ z_0\operatorname{ctg}\chi+D\operatorname{ctg}\chi_1=x, \tag{70} \]

whose geometrical meaning is obvious.

To determine the amplitude of the wave at the point \(S\), let us construct the ray \(OT'Q\) (Fig. 12), lying in the plane \(OTS\) and incident on the boundary at a grazing angle \(\chi-\Delta\chi\) close to \(\chi\). Let us now consider the rays enclosed between the cones formed by rotating the rays \(OTS\) and \(OT'Q\) about the \(z\)-axis. The energy carried by these rays is partly reflected at the boundary, and partly passes into the lower medium and propagates there between \(TS\) and \(T'Q\). Let the amplitude of the wave at the point \(T\) in the lower medium be \(A(T)\). To determine \(A(S)\)—the amplitude of the wave at \(S\)—it is necessary to establish the relation between the length of the segment \(SQ\), which is perpendicular to \(T'Q\) at \(S\), and the length of the segment \(T'Q'\), which is perpendicular to the same line at the point \(Q'\). The energy flux through the two rings formed by the rotation of \(Q'T'\) and \(QS\) is the same. Therefore, the ratio of the wave amplitudes at the points \(T\) and \(S\) is equal to the square root of the inverse ratio of the areas. The area of the ring formed by rotating \(Q'T'\) is equal to \(2\pi(CT)(Q'T')\), while the area of the ring formed by rotating \(SQ\) will be \(2\pi x(SQ)\). Consequently,

\[ \frac{A(S)}{A(T)}=\sqrt{\frac{(CT)(Q'T')}{x(SQ)}}. \tag{71} \]

But

\[ CT=z_0\operatorname{ctg}\chi, \]

\[ TT'=z_0\frac{d\chi}{\sin^2\chi}, \]

\[ Q'T'=TT'\sin\chi_1=\frac{z_0\sin\chi_1\,d\chi}{\sin^2\chi}. \tag{72} \]

\(SQ\) is equal to the length of the perpendicular \(Q'T'\) at the boundary plus the additional distance arising from the rotation of the ray in the plane of incidence:

\[ SQ=Q'T'+TS\,d\chi_1=Q'T'+D\frac{d\chi_1}{\sin\chi_1}. \tag{73} \]

\(d\chi_1\) is found from equation (68), which, after being differentiated, gives:

\[ \sin\chi\,d\chi=n\sin\chi_1\,d\chi_1. \]

The field \(\psi_1\) at the point \(T\) below the interface is related to the field \(\psi\) above the interface by the boundary condition [see (6)]

\[ \psi_1=\frac{1}{m}\psi. \]

But above the interface the amplitude of the wave will be:

\[ |\psi|=\frac{1}{R}\left[1+V(\chi)\right]. \tag{74} \]

Substituting this into the preceding formula and using (71), we obtain the amplitude of the wave at the point \(S\). Then taking into account the phase increment given by formula (69), after some transformations we obtain the complete value of the vertical component of the Hertz vector or of the acoustic potential in the approximation of geometrical optics:

\[ \psi_1(S)= \frac{ 2\sqrt{\cos\chi}\, e^{ik\left[\frac{z_0}{\sin\chi}+\frac{nD}{\sin\chi_1}\right]} }{ \sqrt{ x\left(z_0\sin^{-3}\chi-\frac{1}{n}D\sin^{-3}\chi_1\right) (m\sin\chi+n\sin\chi_1) } }, \tag{75} \]

where, according to (7),

\[ \sin\chi_1=\frac{1}{n}\sqrt{\,n^2-\cos^2\chi\,}, \]

and the angle \(\chi\) is found from equations (70) and (68).

Expression (75) is valid if the distances of the radiator and the receiver from the interface are sufficiently large in comparison with the wavelength, as will be seen from the more exact solution given in the next paragraph.

12. Corrections to geometrical optics for refracted waves

A more exact determination of the field of the refracted wave can be carried out by investigating the integral \((21')\) by the method of steepest descent, analogously to how this was done in § 5 for the reflected wave.

To elucidate the physical meaning of the corrections thus obtained, let us consider Fig. 13, where two cases of wave refraction are shown: for \(n>1\) and \(n<1\). In both cases \(OTS\) is a ray constructed according to the laws of geometrical optics. The corresponding expression for the Hertz vector or acoustic potential is given by formula (75).

As in the case of the reflected wave, the refinement of geometrical optics proceeds along two lines:

1. Taking into account the second approximation in the method of steepest descent, owing to which one more term of the next order of smallness with respect to the small quantity \(\frac{1}{kR}\) is added to expression (75).

1. The appearance of a wave of a new type, obtained by integration along the edges of the cut and analogous to the lateral wave in the case of reflected waves. The transfer of energy from the emitter to the receiver by this wave is effected in a way essentially different from that in geometrical optics. In Fig. 13 these are the paths \(OMS\). In case (b) the meaning of this wave is especially simple. Here it is the well-known wave, exponentially attenuating with depth into the “lower” medium, which is obtained when the ray \(OM\), incident on the interface at an angle more grazing than the angle of total internal reflection \((\cos \beta > n)\), is reflected. In case (a) it arises because the inhomogeneous plane waves, exponentially attenuating in \(z\),

Fig. 13. Two paths by which the energy of the radiation reaches, from the emitter \(O\), the point \(S\) of the lower medium. The path \(OTS\) corresponds to the ordinary ray constructed according to the laws of geometrical optics; the path \(OMS\) is foreign to geometrical optics. The system of horizontal dashes represents the inhomogeneous wave.

present in the expansion of the spherical wave issuing from \(O\) (in Fig. 13 they are represented by a system of horizontal dashes), upon falling on the boundary, excite in the lower medium ordinary plane waves propagating at all grazing angles \(\beta\) satisfying the condition \(\cos \beta > \dfrac{1}{n}\).

Let us note that case (a) is obtained from (b) if, in the latter, the emitter and receiver are interchanged.

Without giving the detailed investigation in the complex plane, which is of exactly the same form as in the case of reflected waves, we write the final results for the vertical component of the Hertz vector and of the acoustic potential.

Case \(n > 1\). Expression (75), in a refined form, is written as:

\[ \psi_1(S)=\frac{2}{x}\,e^{ik\left(x+D\sqrt{n^2-\cos^2\gamma}\right)} \left( \sqrt{\frac{x}{z_0}}\, \frac{\sin^{3/2}\gamma}{m\sin\chi+\sqrt{n^2-\cos^2\chi}} + \frac{im}{(n^2-1)kx} \right), \tag{76} \]

REFLECTION AND REFRACTION OF SPHERICAL WAVES

where it is assumed that \(\chi \ll 1,\ m\chi \ll \sqrt{n^2-1}\). If these assumptions are also taken into account in equalities (68) and (70), then we obtain:

\[ \chi \simeq \frac{z_0}{x-\dfrac{D}{\sqrt{n^2-1}}}. \tag{77} \]

In those cases where it is meaningful to refine the result of geometrical optics, the assumption of the smallness of \(\chi\) is always satisfied.

If in expression (75) one neglects \(D\sin^{-3}\chi_1\) in comparison with \(z_0\sin^{-3}\chi\) and transforms the exponent in a simple way, then it passes into (76) with the second term in parentheses discarded, as indeed it should, since this term gives the correction to geometrical optics. As \(\chi\to 0\), when the first term corresponding to geometrical optics vanishes, the second (correction) term becomes the principal one.

To expression (76) one must also add a wave of the type of a lateral wave, obtained from integration along the edges of the cut in the complex plane. It occurs only for sufficiently small angles \(\beta\) (Fig. 13), satisfying the condition

\[ \cos\beta > \frac{1}{n}. \]

Its analytic expression has the form

\[ \frac{2n}{R}\, e^{ik_1R-kz_0\sqrt{\,n^2\cos^2\beta-1\,}} \left[ \frac{\sin\beta}{n\sin\beta+im\sqrt{\,n^2\cos^2\beta-1\,}} + \frac{i}{m(1-n^2)kR} \right], \tag{78} \]

where

\[ R=\sqrt{x^2+D^2}. \]

As we see, the amplitude of this wave decreases exponentially as the emitter is moved away from the boundary of separation.

In the case \(n<1\), the corresponding formulas will be:

\[ \psi_1(S)= \frac{2n}{x}\, e^{ik_1x+ikz_0\sin\chi_1} \left[ \sqrt{\frac{x}{z}}\, \frac{\sin^{3/2}\chi_1}{n\sin\chi_1+im\sqrt{\,1-n^2\cos^2\chi_1\,}} + \frac{i}{m^2(1-n^2)kx} \right]. \tag{79} \]

Here it is assumed that \(\chi_1\ll 1\) and \(n\chi_1 \ll m\sqrt{1-n^2}\). From equations (68) and (70) we have:

\[ \chi_1 \simeq \frac{D}{x-\dfrac{nz_0}{\sqrt{1-n^2}}}. \tag{80} \]

The additional wave of the type of a lateral wave, which occurs for \(\cos\beta>n\), will be:

\[ \frac{2}{R}e^{ikR-kD\sqrt{\cos^2\beta-n^2}} \left[ \frac{\sin\beta}{m\sin\beta+i\sqrt{\cos^2\beta-n^2}} + \frac{im}{kR(n^2-1)} \right], \tag{81} \]

where \(R=\sqrt{x^2+z_0^2}\).

Here it has been assumed that \(m\beta \ll \sqrt{1-n^2}\). If, on the contrary, \(m\beta \gg \sqrt{1-n^2}\), then for the second term in brackets one would obtain \(2/kRm^2\sqrt{1-n^2}\sin^3\beta\).

It is seen from Fig. 13 that one case passes into the other under the substitution \(z_0 \rightleftarrows D\), \(n\to \frac{1}{n}\), \(m\to \frac{1}{m}\), \(\chi \rightleftarrows \chi_1\). It follows that, under such a substitution, formulas (76) and (78) must pass respectively into formulas (79) and (81), and conversely. This indeed takes place, as is not difficult to verify. In doing so one must also take into account the necessity of division by \(m\) of each of the formulas under such a substitution, which follows, for example in acoustics, from the fact that here we are basing ourselves on the principle of reciprocity, valid for sound pressure, whereas our \(\psi\) denotes the sound potential, differing from the pressure by the factor \(\rho\) in the upper medium and \(\rho_1\) in the lower.

Having obtained more accurate expressions for the Hertz vector and the sound potential, we can from them determine the limits of applicability of the geometrical-optics approximation.

In the case \(n>1\), from the condition that the second term in brackets in (76) be negligible, as also in § 7, we obtain:

\[ kz_0 \gg \frac{m}{\sqrt{n^2-1}}. \tag{82} \]

Thus, for geometrical optics to be applicable, it is necessary that the elevation of the emitter above the interface be sufficiently large in comparison with the wavelength. Let us note that, as \(z_0\) increases, the amplitude of wave (78) also tends to zero, as it should, since in geometrical optics this wave does not occur.

Similarly, in the case \(n<1\), the condition of smallness of the second term in (79) in comparison with the first gives:

\[ kz \gg \frac{1}{m\sqrt{1-n^2}}, \tag{83} \]

and this is the condition of applicability of geometrical optics for this case.

Thus, for the applicability of geometrical optics when \(n>1\), it is necessary that the emitter be sufficiently far removed from the boundary; the position of the receiver, however, plays no essential role (we recall, however, that the case \(z\ll x,\ z_0\ll x\) is being considered). For \(n<1\), on the contrary, it is necessary that the receiver be sufficiently far removed from the boundary.

13. The case when one of the media has appreciable absorption

Let us consider the case when the receiver is in an absorbing medium, and the emitter in a nonabsorbing medium. In electrodynamics this case is realized when the receiving antenna is placed in the earth or in sea water. In hydroacoustics, a situation analogous to this arises if the receiver is placed in water saturated with bubbles.

For a sufficiently large distance of the emitter and receiver from the interface, the problem is solved elementarily by means of geometrical optics (see formula (75), in which \(n\) must be regarded as a complex quantity, which, as is known, corresponds to the presence of absorption). We shall consider the case when these distances are not large in comparison with the wavelength.

When absorption is present in the lower medium, of the two waves considered above, the only one that will have appreciable amplitude at the point of reception is the wave whose principal part of the path lies in the upper, nonabsorbing medium. This will be the wave \(OTS\) in case (a) and the wave \(OMS\) in case (b) (Fig. 13).

Thus, in these two cases the Hertz vector, or the sound potential, will be given respectively by formulas (76) and (81). The latter differ from one another only in that one contains the angle \(\chi\), and the other \(\beta\). However, if in formula (77) for \(\chi\) we neglect \(z\) in comparison with \(x\), then we obtain \(\chi \simeq \dfrac{z_0}{x}\simeq \beta\).

Thus, for our case,

\[ m\psi_1(x,D)=\frac{2}{R}e^{ikR+ikD\sqrt{n^2-\cos^2\chi}} \left[ \frac{\sin\chi}{m\sin\chi+\sqrt{n^2-\cos^2\chi}} + \frac{im}{(n^2-1)kR} \right], \tag{84} \]

where \(\sin\chi\simeq \dfrac{z_0}{x}\).

The form of formula (84) permits one important remark to be made concerning the field in the absorbing medium. Let us use formulas (15), (36), and (38) and compute the field \(\psi(x,0)\) in the upper medium near the very interface \((z=0)\). Making the corresponding approximations and comparing the result obtained with expression (84), we obtain:

\[ m\psi_1(x,D)=\psi(x,0)e^{-kD\sqrt{\cos^2\chi-n^2}}. \]

This formula simply relates the field in the lower medium at the point \(S\) (Fig. 12) with the field in the upper medium, taken at the very boundary of separation (point \(A\)). It shows that the depth of the receiver affects only the factor \(e^{-kD\sqrt{\cos^2\chi-n^2}}\). Taking the modulus of both sides of the last formula, we obtain the following law of decrease of the wave amplitude with depth:

\[ m\left|\psi_1(x,D)\right|=\left|\psi(x,0)\right|e^{-\varkappa D}, \tag{85} \]

where

\[ \varkappa=\frac{2\pi}{\lambda}\operatorname{Re}\sqrt{\cos^2\chi-n^2}. \]

This formula was first obtained by M. A. Leontovich and V. V. Vladimirsky.

In acoustics, formula (85), according to (3), can be rewritten directly for the sound pressure:

\[ \left|p_1(x,D)\right|=\left|p(x,0)\right|e^{-\varkappa D}. \tag{86} \]

Similarly, in electrodynamics, instead of formula (85) one may use directly the formulas for the components of the field, which have the form[^15]

\[ |n|^2\left|E_{1z}(x,D)\right|=\left|E_z(x,0)\right|e^{-\varkappa D}, \tag{87} \]

\[ \left|E_{1x}(x,D)\right|=\left|E_x(x,0)\right|e^{-\varkappa D}, \tag{88} \]

and a formula analogous to (88) for \(H\).

Let us note that the results used here [in particular, relation (86)] are valid only under the condition that the pole does not lie too close to the saddle point (see § 6). This means that, for given parameters of the media, the wavelength must not be too large. However, by means of the modified saddle-point method set forth in § 6, it can be shown[^15] that formulas (85), (87), and (88) are valid in any case.

14. Sound field in water from a radiator located in air

The sound field in water, produced by a radiator located in air, can be calculated from formulas (79) and (81), and for this particular case \(m \simeq 800\), \(n \simeq \dfrac{2}{9}\). Since \(m\) is very large, the second terms in the brackets in (79) and (81) may be neglected [see what was said after formula (81)].

As a result, the sound potential proves to consist of two parts. The first corresponds to geometrical acoustics and, according to formula (79), taking also formula (80) into account, has the following amplitude:

\[ |\psi_1(x,D)|=\frac{2nD}{m\sqrt{x(1-n^2)}\left(x-\dfrac{nz_0}{\sqrt{1-n^2}}\right)^{3/2}} . \tag{89} \]

The second part of the sound potential is different from zero only when the condition

\[ \cos\beta=\frac{x}{\sqrt{x^2+z_0^2}}>n,\quad \text{i.e. } \beta<77^\circ . \]

is satisfied.

From (81), assuming that \(m\beta\gg 1\), we obtain for the amplitude of this part of the field:

\[ \frac{2e^{-kD\sqrt{\cos^2\beta-n^2}}}{m\sqrt{x^2+z_0^2}}, \tag{90} \]

We see that this part of the sound potential decays exponentially with depth into the water. However, for small \(D\) the amplitude of this wave may exceed many times the amplitude of the wave (89), corresponding to geometrical optics.

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