An analysis is presented of reflection from a marine sediment consisting of a homogeneous mud layer overlying a sand-mud basement, the latter with an upward-refracting, inverse-square sound speed profile. Such layering is representative of the sediment at the New England Mud Patch (NEMP). By applying appropriate integral transforms and their inverses to the Helmholtz equations for the ocean and the two sediment layers, along with the boundary conditions, a Sommerfeld–Weyl type of wavenumber integral is obtained for the cylindrical-wave reflection coefficient of the sediment, *R*. A stationary phase evaluation of this integral yields a closed-form expression for the plane-wave reflection coefficient, *R*_{0}. In the absence of attenuation, the plane-wave solution exhibits total reflection up to a critical grazing angle, *a*_{c}, but when attenuation in the sediment is introduced, the region of total reflection in |*R*_{0}| is replaced by a sequence of contiguous peaks. With realistic levels of sediment attenuation, the cylindrical-wave solution, |*R*|, exhibits a quasi-critical grazing angle, less than *a*_{c}, which is strongly dependent on the source-plus-receiver height above the seabed, which is mildly dependent on the depth of the mud layer but is essentially independent of frequency. Such behavior is consistent with independent experimental observations at the NEMP.

## I. INTRODUCTION

This paper is one in a series on the reflection coefficient of marine sediments. The first of the previous papers on the topic addresses semi-infinite sediments exhibiting an upward refracting sound speed profile, either linear or inverse-square.^{1} This was followed by an analysis of plane-wave reflection from a two-layer sediment, the top layer with an upward-refracting (linear or inverse-square) sound speed profile above a homogeneous basement.^{2,3} An unexpected phenomenon that emerged from that particular two-layer analysis was the presence of acoustic “glint,” whereby, in the kHz frequency range very narrow spikes of total reflection occurred at several discrete grazing angles. The distribution across grazing angles of the glint is heavily dependent on the depth of the top layer, the frequency, and to a lesser extent the gradient or shape of the profile in the layer.

Theoretical investigations of sound waves in stratified fluids have been pursued by previous authors, notably Tolstoy,^{4,5} Rutherford and Hawker,^{6} and Robins,^{7} who considered reflection from a fluid layer of varying density and sound speed sandwiched between two homogeneous fluids.

In 2017, the Office of Naval Research sponsored a research program to investigate the geo-acoustic and geo-physical properties of fine-grained sediments. In support of this effort, the international, multi-institutional Seabed Characterization Experiment^{8} (SBCEX17) was conducted on the New England Mud Patch (NEMP), located about 95 km south of Martha's Vineyard, MA. The sediment at the NEMP^{9–12} consists of a near-homogeneous layer of fine-grained material (mud), which increases in thickness from roughly 2 m to 12 m along a northwest to southeast transect approximately 15 km long. Beneath the mud, the basement consists of an initial admixture of sand, which builds in concentration steadily with increasing depth, eventually becoming a medium-to-coarse sand layer^{13} with little if any mud present. This description of the sediment stratification at the NEMP is, it should be noted, a considerably simplified version of the detailed surveys presented by Twitchell *et al.*,^{12} Goff *et al.*,^{9} and Chaytor *et al.*^{13}

At site SC2 in the southeast of the NEMP, where the near-homogeneous mud layer is relatively thick, extending to approximately 10.5 m beneath the seabed, Jiang *et al.*^{14} conducted acoustic reflection experiments, which returned data that they inverted to recover the sediment geo-acoustic properties: the sound speed, density, and attenuation as functions of depth. From their Table III,^{14} the results are as depicted by the solid blue circles in Fig. 1. Their values for the sound speed and attenuation in Figs. 1(a) and 1(c) are for a frequency of 1175 Hz. According to Table I in Jiang *et al.*,^{14} the sound speed at the base of the water column was 1472.3 m/s, which is substantially less than 1969 m/s, the deepest data point shown in Fig. 1(a), suggesting that the plane-wave reflection coefficient might be expected to show a critical grazing angle somewhere in the region of 41.6°.

Parameter . | Symbol . | Value . |
---|---|---|

Sound speed, seawater (m/s) | c_{1} | 1472.30 |

Density, seawater (kg/m^{3}) | ρ_{1} | 1023 |

Sound speed, mud layer (m/s) | c_{2} | 1477.78 |

Density, mud layer (kg/m^{3}) | ρ_{2} | 1750 |

Thickness, mud layer (m) | d | 10.5 |

Attenuation, mud layer (dB/m/kHz) | $\sigma \xaf2$ | 0.05 |

Density, basement (kg/m^{3}) | ρ_{3} | 2270 |

Source-plus-receiver height above seabed (m) | W | 36 |

Parameter . | Symbol . | Value . |
---|---|---|

Sound speed, seawater (m/s) | c_{1} | 1472.30 |

Density, seawater (kg/m^{3}) | ρ_{1} | 1023 |

Sound speed, mud layer (m/s) | c_{2} | 1477.78 |

Density, mud layer (kg/m^{3}) | ρ_{2} | 1750 |

Thickness, mud layer (m) | d | 10.5 |

Attenuation, mud layer (dB/m/kHz) | $\sigma \xaf2$ | 0.05 |

Density, basement (kg/m^{3}) | ρ_{3} | 2270 |

Source-plus-receiver height above seabed (m) | W | 36 |

In passing, it is worth noting that Jiang *et al.*,^{14} in their Figs. 6 and 7, show a sediment stratification at the SC2 site with interfaces at 11.7 m (mud base), 12.8 m (sand base), and 14.4 m (deep base). Above the mud base is a “geo-acoustic transition layer” between depths of 10.8 and 11.7 m. Although inferred from inversions yielding marginal posterior probability profiles, none of these boundaries is visible in the sediment sound speed data in Fig. 1(a), where at depths greater than 10.5 m, it is evident that the monotonic-increasing inverse-square profile displays a conspicuously good match to the data points.

The purpose of this paper is to develop an analysis of the plane wave and cylindrical-wave reflection coefficients of a two-layer sediment representative of the NEMP, based upon the idealized profiles depicted by the red lines in Fig. 1. The theoretical approach to be followed is similar to that in the previous analyses,^{1–3} but with obvious differences in the geo-acoustic properties of the layers. It is assumed that the ocean and upper sediment (mud) layers are homogeneous and that the lower (basement) layer supports an inverse-square sound speed profile [Fig. 1(a)] but a depth-independent density [Fig. 1(b)]. Shear is considered to be negligible in the fine-grained material at the NEMP. Attenuation in the ocean is treated as finite but vanishingly small, and initially, to clarify the discussion of the plane wave reflection coefficient, attenuation in both sediment layers is neglected. A little later, a realistic level of attenuation is introduced into the mud and basement layers, as illustrated by the red line in Fig. 1(c), which has a significant effect on both reflection coefficients.

In the theoretical development, the wave (Helmholtz) equations are set up for the ocean and the two sediment layers, and spatial Fourier transforms with respect to horizontal distance are applied over all three domains. A further spatial Fourier transform is performed over elevation in the (semi-infinite) ocean, followed by the corresponding inverse transform; and the partial differential equation for the basement, which supports the inverse-square profile, is transformed into the modified Bessel equation^{15} by an appropriate mapping of the depth coordinate. After evaluating the various constants of integration that are involved with the aid of the boundary conditions, a Sommerfeld–Weyl^{16,17} type of horizontal wavenumber integral for the field in the water column is obtained. A numerical evaluation of this integral yields the cylindrical-wave reflection coefficient, which incorporates the curvature of the wavefronts due to the proximity of the source and receiver to the seabed.

By setting the source and receiver far above the seabed and then applying a stationary phase analysis to the wavenumber integral, a closed-form expression emerges for the plane-wave reflection coefficient, which, with no losses present, exhibits a critical grazing angle, *α*_{c}, when evaluated under the geo-acoustic conditions of site SC2 at the NEMP. An apparent, or quasi, critical grazing angle,^{18} *α*_{q}, is predicted by the numerical integration for the cylindrical-wave reflection coefficient, which is less than *α*_{c} by an amount that depends on the source-plus-receiver elevation above the seafloor. Unlike the case of a top layer with an inverse-square profile,^{3} no glint is predicted when a similar profile is present in the basement layer.

## II. COORDINATE SYSTEMS AND THE INVERSE-SQUARE PROFILE

As in the previous analyses,^{1–3} two coordinate systems are used to characterize the acoustic field in the vertical: *z*, in the downward direction with its origin at a distance, *h*, above the mud-basement interface; and $z\xaf$ directed upwards with its origin at a distance, *d*, beneath the seabed at the bottom of the mud layer. These vertical coordinate systems, along with the sound speeds in the three domains, are illustrated in the schematic of Fig. 2.

^{3}the inverse-square profile, as depicted in Fig. 2, is given by the expression

*z*

_{T}is the transition depth and

*c*

_{∞}is the asymptotic value of

*c*

_{3}(

*z*) at great depth. Three parameters characterize the inverse-square profile:

*c*

_{∞},

*z*

_{T}, and

*h*, the latter being the value of the

*z*-coordinate at the top of the basement, where the sound speed is

*c*

_{3}(

*h*) =

*c*and the sound speed gradient is $\gamma =c3\u2032(h)=(dc3/dz)|z=h$. Some elementary algebra shows that

_{h}*z*and

_{T}*h*can be expressed in terms of the more intuitive

*c*,

_{h}*c*

_{∞}, and

*γ*as follows:

*c*,

_{h}*c*

_{∞}, and

*γ*.

## III. STRUCTURE OF THE SOLUTION FOR THE REFLECTION COEFFICIENT

*x*-

*z*plane at

*x*= 0 and elevation $z\xaf\u2032$ above the seabed is

*Q*is the source strength and

*δ*(.) is the Dirac delta function. (A line source embodies much the same physics as a point source but with somewhat simpler algebra.) The corresponding homogeneous Helmholtz equations for the mud layer and the basement are, respectively,

*j*= 1, 2, or 3 on a variable identifies the domain with which it is associated. The frequency dependent fields in the three domains are denoted by

*Ψ*, and the acoustic wavenumbers are

_{j}*k*=

_{j}*ω*/

*c*, where

_{j}*ω*is the angular frequency.

*x*is now applied to all three Helmholtz equations, and, on the equation for the ocean domain, which occupies the semi-infinite depth interval (

*d*, ∞), a further Fourier transform is performed with respect to $z\xaf$. An inverse Fourier transform with respect to vertical wavenumber then returns the following explicit expression for the field in the ocean as a function of angular frequency,

*ω*, horizontal wavenumber,

*p*, and elevation above the bottom of the mud layer, $z\xaf$:

*p*indicates that the associated field variable is a Fourier transform with respect to horizontal distance,

*x*, bearing in mind that

*x*and

*p*are a Fourier transform pair. This convention simplifies the notation considerably, especially when several spatial transforms are involved. The condition on the imaginary part of

*η*

_{1}in Eq. (8) implies a finite but infinitesimal attenuation in the ocean.

*ω*and

*p*by solving the associated differential equation directly. The solution is

*c*

_{3}, as expressed in Eq. (1), by making the transformation

^{15}$Iiv(|p|z)$ and $Kiv(|p|z)$, respectively, where

^{15}

*z*=

*h*. Eventually, this procedure leads to an expression for the wavenumber field in the water column, $\Psi 1p(z\xaf)$, containing the familiar sum of two terms representing the direct arrival and the bottom-reflected wave. On performing the Fourier inversion over horizontal wavenumber,

*p*, the frequency dependent field in the water column is found to be

^{19}the direct arrival, can be expressed explicitly as a Hankel function of the second kind of order zero: $\pi H0(2)(k1|z\xaf\u2212z\xaf\u2032|2+x2)$. The radical in this expression is the distance between the source and receiver; and the Hankel function itself appears because of the two-dimensional, (

*x*,

*z*), nature of the problem. The second integral, in effect a Sommerfeld–Weyl integral

^{16,17}representing the plane wave expansion of a cylindrical wave, takes account of the wavefront curvature due to the proximity of the source and receiver to the seabed. In the integrand, the plane wave reflection coefficient for horizontal wavenumber,

*p*, takes the form

*M*functions are

*b*in these expressions are density ratios,

_{jk}*z*, evaluated at

*z*=

*h*.

The second wavenumber integral in Eq. (15), taken over the real axis, includes all contributions, notably the normal modes in the seawater-mud-basement waveguide, to the reflected field in the water column. The characteristic equation for the modal eigenvalues may be obtained from Eq. (16) by setting the denominator to zero, which yields a transcendental equation whose solutions are the poles of *R*_{0}(*p*) in the complex *p*-plane. Also accommodated by the second integral in Eq. (15) is the curvature of rays due to the upward refracting, inverse-square profile in the basement, including the effects of any caustic that may appear in the water column as a result of such curvature.^{18}

## IV. DEFINING THE REFLECTION COEFFICIENTS

*R*

_{0}(

*p*) is as specified in Eq. (16). In the last term of Eq. (20) the integral in the denominator has been replaced by its explicit expression, the Hankel function of the second kind of zero order,

^{19}in the argument of which the radical is the distance between the image source and the receiver. In effect, the normalization removes the geometrical spreading of the reflected wave. The normalized integral in Eq. (20) is the generally valid (in that it includes wavefront curvature) cylindrical-wave reflection coefficient,

*R*, of a sediment with layering, as illustrated by the idealized profiles in Figs. 1 and 2.

*W*is indefinitely large, the integrand in the numerator of Eq. (20) is a rapidly varying function of

*p*, and the integral in the numerator may be evaluated by the method of stationary phase. With

*x*=

*W*/tan(

*α*), where

*α*is the specular grazing angle, the stationary point is at

*p*=

*p*

_{s}= −

*k*

_{1}cos(

*α*). From a standard stationary phase argument

^{20}in conjunction with the asymptotic form for the Hankel function

^{15}in the denominator, the reflection coefficient in Eq. (20) reduces identically to

*s*signifies that the associated variable is to be evaluated at the stationary point. For example. the vertical wavenumbers in Eq. (22) are

The function *R*_{0}(*p*_{s}) in Eq. (22) is the plane wave reflection coefficient of a sediment with idealized layering, as illustrated in Figs. 1 and 2. As a simple check, it is a straightforward matter to show that in the limit as *d* goes to zero, Eq. (22) reduces to the correct form for a semi-infinite basement with an inverse-square profile and no overlying homogeneous mud layer.^{1}

## V. MACDONALD'S FUNCTION OF COMPLEX ORDER

*N*in Eq. (18), has the integral representation

^{15}

*y*) > 0,

*ν*arbitrary, and in the present case,

*y*=

*ξ*

_{∞}

*h*. The derivative of this expression with respect to the argument

*y*, denoted by the prime, is

These two integrals may be computed numerically by a Simpson's rule or similar algorithm,^{21} taking care with the choice of the upper limit and the sampling rate, both of which may be determined from a visual inspection of the integrands. The function *N* in Eq. (18) may then be evaluated, along with *M*_{1} and *M*_{2} in Eqs. (17a) and (17b), respectively, thereby facilitating the determination of the cylindrical-wave [Eq. (20)] and plane-wave [Eq. (22)] reflection coefficients, *R* and *R*_{0}, respectively.

^{22}Although there are several such expansions, of greatest interest here is for

*y*=

*iw*, where, from Eq. (14),

*π*/2 < arg (

*y*) <

*π*, Macdonald's function can be expressed as

*iν*. The Debye asymptotic expansion of this Hankel function is

^{22}

*N*in Eq. (18) is obtained:

*M*functions in Eq. (17) at the stationary point, which in turn provide an approximation for the plane-wave reflection coefficient,

*R*

_{0}, in Eq. (22) This approximation, based on the Debye asymptotic expansion of the Hankel function in Eqs. (29), is quicker to compute and is almost indistinguishable from its numerically evaluated counterpart under the geo-acoustic conditions of site SC2 at the NEMP.

## VI. GEOACOUSTIC PROPERTIES OF SITE SC2 AT THE NEMP

Site SC2 is located in the southeast of the NEMP, where the mud layer is approximately 10.5 m thick. Figure 1 shows data on the sound speed structure, density, and attenuation in the mud layer and the basement, as reported by Jiang *et al.*^{14} in their Table III, along with the associated idealized profiles used here to evaluate the cylindrical-wave and plane-wave reflection coefficients in Eqs. (20) and (22), respectively. As an adjunct to Fig. 1, Table I summarizes the numerical values of the geo-acoustic parameters for the site, as specified in Jiang *et al.*;^{14} and Table II lists the inverse-square parameter values used here to characterize the sound speed and attenuation profiles in the basement, as illustrated by the red curves in Figs. 1(a) and 1(c). With the aid of the numerical values in Tables I and II, the expressions for the reflection coefficients, *R* and *R*_{0}, in Eqs. (20) and (22), respectively, may be evaluated as functions of grazing angle, *α*, with the thickness of the mud layer, *d*, the frequency, *f*, and the source-plus-receiver elevations above the seabed, *W*, treated as parameters.

Parameter . | Symbol . | Value . |
---|---|---|

Sound speed, top of basement (m/s) | c _{h} | 1477.78 |

Gradient, top of basement (s^{−1}) | γ | 300 |

Sound speed, infinite depth (m/s) | c_{∞} | 2050 |

Attenuation, top of basement (dB/m/kHz) | $\sigma \xafh$ | 0.347 |

Attenuation, infinite depth (dB/m/kHz) | $\sigma \xaf\u221e$ | 0.25 |

Offset of origin of z (m) | h | 2.366 |

Transition depth (m) | z _{T} | 2.275 |

Parameter . | Symbol . | Value . |
---|---|---|

Sound speed, top of basement (m/s) | c _{h} | 1477.78 |

Gradient, top of basement (s^{−1}) | γ | 300 |

Sound speed, infinite depth (m/s) | c_{∞} | 2050 |

Attenuation, top of basement (dB/m/kHz) | $\sigma \xafh$ | 0.347 |

Attenuation, infinite depth (dB/m/kHz) | $\sigma \xaf\u221e$ | 0.25 |

Offset of origin of z (m) | h | 2.366 |

Transition depth (m) | z _{T} | 2.275 |

## VII. PLANE-WAVE REFLECTION

### A. Lossless sediment

*c*

_{1}<

*c*

_{∞}, the radical in Eq. (24) is real for grazing angles 0 <

*α*<

*α*, where

_{c}*α*is given by

_{c}*α*is a critical grazing angle, which depends on the sound speeds in the water column and deep in the basement but is independent of the sound speed in the mud layer. If

_{c}*α*is a critical grazing angle, then |

_{c}*R*

_{0}| is expected to exhibit total reflection over the angular region 0 <

*α*<

*α*, and indeed it does. With

_{c}*ξ*

_{∞s}in Eq. (24) real, both integrals in Eqs. (26) are real, so

*N*is real, and so too is

*η*

_{1s}in Eq. (23a). In Eq. (23b),

*η*

_{2s}is real, imaginary, or zero, depending on whether cos(

*α*) is less than, equal to, or greater than

*c*

_{1}/

*c*

_{2}. It is easy to show that in all three cases the numerator and denominator of Eq. (22) are complex conjugates, yielding the result |

*R*

_{0}| = 1 for all 0 <

*α*<

*α*, representing total reflection. For the conditions of site SC2 at the NEMP, the critical grazing angle is $\alpha c=cos\u22121(c1/c\u221e)=44.09\u2218.$

_{c}The region of total reflection is illustrated in Fig. 3, which shows |*R*_{0}| from Eq. (22) for site SC2, under lossless conditions, plotted for two frequencies, 1175 and 2975 Hz (corresponding to the lowest and highest frequencies, respectively, used by Jiang *et al.*^{14} in the inversion of their SC2 data.) Beyond *α _{c}*, the red and blue curves represent, respectively, the numerical evaluation of the integrals representing the modified Bessel functions in Eqs. (26) and their Debye asymptotic approximations in Eqs. (29) and (30).

The red and blue curves in Fig. 3 are extremely well matched across the range of grazing angles from the critical up to normal incidence, with the blue curve almost completely masking the red curve. As the frequency increases, the rate of the oscillations rises but the peak-to-trough height across grazing angle remains essentially the same.

### B. Sediment attenuation

*k*

_{2}complex with the imaginary part, like the real part, independent of depth,

*β*

_{2}is the dimensionless loss tangent. Attenuation that is independent of depth in a fine-grained sediment, the mud layer in the present case, is consistent with the idea that the grain-to-grain contacts, which give rise to the losses in the viscous grain shearing (VGS) theory,

^{23–26}are maintained by electro-chemical or intermolecular forces rather than the overburden pressure due to gravity. Electro-chemical bonding and the associated attenuation are expected to be uniform in depth, unaffected by gravity, as in Fig. 1(c), whereas the overburden pressure would lead to a gravity-driven monotonic increasing attenuation with increasing depth,

^{26}which is not observed in the mud at SC2.

*k*

_{∞}is made complex so that, with

*β*

_{3}the loss tangent,

*k*

_{3}(

*z*), takes the form

^{24,26}

Figure 4 shows the plane-wave reflection coefficient, |*R*_{0}|, from Eq. (22), as a function of grazing angle for the two frequencies 1175 and 2975 Hz, computed using the geo-acoustic and inverse-square parameter values in Tables I and II, respectively. It is evident in Fig. 4 that the well-defined region of total reflection of the lossless case is no longer featured at kHz frequencies when attenuation is present. Instead, over the range of grazing angles up to the critical (i.e., the critical grazing angle that would be present in the absence of attenuation), the modulus of the reflection coefficient (red) exhibits a set of contiguous peaks of approximately uniform width, with peak values below unity. The width and level of the peaks both depend on frequency. Above the critical grazing angle, the numerical evaluation of the integrals in Eqs. (26) yields the red curve, which is masked by the almost identical blue curve from the Debye asymptotic approximation in Eq. (29).

## VIII. CYLINDRICAL-WAVE REFLECTION

At site SC2 on the NEMP the observed reflection coefficient, reported in Fig. 3 of Jiang *et al.*,^{14} exhibits a quasi-critical grazing angle, *α _{q}*, of approximately 21°, where the character of the reflectivity changes abruptly. Similarly, at two other sites on the NEMP, designated SWAMI (

*d*≈ 10 m) and VC31-2 (

*d*≈ 2 m), quasi-critical grazing angles of approximately 25° and 30°, respectively, have been observed.

^{27,28}These quasi-critical grazing angles show only a weak, if any, dependence on frequency, and in all three cases,

*α*is significantly less than the genuine critical grazing angle,

_{q}*α*= 44.09°, featured in the lossless, plane-wave reflection coefficient for SC2 (see Fig. 3).

_{c}The discrepancy between *α _{c}*, as predicted by the plane-wave analysis, and the quasi-critical grazing angles is consistent with the idea, originally proposed by Stickler,

^{18}that wave-front curvature, arising from the proximity of the source and receiver to the seabed, is responsible for the presence of an apparent critical grazing angle, which appears below and instead of

*α*. Under such circumstances, when the source and receiver are close to the seabed, the plane-wave condition

_{c}*W*➛ ∞ underpinning the stationary phase evaluation of the second wavenumber integral in Eq. (15) ceases to apply, in which case the expression for the plane-wave reflection coefficient in Eq. (22) no longer holds. It is then necessary to perform a numerical integration in order to evaluate the cylindrical-wave reflection coefficient

*R*in Eq. (20) using a Simpson's rule

^{21}or similar algorithm. As with the integrals in Eqs. (26) for Macdonald's function and its derivative, the upper limit and sampling rate must be chosen with care, in the present case based on a visual inspection of the integrand. Holland and colleagues

^{28,29}have embedded an integral, analogous to that on the right of Eq. (20), into their inversions for the geo-acoustic parameters of the fine-grained sediments at two sites, the NEMP and the Malta Plateau south of Sicily in the Mediterranean Sea.

The effect of wave-front curvature on the cylindrical reflection coefficient in Eq. (20) is illustrated in Fig. 5 for four source-plus-receiver elevations, *W*, above the seabed, under the environmental conditions of site SC2. As *W* rises, the quasi-critical grazing angle increases from 10° in panel (a), reaching 39° in panel (d). It is clear in Fig. 5 that, with increasing *W*, the quasi-critical grazing angle, *α _{q}*, approaches the true value,

*α*. Moreover, the cylindrical reflection coefficient in panel (d) is starting to resemble closely its plane-wave counterpart in Fig. 4(a).

_{c}As with the observations of Jiang *et al.*^{14} at site SC2, the cylindrical reflection coefficient in Eq. (20) hardly varies with frequency, at least in the low-to-medium kHz range. This is illustrated in Fig. 6, which shows |*R*| for the two frequencies 1175 and 2975 Hz, with *W* set equal to 36 m, corresponding to the experimental arrangement reported by Jiang *et al.*^{14} The two theoretical values of *α _{q}*, marked by the vertical dashed lines in Fig. 6, as identified by visual inspection, are almost indistinguishable at 21.7°. This frequency independent quasi-critical grazing angle closely matches that observed experimentally by Jiang

*et al.*,

^{14}as shown in their Fig. 3.

As mentioned earlier, the depth of the mud layer increases from about 2 m to 12 m along a transect from northwest to southeast at the NEMP. Although the quasi-critical grazing angle is sensitive to the depth of the mud layer, its effect is mild, as shown in Figs. 7(a) and 7(b), where the difference in the depths is 7.5 m, giving rise to a shift in *a _{q}* of about 5.4°. A more obvious difference between Figs. 7(a) and 7(b) is in the rate of the oscillations, which is significantly slower across all grazing angles in the case of the shallower mud layer. It is worth noting that in Fig. 7, several of the peaks exceed unity, resulting from refraction in the inverse-square basement, giving rise to the formation of one or more caustics,

^{18}but this does not violate the principle of conservation of energy.

## IX. CONCLUDING REMARKS

The fine-grained sediment at the NEMP, located on the continental shelf off the East Coast of the United States, consists of a near-homogeneous mud layer overlying a sand-mud basement. The proportion of sand increases with depth in the basement, giving rise to a monotonic increasing sound speed, which is well characterized by an upward-refracting, inverse-square profile. Along a 15 km transect from northwest to southeast on the NEMP, the thickness of the mud layer increases from approximately 2 m to 12 m.^{9,12,13}

An analysis of acoustic reflection from such a two-layer sediment is developed in this paper, based on a sequence of integral transforms as applied to each of the three layers (homogeneous) seawater, mud, and sand-mud basement. With aid of the boundary conditions, continuity of pressure and continuity of the normal component of particle velocity, an expression is developed in the form of a Sommerfeld–Weyl^{16,17} wavenumber inversion integral for the cylindrical reflection coefficient, *R*, of the sediment [(Eq. (20)]. Under the condition where the source and/or receiver are very far above the seabed, a stationary phase evaluation of the wavenumber integral leads to the analytical, closed-form expression for the plane wave reflection coefficient, *R*_{0}, in Eq. (22). The expressions for both *R* and *R*_{0} involve a modified Bessel function of the third kind, otherwise known as Macdonald's function, which is evaluated in two ways: numerically from its integral representation and in terms of the Debye asymptotic approximation.^{22} It turns out that these two approaches yield results for |*R*_{0}| that are visually almost indistinguishable over the range of grazing angles where the Debye approximation holds.

With values for the geo-acoustic parameters that are representative of site SC2 at the NEMP, and in the absence of attenuation in the sediment, the plane-wave reflection coefficient, *R*_{0}, exhibits total reflection at grazing angles up to the critical, *α _{c}* ≈ 44.09°, beyond which |

*R*

_{0}| shows oscillations with peak values of approximately 0.4. When realistic levels of attenuation are introduced, the region of total reflection is replaced by a succession of contiguous peaks, the widths of which are very sensitive to the frequency and to the depth of the mud layer. It is hypothesized that these peaks are directly associated with the normal modes in the waveguide formed by the seawater-mud-basement system.

With the source and receiver closer to the seabed but otherwise under the same SC2 conditions, the cylindrical-wave reflection coefficient, *R*, exhibits a quasi-critical grazing angle that is less than the genuine *α _{c}* by an amount that depends on the source-plus-receiver height above the seabed,

*W*, and the depth of the mud layer,

*d*. For example, with

*W*= 36 m and

*d*= 10.5 m, the predicted quasi-critical grazing angle is 21.7°, which is in excellent agreement with its experimentally determined counterpart, as reported by Jiang

*et al.*

^{14}in their Fig. 3. Moreover, the theoretical quasi-critical grazing angle is found to be essentially independent of frequency, at least over the low-to-mid kHz range, which is again in accord with experimental observations.

^{14}

To conclude, a few comments on density and attenuation profiles are in order. The density data in the basement [Fig. 1(b)], although treated as uniform in the present analysis, are well matched by an inverse-square profile. Such a density profile, however, appears to be intractable, prohibiting the development of an analytical solution for the reflection coefficient. In a discussion of density profiles, Robins^{30} has introduced various coordinate transformations, which lead to solutions for specific cases, but not including the inverse-square. It seems that an analytical solution for the reflection coefficient associated with an inverse-square density profile is a problem that must be left for another time.

With regard to attenuation, the basement data shown in Fig. 1(c) are widely spread, over 2 orders of magnitude from 0.01 to 1 dB/m/kHz, presumably because they represent the *effective* attenuation due to scattering and other such mechanisms and not the irreversible *intrinsic* attenuation, in which acoustic energy is converted into heat.^{24} Be that as it may, it is evident that the data could be equally poorly fitted by an increasing, a constant, or a decreasing profile (the author is indebted to an anonymous reviewer for this choice of words). The decreasing inverse-square profile shown in Fig. 1(c) was selected as a matter of expediency since it makes the analysis of the reflection coefficient tractable.

## ACKNOWLEDGMENTS

This research was supported by the Office of Naval Research, Ocean Acoustics Code 322OA, under Grant No. N00014-22-1-2598. The author would like to thank Dr. Charles Holland for several constructive discussions concerning the reflection coefficient of fine-grained sediments.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

*Special Functions and Their Applications*

*Partial Differential Equations in Physics: Lectures on Theoretical Physics*

*Tables of Integral Transforms*

*Mathematical Methods for Physicists*

*Formulas and Theorems for the Special Functions of Mathematical Physics*