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, R0. In the absence of attenuation, the plane-wave solution exhibits total reflection up to a critical grazing angle, ac, but when attenuation in the sediment is introduced, the region of total reflection in |R0| 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 ac, 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.

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 Experiment8 (SBCEX17) was conducted on the New England Mud Patch (NEMP), located about 95 km south of Martha's Vineyard, MA. The sediment at the NEMP9–12 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 layer13 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°.

FIG. 1.

(Color online) (a) Sound speed, (b) density, and (c) attenuation representative of site SC2 at the NEMP. The blue circles are the data from Table III in Jiang et al. (Ref. 14) and the red lines depict the idealized profiles used in the theoretical development of the reflection coefficient. The bottom of the mud layer is coincident with the top of the basement at a depth of 10.5 m.

FIG. 1.

(Color online) (a) Sound speed, (b) density, and (c) attenuation representative of site SC2 at the NEMP. The blue circles are the data from Table III in Jiang et al. (Ref. 14) and the red lines depict the idealized profiles used in the theoretical development of the reflection coefficient. The bottom of the mud layer is coincident with the top of the basement at a depth of 10.5 m.

Close modal
TABLE I.

Geo-acoustic parameters representative of site SC2 at the NEMP.

Parameter Symbol Value
Sound speed, seawater (m/s)  c1  1472.30 
Density, seawater (kg/m3 ρ1  1023 
Sound speed, mud layer (m/s)  c2  1477.78 
Density, mud layer (kg/m3 ρ2  1750 
Thickness, mud layer (m)  d  10.5 
Attenuation, mud layer (dB/m/kHz)  σ¯2  0.05 
Density, basement (kg/m3 ρ3  2270 
Source-plus-receiver height above seabed (m)  W  36 
Parameter Symbol Value
Sound speed, seawater (m/s)  c1  1472.30 
Density, seawater (kg/m3 ρ1  1023 
Sound speed, mud layer (m/s)  c2  1477.78 
Density, mud layer (kg/m3 ρ2  1750 
Thickness, mud layer (m)  d  10.5 
Attenuation, mud layer (dB/m/kHz)  σ¯2  0.05 
Density, basement (kg/m3 ρ3  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 equation15 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–Weyl16,17 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, αc, when evaluated under the geo-acoustic conditions of site SC2 at the NEMP. An apparent, or quasi, critical grazing angle,18  αq, is predicted by the numerical integration for the cylindrical-wave reflection coefficient, which is less than αc 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.

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¯ 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.

FIG. 2.

Schematic of the vertical coordinate systems and sound speed structure used in the analysis of the reflection coefficient of the seabed. The inverse-square profile, c3(z), is asymptotic to c (dotted line) at great depth in the basement and is equal to ch at the mud-basement interface, where z = h.

FIG. 2.

Schematic of the vertical coordinate systems and sound speed structure used in the analysis of the reflection coefficient of the seabed. The inverse-square profile, c3(z), is asymptotic to c (dotted line) at great depth in the basement and is equal to ch at the mud-basement interface, where z = h.

Close modal
Following the nomenclature introduced previously,3 the inverse-square profile, as depicted in Fig. 2, is given by the expression
(1)
where zT is the transition depth and c is the asymptotic value of c3(z) at great depth. Three parameters characterize the inverse-square profile: c, zT, and h, the latter being the value of the z-coordinate at the top of the basement, where the sound speed is c3(h) = ch and the sound speed gradient is γ=c3(h)=(dc3/dz)|z=h. Some elementary algebra shows that zT and h can be expressed in terms of the more intuitive ch, c, and γ as follows:
(2)
and
(3)
With these expressions, the inverse-square profile can be fully specified by selecting appropriate values for ch, c, and γ.
The inhomogeneous Helmholtz equation for the (velocity potential) field in the water column from a line source normal to the x-z plane at x = 0 and elevation z¯ above the seabed is
(4)
where 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,
(5)
and
(6)
In Eqs. (4)(6), the subscript 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 Ψj, and the acoustic wavenumbers are kj = ω/cj, where ω is the angular frequency.
As outlined in the Introduction, a Fourier transform with respect to horizontal distance 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¯. 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¯:
(7)
where i=1 and
(8)
In Eq. (7), the subscript 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 η1 in Eq. (8) implies a finite but infinitesimal attenuation in the ocean.
For domain 2, the homogeneous mud layer, the field is obtained as a function of ω and p by solving the associated differential equation directly. The solution is
(9)
where
(10)
Similarly for the basement, domain 3, the differential equation may be solved directly, but first it is necessary to accommodate the inverse-square depth dependence of c3, as expressed in Eq. (1), by making the transformation
(11)
This leads to the modified Bessel equation, whose solutions are the modified Bessel functions of the first and third kinds,15  Iiv(|p|z) and Kiv(|p|z), respectively, where
(12)
Now, Iiv(|p|z) may be disregarded because it diverges in the limit of great depth, leaving as the solution for the field in the basement,
(13)
where
(14)
In the interests of brevity, the modified Bessel function of the third kind will henceforth be referred to as Macdonald's function.15 
The solutions in Eqs. (7), (9), and (11) for the field in each of the three domains contain several unspecified constants, which may be determined from the boundary conditions, continuity of pressure and continuity of normal component of particle velocity, at z¯=d, z¯=0, and z = h. Eventually, this procedure leads to an expression for the wavenumber field in the water column, Ψ1p(z¯), 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
(15)
where the first integral,19 the direct arrival, can be expressed explicitly as a Hankel function of the second kind of order zero: πH0(2)(k1|z¯z¯|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 integral16,17 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
(16)
where the M functions are
(17)
(17a)
(17b)
and
(18)
The bjk in these expressions are density ratios,
(19)
and the prime on Macdonald's function in Eq. (18) denotes a derivative with respect to 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 R0(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 

The cylindrical-wave reflection coefficient of the stratified sediment is defined as the normalized quantity
(20)
where
(21)
is the sum of the source and receiver elevations above the seabed and R0(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.
When the source and receiver are far above the seabed, 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 = ps = −k1cos(α). From a standard stationary phase argument20 in conjunction with the asymptotic form for the Hankel function15 in the denominator, the reflection coefficient in Eq. (20) reduces identically to
(22)
where the subscript s signifies that the associated variable is to be evaluated at the stationary point. For example. the vertical wavenumbers in Eq. (22) are
(23a)
(23b)
and
(24)
In these expressions,
(25a)
(25b)

The function R0(ps) 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 

One issue remains to be addressed before the expressions for the reflection coefficients in Eqs. (20) and (22) can be evaluated. It concerns the function N in Eq. (18), which involves Macdonald's function and its derivative with respect to the argument.

Macdonald's function, which appears in Eq. (20) for the general reflection coefficient through the function N in Eq. (18), has the integral representation15 
(26)
(26a)
valid for Re(y) > 0, ν arbitrary, and in the present case, y = ξh. The derivative of this expression with respect to the argument y, denoted by the prime, is
(26b)

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 M1 and M2 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 R0, respectively.

When the order and the argument are both large, Macdonald's function may be approximated by an asymptotic expansion of the Debye type.22 Although there are several such expansions, of greatest interest here is for y = iw, where, from Eq. (14),
(27)
In this case, under the condition −π/2 < arg (y) < π, Macdonald's function can be expressed as
(28)
where Hiv(2)(w) is the Hankel function of the second kind of order . The Debye asymptotic expansion of this Hankel function is22 
(29)
(29a)
where
(29b)
Only the first term in the Debye asymptotic series has been included in Eqs. (29a) and (29b). On taking the derivative of Eq. (29a) with respect to the argument, which involves some elementary algebra, and forming the ratio Hiv(2)(w)/Hiv(2)(w), the following approximation for N in Eq. (18) is obtained:
(30)
As discussed below, this elementary algebraic expression can be used to evaluate the M functions in Eq. (17) at the stationary point, which in turn provide an approximation for the plane-wave reflection coefficient, R0, 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.

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 R0, 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.

TABLE II.

Inverse-square parameters representative of site SC2 at the NEMP.

Parameter Symbol Value
Sound speed, top of basement (m/s)  ch  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)  σ¯h  0.347 
Attenuation, infinite depth (dB/m/kHz)  σ¯  0.25 
Offset of origin of z (m)  h  2.366 
Transition depth (m)  zT  2.275 
Parameter Symbol Value
Sound speed, top of basement (m/s)  ch  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)  σ¯h  0.347 
Attenuation, infinite depth (dB/m/kHz)  σ¯  0.25 
Offset of origin of z (m)  h  2.366 
Transition depth (m)  zT  2.275 
With an absence of attenuation in the sediment layers, and for the sound speed profile as depicted in Fig. 2 with c1 < c, the radical in Eq. (24) is real for grazing angles 0 < α < αc, where αc is given by
(31)
The implication here is that αc 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 αc is a critical grazing angle, then |R0| is expected to exhibit total reflection over the angular region 0 < α < αc, and indeed it does. With ξ∞s in Eq. (24) real, both integrals in Eqs. (26) are real, so N is real, and so too is η1s in Eq. (23a). In Eq. (23b), η2s is real, imaginary, or zero, depending on whether cos(α) is less than, equal to, or greater than c1/c2. It is easy to show that in all three cases the numerator and denominator of Eq. (22) are complex conjugates, yielding the result |R0| = 1 for all 0 < α < αc, representing total reflection. For the conditions of site SC2 at the NEMP, the critical grazing angle is αc=cos1(c1/c)=44.09.

The region of total reflection is illustrated in Fig. 3, which shows |R0| 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 αc, 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).

FIG. 3.

(Color online) Lossless plane-wave reflection coefficient from Eq. (22) for the parameter values in Tables I and II and frequencies (a) 1175 Hz and (b) 2975 Hz. The red lines are from the numerical evaluation of the integrals for the modified Bessel functions in Eqs. (26), and the blue curves represent the Debye asymptotic approximation in Eqs. (29) and (30).

FIG. 3.

(Color online) Lossless plane-wave reflection coefficient from Eq. (22) for the parameter values in Tables I and II and frequencies (a) 1175 Hz and (b) 2975 Hz. The red lines are from the numerical evaluation of the integrals for the modified Bessel functions in Eqs. (26), and the blue curves represent the Debye asymptotic approximation in Eqs. (29) and (30).

Close modal

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.

For the mud layer, the attenuation data in Fig. 1(c), although sparse, appear to be essentially uniform in depth, allowing the theoretical attenuation to be accommodated by the usual expedient of making the wavenumber k2 complex with the imaginary part, like the real part, independent of depth,
(32)
where σ2=ωβ2/c2 is the attenuation coefficient with units of Nepers/m and β2 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.
The attenuation in the basement is treated in such a way as to make the analysis tractable: The wavenumber k is made complex so that, with β3 the loss tangent, k3(z), takes the form
(33)
It follows that the attenuation exhibits an inverse-square dependence on depth, varying as
(34)
which is a monotonic decreasing function, as illustrated by the red curve in Fig. 1(c). The widely spread data points for the basement straddle the red line in Fig. 1(c), suggesting that the latter, with its weak negative gradient, should provide a reasonable representation of the attenuation, even though a mild positive gradient due to the overburden pressure would be expected in practice.24,26

Figure 4 shows the plane-wave reflection coefficient, |R0|, 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).

FIG. 4.

(Color online) Plane-wave reflection coefficient from Eq. (22), including attenuation in the mud and the sand-mud basement, for the parameter values in Tables I and II and frequencies (a) 1175 Hz and (b) 2975 Hz with the depth of the mud layer set at 10.5 m. The red and blue lines are as in Fig. 3.

FIG. 4.

(Color online) Plane-wave reflection coefficient from Eq. (22), including attenuation in the mud and the sand-mud basement, for the parameter values in Tables I and II and frequencies (a) 1175 Hz and (b) 2975 Hz with the depth of the mud layer set at 10.5 m. The red and blue lines are as in Fig. 3.

Close modal

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, αq, 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, αq is significantly less than the genuine critical grazing angle, αc = 44.09°, featured in the lossless, plane-wave reflection coefficient for SC2 (see Fig. 3).

The discrepancy between αc, 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 αc. Under such circumstances, when the source and receiver are close to the seabed, the plane-wave condition 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 rule21 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 colleagues28,29 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, αq, approaches the true value, αc. Moreover, the cylindrical reflection coefficient in panel (d) is starting to resemble closely its plane-wave counterpart in Fig. 4(a).

FIG. 5.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for (a) W = 10 m, (b) W = 50 m, (c) W = 100 m, and (d) W = 500 m. The frequency and depth of the mud layer are shown in each panel. The remaining parameters are as specified in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

FIG. 5.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for (a) W = 10 m, (b) W = 50 m, (c) W = 100 m, and (d) W = 500 m. The frequency and depth of the mud layer are shown in each panel. The remaining parameters are as specified in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

Close modal

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 αq, 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.

FIG. 6.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for two frequencies: (a) f =1175 Hz and (b) f = 2975 Hz. All other parameters are as shown in the panels or in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

FIG. 6.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for two frequencies: (a) f =1175 Hz and (b) f = 2975 Hz. All other parameters are as shown in the panels or in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

Close modal

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 aq 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.

FIG. 7.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for two mud-layer depths: (a) d = 3 m and (b) d = 10.5 m. All other parameters are as shown in the panels or in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

FIG. 7.

(Color online) Cylindrical-wave reflection coefficient from Eq. (20) for two mud-layer depths: (a) d = 3 m and (b) d = 10.5 m. All other parameters are as shown in the panels or in Tables I and II. The vertical dashed lines depict the positions of the quasi-critical grazing angle.

Close modal

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–Weyl16,17 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, R0, in Eq. (22). The expressions for both R and R0 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 |R0| 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, R0, exhibits total reflection at grazing angles up to the critical, αc ≈ 44.09°, beyond which |R0| 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 αc 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, Robins30 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.

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.

The author has no conflicts to disclose.

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

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