The behavior of sound near an ocean front in a region with wedge bathymetry is examined. The front is parameterized as a zone of variation with inshore and offshore boundaries parallel to a straight coastline. The importance of frontal width and frontal sound speed on the ducting of acoustic energy is examined. Previous analytical studies of sound propagation and parameter sensitivity in an idealized wedge environment use an unphysical but convenient single interface front representation, which is here replaced by a continuous sound speed profile. The continuous profile selected is convenient for analytical investigation, but encourages the use of asymptotic approximation methods which are also described. The analytical solution method is outlined, and numerical results are produced with an emphasis on comparing to the single interface front. These comparisons are made to highlight the strengths and weaknesses of the idealized model for capturing the horizontal ducting effects.

Over the sloping bathymetry of a coastal shelf, sound speed fronts occur due to temperature, density, and salinity gradients.1–5 In a shelf-break front environment, the ocean can be generally divided into three regions characterized by sound speed behavior: inshore, offshore, and frontal. Within the front, the sound speed gradient in range is larger when compared to the more slowly varying inshore and offshore regions. This kind of ocean feature can influence acoustic propagation through along-shore ducting and refraction of acoustic energy by the front. Using mathematical models of a curved coastal front, the importance of front width can be determined, with prior analyses of a similar environment from Lin and Lynch providing a baseline for comparisons.6–8 

To examine the ducting effects of front width over a sloping bottom, the idealized front of Lin and Lynch is considered.6 This model describes the front as an instantaneous change between isospeed inshore and offshore regions. While convenient for analytic calculations, the effects of a more physical continuously varying ocean are lost. This paper will introduce a range dependent continuous inverse-square wavenumber profile, which has the advantages of representing a continuously varying sound speed front in a mathematically convenient form, as well as having some established applications in ocean acoustics literature.9,10 The method of normal modes is applied, with the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation introduced for the front region to facilitate numerical calculation of radial modes. Horizontal wavenumber eigenvalues, radial mode eigenfunctions, and transmission loss fields are calculated for various front widths to illustrate front width influence on acoustic pressure. While this paper seeks to address one unphysical aspect of the idealized wedge model, other properties such as bottom attenuation and variability in the front structure have been examined by other authors.11–14 

The solution method presented here will follow closely those methods outlined by Frisk15 and Lin and Lynch6 for acoustic pressure in a wedge with an impenetrable bottom and a pressure release surface. This formulation will introduce a continuously varying front region of constant curvature with an inshore point source located at (r0, θ0, y0) in cylindrical coordinates as illustrated in Fig. 1. Spatial coordinates in the wedge are y which is coincident with the shoreline, θ measuring angular displacement from the surface at θ = 0 and increasing towards the bottom at θ = α, and r which is distance from the shoreline measured perpendicular to the shore. The front is centered at r = rI with a width of s, and the sound speed inshore and offshore are given by the constants c1 and c2 respectively, with a continuous cF(r) satisfying c(rj) = cj at the inshore and offshore interfaces r1 and r2. Acoustic pressure is given by the Helmholtz equation

[2+k2(r)]P(r,θ,y)=4π(rr0)r(θθ0)δ(y),
(1)

where k(r) is the range-dependent wavenumber, the time evolution of the single frequency point source has been accounted for by factoring out eiωt, the source frequency given by ω=2πf,p(r,θ,y,t)=P(r,θ,y)eiωt, and with a unit source strength of 1, with the appropriate implied units of energy per length squared (J/m2). Boundary and interface conditions on P are P(r, 0, y) = 0 (pressure release), P/θ|θ=α=0 (reflecting bottom), P(0, θ, y) <  (pressure is finite at the wedge apex), the Sommerfeld radiation condition as r, and impedance interface conditions at r = rj.

FIG. 1.

(Color online) Model geometry for the shelf-slope wedge with continuous sound speed front. Model parameters are: front location rI, front width s, bottom slope angle α, source frequency f, inshore/offshore sound speeds c1/c2. Boundary conditions are pressure release at the surface θ = 0 and reflection at the bottom θ = α.

FIG. 1.

(Color online) Model geometry for the shelf-slope wedge with continuous sound speed front. Model parameters are: front location rI, front width s, bottom slope angle α, source frequency f, inshore/offshore sound speeds c1/c2. Boundary conditions are pressure release at the surface θ = 0 and reflection at the bottom θ = α.

Close modal

The along-shore spatial coordinate y can be converted to and from the horizontal wavenumber spectrum ky through the Fourier transform pair

P(r,θ,y)=12πG(r,θ,ky)eikyydky,
(2)
G(r,θ,ky)=P(r,θ,y)eikyydy.
(3)

The function G solves the radial mode equation

1rr(rGr)+1r22Gθ2+(k2ky2)G=4π(rr0)r(θθ0),
(4)

and can be represented by a sum of separated eigenfunctions

G=nΦn(θ0)Φn(θ)An(r,ky).
(5)

A set of orthonormal angular eigenfunctions Φn(θ)=(2/α)sin(ηnθ) can be derived, where ηn=(π/α)(n1/2) is a separation constant as well as the dimensionless angular wavenumber. With this definition for the angular eigenfunctions the radial mode equation takes the form of a Bessel equation for the inshore and offshore regions, for which the wavenumber is the constant value kj = 2πf/cj with j = 1 or 2 denoting inshore or offshore, respectively.6,15 Due to the chosen coordinate system and model geometry, G is completely separable as shown in Eq. (5). Range-dependent perturbations to the model that would break the separability of G will require a mode coupling treatment.16 

Within the front region, the square of the wavenumber will be represented by an inverse square wavenumber profile kF(r)=a2+b2/r2 as shown in Fig. 2, which allows the radial mode equation to remain a Bessel type equation,

1rddr(rdAndr)+((a2ky2)(ηn2b2)r2)An=4πδ(rr0)r.
(6)

The constants a and b are chosen such that k(r1) = k1 and k(r2) = k2. The radial mode function An is determined by matching two solutions of the radial mode equation found with the method of variation of parameters, each solving one of the boundary conditions (at r = 0, and at r = r1) on either side of the point source at r = r0,6,15,17 and is expressed as a linear combination of Bessel type functions as

An(r,ky)=i2π21RnJηn(kr1r0)Bn(r,ky),
(7)

where kr1=k12ky2, Rn acts like a reflection coefficient related to energy that returns to the inshore region r < r1 after being refracted back by the front, and Jn is the Bessel function of the first kind. The function Bn is

Bn(r,ky)={12Jηn(kr1r)Jηn(kr1r0)[Hηn(1)(kr1r0)+RnHηn(2)(kr1r0)],0r<r0,12[Hηn(1)(kr1r)+RnHηn(2)(kr1r)],r0<rr1,d1nHiνn(1)(krFr)+d2nHiνn(2)(krFr),r1rr2,TnHηn(1)(kr2r),r2r,
(8)

where kr2=k22ky2,νn=b2ηn2,krF=a2ky2, and Hηn(1,2) are Hankel functions of the first and second kind. Figure 2 illustrates the division of r into four regions I, II, III, and IV relative to the r locations 0, r0, r1, and r2, with I and II representing the inshore region, III the front, and IV offshore. The quantity Rn can be derived by satisfying the conditions of continuity for the radial mode function and its derivative at r1 and r2, and the condition of finiteness at r = 0. It is expressed as

Rn=Δ11E1Δ12E2Δ21E1Δ12E2FD,
(9)

where

Δpq=krqHηn(p)(krqrq)Hηn(p)(krqrq)krFHiνn(1)(krFrq)Hiνn(1)(krFrq)krqHηn(p)(krqrq)Hηn(p)(krqrq)krFHiνn(2)(krFrq)Hiνn(2)(krFrq),
(10)
F=kr1Hηn(1)(kr1r1)Hηn(1)(kr1r1)krFHiνn(2)(krFr1)Hiνn(2)(krFr1)kr1Hηn(2)(kr1r1)Hηn(2)(kr1r1)krFHiνn(2)(krFr1)Hiνn(2)(krFr1).
(11)
Eq=Hiνn(1)(krFrq)Hiνn(2)(krFrq),D=Hηn(1)(kr1r1)Hηn(2)(kr1r1).
(12)

The coefficients d1n, d2n, and Tn can be found simultaneously with Rn by satisfying the continuity conditions on the radial mode function and its derivative.

FIG. 2.

(Color online) Illustration of four radially defined ocean regions of physical importance in the model. The inshore region is comprised of I and II, which in turn lie on either side of the radial location of the inshore point source at r = r0. The front region is denoted by III, while the offshore region is IV.

FIG. 2.

(Color online) Illustration of four radially defined ocean regions of physical importance in the model. The inshore region is comprised of I and II, which in turn lie on either side of the radial location of the inshore point source at r = r0. The front region is denoted by III, while the offshore region is IV.

Close modal

The pressure P can now be expressed as

P(r,θ,y)=i2παnsin(ηnθ0)sin(ηnθ)Jηn(kr1r0)Bn(r,ky)eikyy1Rndky,
(13)

after substituting Eqs. (5), (7), and (8) into Eq. (2). The integral in Eq. (13) can be approximated by complex integration using residue theory. First, the existence of the square roots in Bn introduce branch cuts in the ky plane emanating from the points ±k1, ±k2, and ±a. The Pekeris branch cut will be used for these, which will expose leaky mode eigenvalues in the first quadrant of the ky plane.18 The Hankel functions have a singularity where the argument is zero, occurring at the branch points ±k1, ±k2, and ±a, which will be excluded from the interior of the closed integration path to be chosen in the ky plane. There will be singularities in the integrand where Rn = 1, which are also the locations of the eigenvalues in the ky plane denoted ky,nm, where m represents the m-th radial mode eigenvalue associated with angular mode n.

A closed integration path illustrated in Fig. 3 and as described in Lin and Lynch is used, which includes a segment along the real ky axis that avoids branch points, integration paths that run along either side of the branch lines in the first quadrant of the ky plane, and a path that approaches infinity in the upper half ky plane.6 The integral along the last integration path vanishes, while the branch line integrals can be ignored if only the far field is of interest.18,19 For a closed path Γ as described above, the far-field approximation and the residue theorem20 together state that Γ=2πi(residues),

P(r,θ,y)4π2αnsin(ηnθ0)sin(ηnθ)meiky,nmyRes(Jηn(kr1r0)Bn(r,ky)eikyy1Rn,ky,nm),
(14)

where Res(F(z), zm) is the residue of F at zm, and is defined by (1/2πi)CmF(z)dz=Res(F(z),zm), for a closed contour Cm which contains only zm and no other singularities.20 Because Hankel functions only have singularities21 in the ky plane where ky = ±k1, ±k2, ±a, and as it is unlikely that an eigenvalue will coincide with these values, a path Cm can be defined in a neighborhood of any ky,nm that will not contain any singularities arising from the Hankel functions. There may be locations in the ky plane where d1n, d2n, Tn, or Rn have singularities. Again, these are not expected to coincide with ky,nm in which case there will be a valid contour Cm to calculate the residue.

FIG. 3.

(Color online) The Pekeris branch cut, isolating the integration path around the branch line CB, and exposing the leaky mode eigenvalues (blue diamonds above the horizontal real axis).

FIG. 3.

(Color online) The Pekeris branch cut, isolating the integration path around the branch line CB, and exposing the leaky mode eigenvalues (blue diamonds above the horizontal real axis).

Close modal

The residue in Eq. (14) is calculated under the assumption that the ky,nm are simple poles. A pole at ky,nm will have multiplicity higher than 1 if (/ky)(1Rn)ky=ky,nm=0, in which case the complex integration requires more care.22,23 Due to the complexity of the dispersion relation for this model, its derivative can be tested numerically for each eigenvalue to determine if its multiplicity is higher than 1. For eigenvalues tested and presented in this paper, none have multiplicity higher than 1, so calculations will proceed with this in mind. For simple poles, Eq. (14) can be approximated as

P(r,θ,y)4π2αn,msin(ηnθ0)sin(ηnθ)Jηn(kr1,nmr0)[dRndky|ky=ky,nm]1Ψn(r,ky,nm)eiky,nmy,
(15)

with

Ψn(r,ky,nm)={Jηn(kr1,nmr),0rr1,d1nHiνn(1)(krF,nmr)+d2nHiνn(2)(krF,nmr),r1rr2,TnHηn(1)(kr2,nmr),r2r.
(16)

To aid in numerical computations involving radial behavior in the front, the radial mode equation within the front will be approximated using the WKBJ method, which will yield approximated radial mode functions and an approximated dispersion relation. The first step will be to transform the radial mode equation into a more convenient form by introducing the function β(τ) = r1/2Ψ(r), where τ=rI1r. This leads to the equation

δ2β(τ)h(τ)β(τ)=0,
(17)

where

h(τ)=(1τt21τ2),
(18)

and δ = (ν2 + 1/4)−1/2, and τt = (qFrIδ)−1. The term qF2=krF2 has been introduced, because for lower radial mode numbers which are associated with the trapped modes, a2<ky2 for real ky. For ky within the WG and TR regions, τt ∼ 1, and will grow as radial mode number increases.

The function h(τ) has a zero at τ = τt, so τt is a turning point of Eq. (17) where the behavior of β(τ) transitions between exponential and oscillatory. This information can be used to classify radial mode behavior and types, by using the oscillatory or exponential characteristics of a mode to determine if it is trapped or leaky. Introducing the notation κr(r)=k2(r)ky2(ηn2/r2) to represent the radial wavenumber as the square root of the square of the wavenumber in the water minus the squares of the horizontal wavenumber ky and the scaled azimuthal wavenumber ηn/r, the function h(τ) can be expressed as

h(τ)=rI2δ2(κr2(r)14r2).
(19)

Therefore, the turning point in r, given by rt occurs when κr(rt) = 1/(2rt), for some r1 < rt < r2. Solving for rt gives rt=qF1νn21νn/qF, which is the location at which κr(r) = 0. Because rt is dependent on ky, for a given eigenvalue ky,nm, the radial mode shape will differ as ky changes.

Figure 4 illustrates the expected behavior of the radial mode functions by indicating oscillatory or exponential behavior in the (η2/r2,Re(ky)2) plane, with the zeros of κr(r) shown as thick solid lines. Three clear mode types appear in the illustration: whispering gallery (WG) modes are trapped modes with purely exponential decay offshore for real ky, leaky modes with purely oscillatory behavior offshore, and between the two in the ky spectrum are transition (TR) modes which are not trapped, but which transmit less energy offshore due to the range-dependent nature of the radial wavenumber. This behavior for TR modes is seen in the mixed exponential behavior between the offshore interface of the front at r2, and an offshore turning point at κr2(r) = 0, before they transition to purely oscillatory behavior beyond the turning point. The real ky spectrum is therefore divided into three regions, with WG modes occurring when k2<Re(ky)<k1*, TR modes occurring when k2*<Re(ky)<k2, and leaky modes occurring when 0<Re(ky)<k2*. The quantity kj* denotes locations at which the radial wavenumber will have zero real part at r1 or r2, and is defined kj*=kj2(ηn2/r2).

FIG. 4.

(Color online) Diagram of the general mode behavior in r dependent on horizontal wavenumber ky in the (ηn2/r2,Re(ky2)) plane. Curves for κr(r) = 0 are labeled in the figure, with regions above the curves exhibiting exponential behavior and regions below exhibiting oscillatory behavior. Three mode types are identified based on the exponential/oscillatory distinctions: whispering gallery (WG), transition (TR), and leaky.

FIG. 4.

(Color online) Diagram of the general mode behavior in r dependent on horizontal wavenumber ky in the (ηn2/r2,Re(ky2)) plane. Curves for κr(r) = 0 are labeled in the figure, with regions above the curves exhibiting exponential behavior and regions below exhibiting oscillatory behavior. Three mode types are identified based on the exponential/oscillatory distinctions: whispering gallery (WG), transition (TR), and leaky.

Close modal

For complex ky, Re(τt) > 0, and due to the fact that Re(ky) ≫ Im(ky) in the WG and TR regions (to be shown in Sec. IV, and as seen in Refs. 6 and 7) as well as for a considerable number of leaky modes, the introduction of a nonzero imaginary component of ky will serve to slightly perturb the behavior of the WG mode approximations that assume Im(ky) = 0. For this reason, classification of modes henceforth will be based solely on the real component of ky, and the imaginary component of rt will not be considered.

For small δ in Eq. (17), the uniformly valid WKBJ approximation is24 

β(τ)δ1/6(f(τ)h(τ))1/4[a0Ai(δ2/3f(τ))+b0Bi(δ2/3f(τ))],
(20)

where

f(τ)={[32τtτ|h(t)|dt]2/3,τtτ,[32ττt|h(t)|dt]2/3,ττt.
(21)

For the reference parameter values used for numerical calculations (as described in Sec. IV), δ ∼ ν−1 is around 1/300 to 1/150, so the WKBJ approximation is valid. Transitioning back to the model variables and functions, the radial mode function within the front can be approximated by

AFn(r)a0F1(r)+b0F2(r),
(22)

where the linearly independent functions F1 and F2 are given by

F1(r)=G(r)Ai(γg(r)),F2(r)=G(r)Bi(γg(r)),
(23)

with

G(r)=32γ(g(r)zr21)1/4,
(24)
g(r)={(μ(zr))2/3,rtr,(ξ(zr))2/3,rrt,
(25)
μ(z)=z21cos1(1z),
(26)
ξ(z)=ln(1+1z2z)1z2,
(27)

and where rt = (qFδ)−1, zr = r/rt, and γ=(32ν2+(1/4))2/3.

Now that the radial mode functions have been approximated within the front, these approximations will be applied to the dispersion relation. For ease of calculation, the two linearly independent functions Hiνn(1)(krFr) and Hiνn(2)(krFr) can be replaced in the dispersion relation given by Eq. (9) by the two approximated linearly independent functions F1 and F2. This will require the approximation of the quantities F1(r)/F(r),F2(r)/F(r), and F1(r)/F2(r). After some lengthy calculations, the resulting approximations are

F1(r)F1(r)=16rzr21g(r)3/2(1+4γg(r)Ai(γg(r))Ai(γg(r)))12rzr2zr21,F1(r)F2(r)=Ai(γg(r))Bi(γg(r)),
(28)

and with F2(r)/F2(r) defined as F1(r)/F1(r) with Ai(⋅) replaced with Bi(⋅).

Now consider numerical results for the following parameter values, which agree with those used by Lin and Lynch to allow for easier comparison:6 center of the front at rI = 4000 m, bottom slope α = 3°, inshore sound speed c1 = 1500 ms−1, offshore sound speed c2 = 1520 ms−1, and source frequency f = 25 Hz. The values of the front width s will be given explicitly in the following numerical calculations. To determine the eigenvalues, the dispersion relation as given by Eq. (9) set equal to 1 is solved for ky. The approach used to find these modes locates the curve in the complex ky plane along which |Rn|=1 within a thin strip, and then slices the strip into individual boxes containing intersections of |Rn|=1 and contours of zero phase. The boxes containing eigenvalues are then iteratively refined to at most 10−8 in length along the real and imaginary axes, and the eigenvalue location within the refined box is approximated by calculating the intersection of the |Rn|=1 and arg(Rn) = 0 curves which are assumed to be nearly linear at this scale.

Figure 5(a) shows a portion of the ky plane with eigenvalues for a 100 m front, and angular modes n = 1–3. Dashed vertical lines are placed to illustrate the separation between leaky modes and TR or WG modes, with leaky modes falling to the left of each color-coded dashed line (leaky/trapped division line for n = 1 is the right-most, while n = 3 is the left-most). Figures 5(b) and 5(c) give similar results for front widths of 50 and 0 m respectively.

FIG. 5.

(Color online) Radial mode eigenvalues in the ky plane for angular mode numbers n = 1 (blue circles), n = 2, (red triangles), and n = 3 (brown squares). Vertical dashed lines represent Re(ky) cut-off value separating leaky modes (to the left) from TR and WG modes (to the right). (a) Front width s = 100 m, (b) Front width s = 50 m, (c) Front width s = 0 m. (d) A comparison of angular modes n = 1 only for front widths s = 0 (red circles), s = 50 (blue triangles), s = 100 (black squares).

FIG. 5.

(Color online) Radial mode eigenvalues in the ky plane for angular mode numbers n = 1 (blue circles), n = 2, (red triangles), and n = 3 (brown squares). Vertical dashed lines represent Re(ky) cut-off value separating leaky modes (to the left) from TR and WG modes (to the right). (a) Front width s = 100 m, (b) Front width s = 50 m, (c) Front width s = 0 m. (d) A comparison of angular modes n = 1 only for front widths s = 0 (red circles), s = 50 (blue triangles), s = 100 (black squares).

Close modal

A clear difference between the three illustrated front widths are the appearance and locations of local peaks in the imaginary components of ky. These peaks occur at predictable values, when the differences in phase of the radial modes across the front are close to an integer multiple of π, achieving near-resonance. The phase difference across the front can be approximated by the front width multiplied by an average radial wavenumber in the front, leading to the simple equation

sκr(rI)=Nπ,
(29)

for integers N = 1, 2, 3, …, where any eigenvalues ky,nm for which Eq. (29) is approximately true can be classified as near resonant (NR). A simple relation between s and the ky values for which near-resonance occurs can be achieved, by noting that for a fixed N in Eq. (29), s is inversely proportional to the radial wavenumber, such that an increase in s must be accompanied by a decrease in the real part of κr, which must in turn be accompanied by an increase in the real part of ky by the relation k2(r)=κr2(r)+ky2+ηn2/r2, assuming Im(ky) ≪ Re(ky). Therefore, as the front width increases, the first modes exhibiting near resonance occur at lower indices, and as s goes to zero near-resonance eventually disappears, agreeing with the behavior seen in Fig. 5.

One final difference to note is most clearly shown in Fig. 5(d), in which radial modes are plotted for n = 1 and the three front widths of 100, 50, and 0 m. A general trend of increasing Im(ky) with increasing s is seen, which suggests that larger front widths are associated with “leakier” leaky modes, as higher energy decay along shore is necessarily related to higher transmission of energy offshore.

To see this last point more clearly, consider Fig. 6, which shows how individual modes change with front width. In both Figs. 6(a) and 6(b), all radial mode eigenvalues shown are associated with angular mode n = 1, and similar mode numbers m are connected by dashed lines across the various front widths of 0, 25, 50, 75, 100, and 200 m. The radial modes plotted are m = 12–21 in Fig. 6(a) and m = 32–41 in Fig. 6(b). Modes m = 12–21 are the lowest mode numbers associated with leaky modes at reference parameter values, with modes m = 1–11 being WG and TR modes. For these leaky modes, the general trend observed is an increase in the imaginary component of ky as the front width increases. It is also apparent that the growth of Im(ky) with growing s is sensitive to the initial ky value. In Fig. 6(b), a selection of modes that are near resonance are shown. Modes m = 37–41 do not exhibit a growth in Im(ky) as s increases from 75 m, as the NR status of a mode disrupts the monotonic growth of Im(ky) seen in modes 32–36, as well as all modes in Fig. 6(a).

FIG. 6.

(Color online) Evolution of eigenvalues in the ky plane as front width changes. A single radial mode number m is represented by identical symbols connected by dashed lines. A collection of radial modes for a single front width s share the same color, corresponding to the legend. (a) Radial mode numbers m = 12 to 21, representing leaky modes with the lowest mode numbers. (b) Radial mode numbers m = 32 to 41, illustrating a non-monotonic growth of Im(ky) near NR modes.

FIG. 6.

(Color online) Evolution of eigenvalues in the ky plane as front width changes. A single radial mode number m is represented by identical symbols connected by dashed lines. A collection of radial modes for a single front width s share the same color, corresponding to the legend. (a) Radial mode numbers m = 12 to 21, representing leaky modes with the lowest mode numbers. (b) Radial mode numbers m = 32 to 41, illustrating a non-monotonic growth of Im(ky) near NR modes.

Close modal

Figure 7 illustrates the first five radial modes associated with each of the first three angular modes for a front width of s = 100 m, illustrated by the dashed horizontal lines. The trapping of energy that was seen by Lin and Lynch6 is also present here, with the Whispering Gallery effect manifesting as a trapping of acoustic energy close to the inshore side of the interface. With this in mind, it is pertinent to address the effects of various front widths on the behavior of these WG and other modes to see if the physical simplification of an idealized front loses some of the expected behaviors, or if it is a strong representation of the acoustic ducting effects in this environment.

FIG. 7.

(Color online) Sample radial and angular mode profiles for front width of 100 m: (a) Angular mode n = 1 and radial modes m = 1–5, (b) angular mode n = 2 and radial modes m = 1–5, (c) angular mode n = 3 and radial modes m = 1–5. Real components shown as solid lines, imaginary components as dashed lines.

FIG. 7.

(Color online) Sample radial and angular mode profiles for front width of 100 m: (a) Angular mode n = 1 and radial modes m = 1–5, (b) angular mode n = 2 and radial modes m = 1–5, (c) angular mode n = 3 and radial modes m = 1–5. Real components shown as solid lines, imaginary components as dashed lines.

Close modal

In Fig. 8, two radial modes are shown for angular mode n = 1 and front widths of s = 0, 100, and 200 m. For both modes pictured, the idealized front mode is given by the thick solid lines (blue for real components, red for imaginary), and the location of the idealized front is denoted by a thick vertical black line. For the front width of 100 m, the modes are represented by thin solid lines, as are the inshore and offshore interfaces of the front, and the 200 m front is similarly represented by thin dashed lines. In Fig. 8(a), the WG mode m = 8 is shown, and for all front widths represented it appears to undergo very little perturbation, with the mode shapes for the three front widths sharing enough ink as to make the differences imperceptible in many areas. In contrast, Fig. 8(b) shows the leaky mode m = 16, and in this case the influence of front width is far more visible. Inshore, the mode shapes are in closer agreement for smaller r, with a slight phase shift appearing as the mode approaches the front, as well as a slight growth in mode amplitude. Offshore, the phase and amplitude differences are more exaggerated, with amplitude nearly doubling a short distance front the front. Figure 8(c) shows a closer view of the front region for m = 16, in which it is clear that the divergence from the idealized mode shape is far more exaggerated for the front width of 200 m compared with 100 m. The behaviors seen in these two modes in Fig. 8 are not too surprising, as the eigenvalue behavior shown in Figs. 5 and 6 suggest that Im(ky) increases with increasing front width, and that this increase is greater for larger radial mode numbers, i.e., “leakier” modes. An interpretation of these results is that the trapping behavior exhibited by WG and TR modes which have zero or nearly zero Im(ky,nm) is captured well by the idealized front model, while the leaky modes are not.

FIG. 8.

(Color online) Illustration of the evolution of radial mode shape as front width changes. Blue (dark) lines represent real components, while red (light) represent imaginary components. The thick solid lines represent a reference idealized front with s = 0 m, thin solid lines represent a front width of s = 100 m, and dashed lines represent a front width of s = 200 m. Vertical thick solid, thin solid, and dashed lines represent the idealized, 100 m, and 200 m fronts, respectively. (a) Mode n = 1, m = 8, (b) mode n = 1, m = 16, (c) close-up on the front region of n = 1, m = 16.

FIG. 8.

(Color online) Illustration of the evolution of radial mode shape as front width changes. Blue (dark) lines represent real components, while red (light) represent imaginary components. The thick solid lines represent a reference idealized front with s = 0 m, thin solid lines represent a front width of s = 100 m, and dashed lines represent a front width of s = 200 m. Vertical thick solid, thin solid, and dashed lines represent the idealized, 100 m, and 200 m fronts, respectively. (a) Mode n = 1, m = 8, (b) mode n = 1, m = 16, (c) close-up on the front region of n = 1, m = 16.

Close modal

The goal of the approximation of Eq. (15) which ignores contributions from branch line integrals is to conveniently represent the far-field behavior in y. It is expected that by excluding the branch line integrals, the near-field will lose accuracy, but by including leaky modes that are exposed by the Pekeris branch cut, an approximation of the near-field can be achieved.18 The far-field is described by the WG and TR modes, which propagate long distances along shore in the y direction due to their zero or near-zero Im(ky,nm) values, while leaky modes will have little influence in the far-field due to their larger Im(ky,nm) values. Closer to the source, more leaky modes are required to represent the field, and with larger front widths associated with higher Im(ky) values in the leaky modes, the influence of leaky modes will be less apparent in the far-field for larger width continuous fronts. A rough approximation of the limit on the influence of a single leaky mode can be calculated by assuming that behavior of a single mode in the (r, y) plane is largely governed by the plane wave approximation exp(iκr,nm(r)r+iky,nmy). Using a 1/e decay rate to judge the influence of leaky modes in the (r, y) plane, the range in y that a leaky mode will propagate at a given r and for a given eigenvalues ky,nm is approximately y=ỹ(r;n,m)=[1Im(κr,nm(r))r]/Im(ky,nm), where κr,nm(r)=k2(r)ky,nm2(ηn2/r2). To properly evaluate the acoustic pressure at a location (r*, y*), all modes for which ỹ(r*;n,m)>y* must be included. This leads naturally to the definition

ymax(r;n)=ỹ(r;n,M),
(30)

where M is the last in a finite set of consecutive mode numbers beginning with 1.

Figure 9 gives the transmission loss field in the (r, y) plane located at θ = 1.5° for a source located at (r0, θ0, y0)= (3900 m, 1.5°, 0 m), and with a front width of s = 100 m. Figures 9(a)–9(c) show the contributions from angular modes n = 1, 2, and 3 respectively, with 66 radial modes in Fig. 9(a), 42 modes in Fig. 9(b), and 24 modes in Fig. 9(c) to illustrate the extent that a finite sum of modes can represent the far-field. Figure 9(d) includes all modes represented in Figs. 9(a)–9(c). In all subfigures, the dotted lines represent the ymax value which denotes the limit to the accuracy of a finite sum approximation under the 1/e decay condition. Regions of the (r, y) plane to the right of and above these lines are well represented by the finite sum of modes, as all modes that propagate into that region are included, while the regions to the left of and below these lines cannot be considered accurately represented, as not all modes that contribute to that portion of the field are included. In Fig. 9, only the positive y axis is shown as the field is symmetric across y = 0.

FIG. 9.

(Color online) Transmission loss fields in the (r, y) plane at θ = 1.5°, for a source located at r0 = 3900 m, y0 = 0 m, θ0 = 1.5° denoted by a white star, and a front width of s = 100 m. (a) Contribution from angular mode n = 1 and radial modes m = 1–66. Horizontal solid lines represent the front interfaces, while the horizontal dashed line is at r = rI. The dotted curves represent the locations of e−1 decay for modes at the given n. (b) Angular mode n = 2, radial modes m = 1–42. (c) Angular mode n = 3, radial modes m = 1–24. (d) All modes from (a)–(c).

FIG. 9.

(Color online) Transmission loss fields in the (r, y) plane at θ = 1.5°, for a source located at r0 = 3900 m, y0 = 0 m, θ0 = 1.5° denoted by a white star, and a front width of s = 100 m. (a) Contribution from angular mode n = 1 and radial modes m = 1–66. Horizontal solid lines represent the front interfaces, while the horizontal dashed line is at r = rI. The dotted curves represent the locations of e−1 decay for modes at the given n. (b) Angular mode n = 2, radial modes m = 1–42. (c) Angular mode n = 3, radial modes m = 1–24. (d) All modes from (a)–(c).

Close modal

To best understand the impact of front representation on the acoustic ducting effect, a relative difference between the transmission loss in a 100 m front model compared to an 0 m front model can be considered, by examining D100=|TL0TL100|/TL0, where TLs is the transmission loss calculated for a front of width s. Figure 10(a) illustrates this value in the (r, y) plane at θ0 = 1.5° and r0 = 3900 m, using the same m and n values present in Fig. 9 for the front of 100 m, and analogous mode numbers for the idealized front of 0 m. Four relative difference levels are considered: D100<0.01,D100<0.05,D100<0.1, and D100>0.1, with higher relative difference represented by darker shades. Additionally, the relative field differences are shown for a source located at r = 3900 m (a) and a source at r = 3000 m (b) to illustrate the effects of the front representation for a source located farther inshore, and that the influence of the sharp front approximation is less pronounced for such a source location.

FIG. 10.

Relative difference D100=|TL0TL100|/TL0 between transmission loss fields for front widths of 0 and 100 m. The four color levels represent agreement to within 1%, 5%, 10%, or greater than 10% in order of increasing shades.

FIG. 10.

Relative difference D100=|TL0TL100|/TL0 between transmission loss fields for front widths of 0 and 100 m. The four color levels represent agreement to within 1%, 5%, 10%, or greater than 10% in order of increasing shades.

Close modal

First, consider the regions in Fig. 10(a) which indicate agreement between the two fields within 1%, shown in white in the contour plot. These areas are concentrated largely where the ducting is observed, between the front and the inshore shadow zones. The introduction of the 100 m continuous front does little to alter this behavior inshore, which is dominated by the WG and TR modes. This is expected, as it was previously observed that the WG and TR modes are not as sensitive to the front width compared to leaky modes.

In the darkest shaded regions of Fig. 10(a), the relative difference between the two considered TL fields is greater than 10%. Three major regions of disagreement are observed: the near field, close to y = 0, the shadow zones close to r = 0, and the front region, close to r = rI. For the near-field, disagreement between the 100 and 0 m fronts is not of significant interest, as the approximation given here is for the far-field, and not enough leaky modes are included to arrive at any conclusions for this area. In the shadow zones inshore, energy does not propagate, so disagreement between the two fields is also of little interest, as much of it could be due to limits in numerical accuracy for small values. The primary region of interest for disagreement between the two fields is in the vicinity of the front. It is clear that the interference patterns for the two front widths do not align, so this must be considered when choosing to use an idealized front model to gain insight on interference patterns near the front, and this becomes more important when the source is close to the front.

The effects on acoustic propagation of a curved continuously varying sound speed front over a sloping bottom are examined in comparison to a benchmark model of an idealized front by Lin and Lynch.6 To account for the continuous front, an inverse square wavenumber profile is used in place of a sharp discontinuity, which has the benefit of convenient mathematical interpretations in terms of Hankel functions. With the introduction of a more physically acceptable front representation, the same mode types that appeared in the idealized model are present, with the whispering gallery modes that account for trapping of energy on the inshore side of the front exhibiting nearly identical mode shapes for those front widths considered when comparing with the idealized results. In contrast, leaky modes behave quite differently than their idealized counterparts, propagating shorter distances along shore as more energy passes through the front rather than being refracted by it. In addition to the difference in the amount of ducted energy for the leaky modes, some leaky modes can be classified as near-resonant in the continuous model, which can be characterized by a larger imaginary component in the associated ky eigenvalue, indicating stronger transmission across the front than neighboring modes.

Despite the noted differences, the idealized front model does an excellent job of describing the general acoustic ducting behavior inshore. Notable deviations from the continuous model are exhibited mainly in the interference patterns near the front, due to the reflective nature of the idealized model compared with the refractive nature of a continuous sound speed front. For the purposes of providing insight into the shelf-slope environment, representing the front with an idealized jump in sound speed does not significantly alter the along-front ducting of sound inshore, such that interest in inshore propagation in the shelf-slope front environment can be safely investigated with the convenient idealized front formulation. The structure of the front will understandably play a larger role in behavior in and around the front, in which case it is recommended that a continuous representation is considered.

The authors would like to thank Dr. T. F. Duda for his comments on this work. This work was supported by the Office of Naval Research under grants to Rensselaer Polytechnic Institute (Grant No. N00014-14-1-0372, which is a Special Research Award in Ocean Acoustics for the Ph.D. degree of B.J.D., and also Grant No. N00014-17-1-2370), and to Woods Hole Oceanographic Institution (Grant No. N00014-11-1-0701 which is a Multidisciplinary University Research Initiative, and Grant No. N00014-18-1-2172 which is a Postdoctoral Fellowship for B.J.D.).

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