Scale model observations of coupled vertical modes in a translationally invariant wedge

The study of acoustical wave propagation in non-horizontally stratified waveguides has spanned more than 50 years.^1–6 In underwater acoustics, analytical and experimental investigation of a wedge-shaped waveguide has received substantial attention as it approximates many continental shelf and slope regions. Propagation phenomena in the two-dimensional (2-D) plane of the wedge include vertical mode-cutoff for upslope propagation and mode-capture for downslope propagation which have been described by wavenumber integration,⁷ parabolic equation (PE),⁸ ray,^9,10 and coupled mode^11,12 numerical models.

In the case of the translationally invariant wedge, the wedge cross sectional geometry is extended along the out-of-plane dimension. Early investigations of this problem explained the propagation phenomena through rays, modes, or a combination of both.^4,6,13 In the ray description, a ray bends in the horizontal plane as it undergoes multiple specular reflections between a sloping seafloor and the horizontal sea surface. As described by Harrison,⁶ a ray obliquely propagating upslope will change its heading toward deeper water after each reflection from the seafloor. If the notion of normal mode propagation is also added to the description, then the ray elevation angles at the source are restricted to a set of discrete angles, corresponding to the discrete modal expansion. A modal shadow zone is observed by tracing a fan of rays possessing a fixed elevation angle but varying azimuthal angles, with resulting locations on the wedge where no energy for a given mode will exist (in the adiabatic limit of no coupling between the modes). In this description, Harrison⁶ also notes that constructive and destructive interference will arise in the modal propagation zone from the intersection of multiple ray paths for that mode. This phenomenon was termed intra-mode interference and results in spatial interference patterns which result from a single mode interfering with itself.

Early theoretical treatments of the three-dimensional (3-D) isovelocity linear wedge problem successfully produced the propagation phenomena of horizontal refraction, modal shadow zones, and intra-mode interference under the approximation of adiabatic propagation of the vertical modes. Later, exact solutions were developed using a so-called wedge mode expansion as opposed to a vertical mode expansion. For clarity in this paper a vertical mode is defined as the typical normal mode expansion used in underwater acoustics, namely a mode which depends on the vertical coordinate of the waveguide (i.e., depth). In contrast, a wedge mode possesses a cylindrical wavefront that is centered on the apex of the wedge and is defined in terms of the radial position away from the apex and the vertical depression angle below the air/water interface.^13–16 Consequently, a vertical mode is defined along a line and a wedge mode is defined along an arc. In the case of the 3-D isovelocity linear wedge, the wedge modes propagate adiabatically, an observation which was verified through laboratory tank measurements.^17,18

In this work, laboratory measurements of acoustic propagation over a translationally invariant, isovelocity wedge are presented. The wedge was intentionally designed with a relatively steep slope of $10°$ and high acoustic reflectivity in an effort to (a) generate and (b) observe coupling between vertical modes. Automated and precise positioning of a receiving hydrophone throughout the waveguide permitted the creation of multiple synthetic vertical line arrays (VLAs) such that vertical mode filtering of the measured complex pressure could produce measured modal amplitudes in the along- and across-slope dimensions. Comparisons of the measured pressure fields and vertical mode amplitudes are made to (1) a 3-D adiabatic mode model, (2) a 3-D ray model, and (3) an exact wedge model. The contributions of this paper include the laboratory measurements of the pressure field and the observed vertical mode amplitudes, as well as the three model comparisons for this specific propagation environment.

The remainder of this paper is organized as follows. Section II summarizes the ray-mode analogy in the context of the present work. Section III discusses the scale-model measurement apparatus and describes the experimental design. Measured results of the pressure field and filtered vertical mode amplitude are presented in Sec. IV. For comparison, results from three numerical models are shown and discussed in Sec. V. A discussion of the results is given in Sec. VI.

A brief discussion of the ray-mode analogy will illustrate the underlying propagation phenomena and define terms used in this paper. Figure 1 shows a top-down view of a wedge-shaped waveguide where the across slope range aligns with the x coordinate, the along slope range aligns with the y coordinate, and the z coordinate is the depth below the air/water interface. The wedge apex is located along the line y = 0 m and the deepest water is located along the line y = 0.229 m. An acoustic source is located at (x, y) = (0, 0.229) m.

For an isovelocity water layer, pressure release boundary conditions, and a monochromatic source of wavenumber k, there is an orthonormal set of vertical modes at each (x, y) location in the waveguide whose eigenvalues $kz,n$ depend only on the mode number n and the water depth d, namely $kz,n=nπ/d$⁠. This results in a gradient in the horizontal modal wavenumber $kr,n=k2−kz,n2$ in the along slope dimension up to the point of mode cutoff, where the wavenumber becomes imaginary and the modes abruptly change from propagating to evanescent. As an example, $kr,n$ is shown in Fig. 1 for n = 3. The modal cutoff line for mode three occurs at y = 0.042 m. There should not be any propagating energy for a mode at locations in the waveguide which are shallower than the depth at the modal cutoff line.

At the source location, each mode is associated with a discrete vertical angle $θn=tan−1(kz,n/kr,n)$⁠. In the ray-mode analogy, the acoustical field for mode n can be found by launching a fan of rays with varying azimuthal angle but fixed vertical angles ±θ_n. As a particular ray is launched obliquely up the slope, it will undergo repeated reflections from the waveguide boundaries, with each seafloor reflection causing the ray to turn in the horizontal plane toward deeper water. Two such individual ray paths are shown in Fig. 1. Every ray in the modal ray fan will eventually transition from propagating upslope to propagating downslope, and the boundary formed by the collection of turning points of the modal ray fan is called the modal shadow zone boundary. The modal shadow zone boundary divides the modal propagation zone from the modal shadow zone. Although a mode can theoretically exist in the modal shadow zone (i.e., the region between the shadow zone boundary and the mode cutoff line where the waveguide depth still supports propagation of this particular mode), there should not be any energy from the source in the shadow zone in the adiabatic limit. If energy from the source exists in the modal shadow zone, it would have to arrive there via mode coupling. One of the primary objectives of this work is to create a waveguide environment where the effect of mode coupling is observable in the measured data.

Finally, within the modal propagation zone, multiple ray paths for a given mode can constructively or destructively interfere at a receiver location, resulting in a phenomenon called intra-mode interference. This effect will produce interference patterns in the across slope range, even if only a single mode is present.

This section discusses pertinent aspects of the design of this particular experiment. The experiment was conducted in an indoor laboratory tank using the scale model measurement apparatus shown in Fig. 2(a). Detailed discussions of the measurement apparatus are reported in the literature^19,20 and its basic function is summarized here. A bathymetric part is affixed to a rigid frame and is lowered into a tank to create a waveguide between the air/water interface and the bathymetric part. The source transducer is manually positioned in the waveguide while the location of the receiving transducer is controlled by linear positioning hardware to an accuracy of about 50 μm. The beginning of each sound recording and the transmission of the source waveform are synchronized by means of hardware triggering, which permits both the temporal averaging of repeat transmissions at any receiver location as well as the ability to coherently measure the acoustic field at any point within the waveguide by moving the receiver location.

The bathymetric part utilized in this work was constructed from a closed-cell, machinable polyurethane foam board sold under the trade name Renshape 5030. Reflection coefficient measurements at 1 MHz found that this material was approximately pressure release across all incidence angles.²¹ A $10°$ slope, shown in Fig. 2(b) was machined along the length of the part by a computer numeric control (CNC) end mill. The maximum physical dimensions of the wedge (length × width × depth) are 2.133 m × 0.246 m × 0.043 m.

The source transducer was positioned at (x, y, z) = (0, 0.229, 0.020) m and broadcast a 300 kHz sinusoidal waveform. It was intentionally positioned mid-depth in the waveguide so as to preferentially excite the odd-numbered vertical modes. As depicted in Fig. 3, a grid of observation locations was defined in the across/along slope plane. The observation locations were regularly spaced in the across slope dimension from 0.101 to 1.778 m in 12.7 mm increments and in the along slope dimension from 0.114 to 0.229 m in 12.7 mm increments, resulting in a total of 1330 grid points. At each grid point, the acoustic pressure was sampled in depth from the air/water interface down to 6.3 mm above the bathymetric part with a depth discretization of 1.27 mm. The safe working distance of 6.3 mm above the bottom was imposed so as not to damage the transducer by unintentional contact with the part. This sampling scheme in depth created a synthetic VLA at each observation location, with 27 vertical measurements at the deepest water depth and 11 measurements at the shallowest. The depth sampling was selected such that the first six vertical modes could be resolved from the measured data. Observations at along slope ranges less than y = 0.114 m were not made because the required minimum safe working distance occupied a large enough portion of the waveguide depth so as to preclude filtering of at least six modes.

The theoretical modal shadow zone boundaries for modes 1 through 6 are superimposed on the grid of VLA locations in Fig. 3. Within the domain of the gridded region, the modal propagation zone will exist everywhere for mode 1. However, modal shadow zones should exist in the measurement domain for modes 2 through 6.

The measurements were made in ten batches over a period of seven consecutive days. Each measurement batch consisted of an across slope line of observation points, with either one or two batches measured each day. The water level in the tank was carefully monitored over the seven day period and, when necessary, water was added to the tank between batches to ensure a constant water depth in the waveguide. A thermocouple continuously monitored water temperature during the measurements and the water temperature fluctuated between $18.18 °$C and $18.57 °$C. The total data acquisition time was approximately 87 h across the ten measurement batches.

The sampling frequency of the acoustic data acquisition system was 10 MS/s and 300 temporal averages were made at each receiver location. At the source frequency of 300 kHz, the typical signal-to-noise (SNR) ratio at each receiver exceeded 50 dB. Sufficient time (26 ms) was allowed between each repetition of the source waveform for the acoustic field in the waveguide to decay back to the ambient level.

To provide some context to the dimensionality of the scale model experiment, a few non-scaled quantities are listed here assuming a scale factor of 1:7500. With this scale factor, the frequency of the acoustic source is 40 Hz, and the wedge-shaped waveguide is about 12.8 km long, 1.8 km wide, and 322.5 m at its deepest point. The across- and along-slope spacing between VLA locations is approximately 95 m, and the inter-element spacing along the VLA is about 9.5 m. These representative quantities would change with a different assumed scale factor.

This section presents the measured acoustic intensity and modal amplitudes for the experiment. Figure 4(a) shows the measured steady-state, acoustic intensity in each of the ten planes along the slope. The wedge is visible at the left edge of the figure, the source position is indicated by the star, and the y coordinates of each measurement plane are listed along the upper right edge of the figure. The solid blue section at the bottom of each measurement plane corresponds to regions where the safe working distance of the transducer prohibited data collection.

It is observed that the complexity of the acoustic intensity (i.e., the interference pattern) decreases with across slope range. This is expected since the higher-numbered modes transition from modal propagation to modal shadow zones with increasing across slope range (cf. Fig. 3) and consequently terminate their modal contribution to the total acoustic field. To help see this in Fig. 4(a), the locations where the modal shadow zone boundaries intersect the plane y = 0.229 m are marked by both arrows and mode numbers near the across slope axis labels. It appears that the intensity becomes less complex at the edge of every odd-numbered modal shadow zone boundaries, which suggests that the even-numbered modes were not well excited at the source location and contribute little to the field.

The phenomenon of intra-mode interference is most readily seen in the acoustic intensity for mode 1. In the shallowest plane (y = 0.114 m) only the first mode exists beyond the across slope range of about 0.8 m, while in the deepest plane (y = 0.229 m) only mode 1 exists beyond about 1.1 m. In this portion of the waveguide, there is an interference pattern in the across slope range that resembles a typical two-mode interference pattern for a range-independent waveguide. However, the interference pattern seen here arises because of the interference between more- and less-severely horizontally refracted paths for the same mode. It can also be observed that the cycle distance is increasing and that intra-mode interference pattern is starting to disappear at the most distant ranges in the y = 0.114 m plane—a location which is in the vicinity of the shadow zone boundary for mode 1 (cf. Fig. 3).

Vertical mode amplitudes were obtained at each VLA location through a standard least-squares estimation procedure $A=(Ψ†Ψ)−1Ψ†p*$⁠, where A is a vector of mode amplitudes, Ψ is a matrix of vertical mode shapes, p* is a vector of the demodulated quadrature pressure at each VLA receiver depth z*, and † is the matrix transpose. The quadrature signal was obtained by means of the Hilbert transform and was used in this application to generate the peak modal amplitude as a function of time (as opposed to the momentary modal amplitude). In this waveguide, where the bottom is approximately pressure release, the analytic mode shapes are taken as $Ψ=sin (nπz*/d)$⁠. The use of the analytic mode functions in the mode filtering process is supported by the depth accuracy afforded by the CNC machining of the bathymetric part, the accuracy in the positioning of the receiving transducer, and the temporally invariant isovelocity water layer.

The mode amplitudes for modes 1 − 6 are shown in Fig. 4(b) with the analytic modal shadow zone boundaries superimposed on the data. As was anticipated by the choice of source depth, it appears that the odd-numbered modes were more strongly excited than the even-numbered modes. From the measured data alone, it is not possible to discern if the energy in the even modes was excited at the source or if it arrived there because of mode coupling from odd-numbered modes. This provides one motivation for the modeling results discussed in Sec. V. As general observations, the modes exhibit (1) the highest amplitudes in the modal propagation zone, (2) a caustic-type feature in the vicinity of the modal shadow zone boundary, and (3) the lowest amplitudes in the modal shadow zone. The phenomenon of intra-mode interference is also observable in the mode amplitudes, especially for modes 1, 3, and 5.

Three different acoustic propagation models were employed to describe the measured acoustic field. First, a 3-D analytic model derived in wedge coordinates provides an exact solution for the waveguide. Second, the ray model BELLHOP3D was employed to provide an approximate solution that should include mode coupling effects. It was included in this study as a test of a widely available propagation code on a challenging propagation environment. Finally, the vertical mode/horizontal parabolic equation (VMHPE) method provides an adiabatic solution that helps to quantify the degree of mode coupling in the measured data.

Unlike the physical measurements, there are no inherent sampling limitations in the aforementioned acoustic models. Specifically, there is no need to maintain a transducer safe working distance above the bottom. Vertical sampling in the model space was performed from the air/water interface down to the bottom. Additionally, the depth discretization of the vertical dimension was selected such that mode filtering of six modes could be performed all the way up to the wedge apex.

An analytic model for an isovelocity, translationally invariant wedge with pressure release boundaries was patterned after the derivation of Frisk.¹⁶ The wedge coordinate system for the derivation is shown in Fig. 5(a). Consistent with the variables for the experimental setup, the x dimension is the across slope range. The radial dimension r is measured from the apex to the along slope range and θ is the vertical depression angle from the air/water interface, with θ = β being the slope of the wedge. The only difference between Frisk's derivation and the one utilized here is the change from a rigid to pressure release bottom boundary condition. The pressure field at any point along the wedge, given the source position (r₀, θ₀, x₀ = 0) is

where $αn=nπ/β, Jαn$ is a Bessel function, $Hαn(1)$ is the Hankel function of the first kind, r_< is the lesser of r and r₀, r_> is the greater of r and r₀, and $k2=kr2+kx2$⁠. As an outcome of deriving the solution in the intrinsic wedge coordinates, there is no mode coupling between the wedge modes.

In the wedge coordinate system, the orthonormal mode set is defined along a cylindrical wavefront emanating from the wedge apex instead of a vertical line between the air/water interface and the bottom. As an illustrative example, the difference between vertical and wedge mode shapes at the source position is shown in Figs. 5(b) and 5(c) for modes 3 and 4. The source position used in the experiment is seen to fall directly on a node of the vertical mode shape for mode 4 (and all other even-numbered modes that are not shown). The source position is located near, but not directly on, the node of the mode shape for the wedge mode. Consequently, the even-numbered vertical nodes are not directly excited at the source location and the even-numbered wedge modes are only weakly excited. As illustrated in Figs. 5(b) and 5(c) for mode 3, the odd-numbered vertical and wedge modes would be well excited by the source.

The full pressure field was computed with the analytic wedge model. The mode amplitudes of the modes in wedge coordinates are shown in Fig. 6(a). Again, since the nodes of the wedge modes are not perfectly collocated with the nodes of the vertical modes, the even-numbered wedge modes do not have zero amplitude at the source location. However, the low modal amplitudes of the even-numbered wedge modes indicate that they were not well excited at the source position. The modal shadow zones and intra-mode interference patterns are visible for the odd-numbered wedge modes.

The pressure field in the wedge coordinate system was highly oversampled so that an accurate interpolation onto a Cartesian grid was possible. The analytic solution in Cartesian coordinates was then mode filtered using vertical modes. The mode amplitudes that result from this process are shown in Fig. 6(b). The solution is still exact (up to the error introduced through the interpolation process) but the vertical mode amplitudes are clearly different than the wedge mode amplitudes. Since the even-numbered modes now have relatively more energy throughout their modal propagation zones, despite those modes not being excited at the source location, the explanation is that the energy in these modes results from vertical mode coupling. As a first observation, the amplitude of the even-numbered modes is approximately 20 dB lower than the odd modes. Second, the odd-numbered modes exhibit a caustic-type feature in the vicinity of the shadow zone boundary. Third, the intra-mode interference patterns are clearly visible in the mode amplitudes. All of these features directly correspond to those observed in the measured data. As an example of how the vertical modes contribute to the total field, the acoustic intensity in the across slope plane containing the source is shown in Fig. 6(e). Like the measured data, abrupt changes in the complexity of the acoustic intensity are seen to correspond directly to the locations of the odd-numbered modal shadow zone boundaries [which are visible in Fig. 6(b)].

BELLHOP3D is a 3-D extension of the widely available BELLHOP ray tracing program. This model was utilized with the geometric Gaussian beam option. A relatively dense ray fan of 250 vertical launch angles (between −$80°$ and $80°$⁠) and 250 horizontal launch angles (between −$80°$ and $0°$⁠) was required to obtain convergence of the pressure field. Because the native acoustic field output is in a cylindrical coordinate system, oversampling was again employed so that interpolation to a Cartesian coordinate system could be accomplished with minimal error. With the field in Cartesian coordinates, vertical mode filtering was performed to obtain the mode amplitudes shown in Fig. 6(c). Comparison of the BELLHOP3D mode amplitudes with those of the analytic model show good agreement over much of the domain. The largest errors tend to occur at ranges closest to the source, and can be ascribed to neglecting the last $10°$ in both the vertical and horizontal launch angles. The acoustic intensity in the across slope plane of the source is shown in Fig. 6(f) and compares well to the analytic solution.

The method of vertical modes/horizontal parabolic equation separates the three-dimensional Helmholtz equation into vertical and horizontal contributions. The vertical contribution is expanded in terms of vertical normal modes with the modal amplitudes given by a horizontal refraction equation. The horizontal refraction equation is solved by the parabolic equation method with the horizontal modal wavenumber taking the usual place of the vertical sound speed field. Absorbing boundary layers are added to the edges of the computational domain for the parabolic equation solution, effectively eliminating reflections that would otherwise occur. The resulting pressure field includes the 3-D phenomena of horizontal refraction and intra-mode interference but neglects any coupling between vertical normal modes. The mode amplitudes and acoustic intensity computed from this model are, respectively, shown in Figs. 6(d) and 6(g).

The even-numbered modes are not excited at the source location and also receive no energy in the modal propagation zone because the VMHPE model explicitly neglects mode coupling effects. It is also observed that the caustics near the shadow zone boundary are not quite as intense as they are in the analytic solution. The acoustic intensity of the VMHPE solution contains small differences compared to the analytic and BELLHOP3D solutions.

A comparison of the vertical mode amplitudes from the analytic model and the adiabatic VMHPE model gives insights regarding the degree of mode coupling for this waveguide. Figure 7 compares the modal amplitudes from these two models in the plane y = 0.229 m. The reader will note that the VMHPE amplitudes for the even-numbered modes do not appear in the figure because the source was located at a node of these modes and mode coupling is neglected in this model. The location of the modal shadow zone boundaries in the plane of the figure are indicated by the vertical lines and are labeled above the figure.

Several observations are mentioned and discussed. First, the odd-numbered modes exhibit the typical spherical spreading loss at very short ranges with a transition to cylindrical spreading within about half of a water depth (0.02 m). The even-numbered analytic modes do not exhibit the same behavior. In fact, their modal amplitudes start at a local minimum at the source location and begin to increase with range, suggesting that instead of a undergoing a geometric spreading loss, they are acquiring energy that is being scattered out of other modes.

Second, the modal amplitudes for the odd-numbered modes have good agreement to ranges past the modal shadow zone boundary. Less attention is given here to the depths of the intra-mode interference nulls, as the computational efficiency of the VMHPE model allowed range sampling about eight times greater than the analytic model. The phasing of the intra-mode interference pattern is almost identical between the two models, with the largest discrepancy in the cycle-distance occurring for mode 1. There is an across slope range (approximately 1.35 and 0.78 m for modes 3 and 5, respectively) at which the analytic solution deviates abruptly from the adiabatic solution. Instead of the modal amplitude continuing to decay with range, it levels off as the energy scattered into the mode begins to dominate the amplitude curve. This theoretical result shows that energy can exist in a modal shadow zone for a given mode, but that it has to be coupled from other modes. In this particular waveguide, the analytic mode amplitudes in the shadow zone are approximately 35 dB lower than they are in the vicinity of the modal shadow zone boundary.

Third, the even-numbered modes of the analytic model exhibit interference patterns with range that are fundamentally different than the intra-mode interference patterns of the odd-numbered modes. For example, the mode 2 amplitude in Fig. 7 exhibits four cycles in the first 1.2 m that are much longer than the cycle distances of the adjacent modes. This feature is also visible in Fig. 6 for other along slope ranges and in the measured data of Fig. 4(b) for the second mode. It is presumed that this different interference pattern arises due to the mechanics of coupling energy into the even modes, as opposed to the intra-mode interference that results for energy radiated by the source.

Fourth, one particular energy transfer path between modes is inferred from Fig. 7. Consider the shape of the mode amplitude curve for mode 3 in the vicinity of its shadow zone boundary. As this mode is being refracted down the slope, it is coupling energy into modes 4, 5, and 6, which is inferred by the shape of the third mode amplitude curve in the mode amplitude curves of higher angle modes. A similar feature can be observed between modes 5 and 6 in the vicinity of the modal shadow zone boundary of mode 5. The scattering of energy into higher angle modes appears to be similar to that which occurs during upslope propagation in the vicinity of mode cutoff. Any cascade of energy from higher angle modes into lower angle modes is not readily apparent in Fig. 7.

This work was supported by Office of Naval Research Grant Nos. N00014-15-1-2017 and N00014-16-1-2397.