underwater.propagation.weston_regimes

Weston’s shallow-water propagation regimes (flux theory).

A source in a shallow-water waveguide loses energy in four successive range regimes, each with its own power law. The boundaries between them follow from the seabed reflectivity alone, which makes the set an inexpensive analytic reference for any numerical propagation model:

• spherical spreading — $F=1/r^2$ ($20{\log }_{10}r$), while the sound has not yet felt the boundaries;
• cylindrical spreading — $F=2{\psi }_{\mathrm{c}}/(rH)$ ($10{\log }_{10}r$), once the energy is confined to a cylinder of height H and only rays within the critical angle ${\psi }_{\mathrm{c}}$ survive;
• mode stripping — $F=(\pi /(\eta H){)}^{1/2}{r}^{-3/2}$ ($15{\log }_{10}r$), once the accumulated reflection loss has eroded the steep paths;
• single mode — an exponential decay dominated by the lowest-order mode.

Everything here is implemented clean-room from Ainslie, Principles of Sonar Performance Modelling (Springer 2010), §9.1.1.2 (printed pp. 452-458): Equations (9.42) to (9.61) and the seabed properties of Table 9.1 (WESTON_SEABEDS). The quantity computed is Ainslie’s propagation factor F (units m⁻²), reported as the propagation loss $\mathrm{PL}=-10{\log }_{10}F$ dB re 1 m², which reduces to spherical spreading for a point source in free water.

The regime formulae are energy-flux (incoherent) results: they describe the range-averaged field, not its modal interference. That is exactly what makes them a usable cross-check for phonometry.underwater.propagation.numerical — the range average of a normal-mode or parabolic-equation field over many interference cycles converges on the cylindrical-spreading law, with ${\psi }_{\mathrm{c}}=\pi /2$ for a totally reflecting (pressure-release) bottom.

Note

Ainslie’s Equation (9.57) for the mode-stripping/single-mode transition is printed as $r_{\mathrm{MS}}\approx k^2{H}_{\mathrm{e}}^3/(9\eta )$. Carrying out the derivation the accompanying text prescribes — “equating ${\theta }_n$ and ${\theta }_{\mathrm{eff}}$ with $n=3/2$” — with the two equations exactly as they are printed, namely ${\theta }_{\mathrm{eff}}=(\pi H/(4\eta r){)}^{1/2}$ (Equation 9.47, with the true water depth H) and ${\theta }_n=n\pi /(k{H}_{\mathrm{e}})$ (Equation 9.56, with the effective depth He), gives $r_{\mathrm{MS}}=k^2{H}_{\mathrm{e}}^{2}H/(9\pi \eta )$ instead. The printed form is larger by $\pi H_{\mathrm{e}}/H$. This module implements the derivation-consistent value, which also keeps ${\theta }_{\mathrm{eff}}$ defined with H everywhere it is used (the composite loss below evaluates Equation 9.47 the same way), and records the discrepancy in docs/ERRATA.md.

Auto-generated from the source docstrings by scripts/generate_api_docs.py (make api-docs). Do not edit by hand.

critical_grazing_angle

critical_grazing_angle(sound_speed_ratio: float) -> float

Critical grazing angle ${\psi }_{\mathrm{c}}=\arccos (c_{\mathrm{w}}/c_{\mathrm{sed}})$, in radians.

A seabed slower than the water ($c_{\mathrm{sed}}\le c_{\mathrm{w}}$, e.g. mud) has no critical angle; the function then returns 0, which correctly switches the reflection-loss gradient to the refracting-sediment branch of reflection_loss_gradient.

Parameters

Name	Description
sound_speed_ratio	$c_{\mathrm{sed}}/c_{\mathrm{w}}$, dimensionless and positive.

Returns: The critical grazing angle, in radians (0 if none exists).

Raises

Exception	When
ValueError	If the ratio is not positive and finite.

effective_depth

effective_depth(
    water_depth: float,
    frequency_hz: float,
    *,
    seabed: str | WestonSeabed = 'sand',
    sound_speed: float = 1500.0,
) -> float

Weston effective water depth He (Ainslie Eq. 9.55), in metres.

$H_{\mathrm{e}}=H+({\rho }_{\mathrm{sed}}/{\rho }_{\mathrm{w}})/((\omega /c_{\mathrm{w}})\sin {\psi }_{\mathrm{c}})$: the depth at which a pressure-release boundary appears to lie, a short distance below the true seabed. Only meaningful for a seabed with a critical angle.

Parameters

Name	Description
water_depth	Water-column depth H, in metres.
frequency_hz	Acoustic frequency, in Hz.
seabed	"sand", "mud" or a WestonSeabed.
sound_speed	Water sound speed c_w, in m/s.

Returns: The effective depth He, in metres.

Raises

Exception	When
ValueError	If the seabed has no critical angle or an input is invalid.

loss_parameter

loss_parameter(attenuation_db_per_wavelength: float) -> float

Sediment loss parameter $\epsilon ={\beta }_{\mathrm{sed}}/(40\pi {\log }_{10}e)$ (Ainslie Eq. 9.23).

Parameters

Name	Description
attenuation_db_per_wavelength	${\beta }_{\mathrm{sed}}$, in dB per wavelength.

Returns: The dimensionless loss parameter $\epsilon$.

Raises

Exception	When
ValueError	If the attenuation is negative or non-finite.

reflection_loss_gradient

reflection_loss_gradient(
    seabed: str | WestonSeabed = 'sand',
    *,
    frequency_hz: float | None = None,
) -> float

Reflection loss gradient $\eta$, in nepers per radian.

The rate at which the seabed reflection loss grows with grazing angle, $|R(\theta )|\approx \exp (-\eta \theta )$ (Ainslie Eq. 9.45). Two branches:

• a reflecting seabed with a critical angle (sand, coarse silt), $\eta =2\epsilon ({\rho }_{\mathrm{sed}}/{\rho }_{\mathrm{w}}){\cos }^2{\psi }_{\mathrm{c}}/{\sin }^3{\psi }_{\mathrm{c}}$ (Eq. 9.51), frequency-independent;
• a refracting seabed with none (mud, clay, fine silt), $\eta =2\omega \epsilon /c^{\prime }$ (Eq. 9.53), proportional to frequency.

Parameters

Name	Description
seabed	"sand", "mud" or an explicit WestonSeabed.
frequency_hz	Acoustic frequency, in Hz; required only for the refracting branch ($c^{\prime }>0$).

Returns: The reflection loss gradient $\eta$, in Np/rad.

Raises

Exception	When
ValueError	If the frequency is missing or invalid for a refracting seabed.

waveguide_cutoff_frequency

waveguide_cutoff_frequency(
    water_depth: float,
    *,
    seabed: str | WestonSeabed = 'sand',
    sound_speed: float = 1500.0,
) -> float

Shallow-water waveguide cut-off frequency fc (Ainslie Eq. 9.60), in Hz.

$f_{\mathrm{c}}=(\pi -{\rho }_{\mathrm{sed}}/{\rho }_{\mathrm{w}})/(2\pi \sin {\psi }_{\mathrm{c}})\cdot c_{\mathrm{w}}/H$ — below it no mode is cut on and ducted propagation does not occur.

Parameters

Name	Description
water_depth	Water-column depth H, in metres.
seabed	"sand", "mud" or a WestonSeabed.
sound_speed	Water sound speed c_w, in m/s.

Returns: The cut-off frequency, in Hz.

Raises

Exception	When
ValueError	If the seabed has no critical angle or an input is invalid.

weston_propagation_loss

weston_propagation_loss(
    range_m: NDArray[np.float64] | list[float] | float,
    frequency_hz: float,
    water_depth: float,
    *,
    seabed: str | WestonSeabed = 'sand',
    sound_speed: float = 1500.0,
    source_depth: float | None = None,
    receiver_depth: float | None = None,
    critical_angle: float | None = None,
    reflection_loss_gradient_value: float | None = None,
) -> WestonPropagationResult

Propagation loss across Weston’s four shallow-water regimes.

Assembles the piecewise loss from Ainslie’s Equations (9.42), (9.49) and (9.54), switching regime at the boundaries of weston_regime_boundaries, and returns each regime’s own law over the whole range grid so the transitions can be drawn.

Parameters

Name	Description
range_m	Range(s) from the source, in metres (scalar or array, strictly positive).
frequency_hz	Acoustic frequency, in Hz.
water_depth	Water-column depth H, in metres.
seabed	"sand", "mud" or a WestonSeabed.
sound_speed	Water sound speed c_w, in m/s.
source_depth	Source depth z0, in metres; defaults to H/2 (used only by the single-mode formula).
receiver_depth	Receiver depth z, in metres; defaults to H/2.
critical_angle	Override ${\psi }_{\mathrm{c}}$, in degrees (90 for an ideal totally reflecting waveguide).
reflection_loss_gradient_value	Override $\eta$, in Np/rad (0 for a lossless bottom: no mode stripping, no single-mode regime).

Returns: A WestonPropagationResult.

Raises

Exception	When
ValueError	If an input is invalid.

weston_regime_boundaries

weston_regime_boundaries(
    frequency_hz: float,
    water_depth: float,
    *,
    seabed: str | WestonSeabed = 'sand',
    sound_speed: float = 1500.0,
    critical_angle: float | None = None,
    reflection_loss_gradient_value: float | None = None,
) -> WestonRegimeBoundaries

Regime boundaries of a shallow-water waveguide (Ainslie §9.1.1.2).

Parameters

Name	Description
frequency_hz	Acoustic frequency, in Hz.
water_depth	Water-column depth H, in metres.
seabed	"sand", "mud" or a WestonSeabed.
sound_speed	Water sound speed c_w, in m/s.
critical_angle	Override the seabed critical angle ${\psi }_{\mathrm{c}}$, in degrees. Use 90 for the ideal totally reflecting waveguide.
reflection_loss_gradient_value	Override $\eta$, in Np/rad. Use 0 for a lossless bottom (no mode stripping, no single-mode regime).

Returns: A WestonRegimeBoundaries.

Raises

Exception	When
ValueError	If an input is invalid.

Note

The two overrides are independent: overriding critical_angle alone leaves $\eta$ computed from the seabed’s own critical angle through Equation (9.51), which mixes two different bottoms. Pass both together (as the ideal-waveguide case critical_angle=90 with reflection_loss_gradient_value=0 does) whenever the intent is a hypothetical seabed rather than a tweak of the tabulated one.

WESTON_REGIMES

Constant (tuple).

WESTON_REGIMES = ('spherical', 'cylindrical', 'mode-stripping', 'single-mode')

WESTON_SEABEDS

Constant (dict).

WESTON_SEABEDS = {'sand': WestonSeabed(name='sand', grain_size=1.5, sound_speed_ratio=1.2, density_ratio=2.1, attenuation_db_per_wavelength=0.88, loss_parameter=0.0161, sound_speed_gradient=0.0), 'mud': WestonSeabed(name='mud', grain_size=8.0, sound_speed_ratio=1.0, density_ratio=1.4, attenuation_db_per_wavelength=0.09, loss_parameter=0.00165, sound_speed_gradient=1.0)}

WestonPropagationResult

WestonPropagationResult(
    range_m: NDArray[np.float64],
    propagation_loss: NDArray[np.float64],
    propagation_factor: NDArray[np.float64],
    regime: NDArray[np.str_],
    spherical: NDArray[np.float64],
    cylindrical: NDArray[np.float64],
    mode_stripping: NDArray[np.float64],
    single_mode: NDArray[np.float64],
    multipath: NDArray[np.float64],
    boundaries: WestonRegimeBoundaries,
    frequency: float,
    water_depth: float,
    source_depth: float,
    receiver_depth: float,
    seabed: str,
)

Weston regime propagation loss versus range.

Attributes

Name	Description
range_m	Ranges from the source, in metres.
propagation_loss	Composite propagation loss $\mathrm{PL}=-10{\log }_{10}F$ per range, in dB re 1 m².
propagation_factor	The composite propagation factor F, in m⁻².
regime	The active regime label at each range (one of WESTON_REGIMES).
spherical	Spherical-spreading loss $20{\log }_{10}r$ at every range, in dB.
cylindrical	Cylindrical-spreading loss (Eq. 9.42) at every range, dB.
mode_stripping	Mode-stripping loss (Eq. 9.49) at every range, dB (nan when the bottom is lossless: without reflection loss there is nothing to strip).
single_mode	Single-mode loss (Eq. 9.54) at every range, in dB.
multipath	Loss from the continuous multipath integral (Eq. 9.46), which joins the cylindrical and mode-stripping regimes smoothly, in dB.
boundaries	The WestonRegimeBoundaries in force.
frequency	Acoustic frequency, in Hz.
water_depth	Water-column depth H, in metres.
source_depth	Source depth z0, in metres.
receiver_depth	Receiver depth z, in metres.
seabed	Name of the seabed used.

WestonPropagationResult.plot()

WestonPropagationResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the composite loss with each regime’s law and the boundaries.

WestonRegimeBoundaries

WestonRegimeBoundaries(
    spherical_to_cylindrical: float,
    cylindrical_to_mode_stripping: float,
    mode_stripping_to_single_mode: float,
    critical_angle: float,
    reflection_loss_gradient: float,
    effective_depth: float,
    cutoff_frequency: float,
    mode_count: float,
)

Range boundaries between Weston’s four propagation regimes.

Attributes

Name	Description
spherical_to_cylindrical	Range at which $1/r^2$ and $2{\psi }_{\mathrm{c}}/(rH)$ are equal, $H/(2{\psi }_{\mathrm{c}})$, in metres.
cylindrical_to_mode_stripping	Ainslie Eq. (9.50) $r_{\mathrm{CS}}=\pi H/(4\eta {\psi }_{\mathrm{c}}^2)$, in metres (inf for a lossless bottom).
mode_stripping_to_single_mode	$r_{\mathrm{MS}}=k^2{H}_{\mathrm{e}}^{2}H/(9\pi \eta )$, in metres (inf for a lossless bottom). See the module note on Eq. (9.57).
critical_angle	Critical grazing angle ${\psi }_{\mathrm{c}}$, in radians.
reflection_loss_gradient	$\eta$, in Np/rad.
effective_depth	Weston effective depth He, in metres.
cutoff_frequency	Waveguide cut-off frequency, in Hz (nan when the seabed has no critical angle).
mode_count	Number of cut-on modes, $(\omega /c_{\mathrm{w}})H_{\mathrm{e}}\sin {\psi }_{\mathrm{c}}/\pi$ (Eq. 9.58), as a real number.

WestonSeabed

WestonSeabed(
    name: str,
    grain_size: float,
    sound_speed_ratio: float,
    density_ratio: float,
    attenuation_db_per_wavelength: float,
    loss_parameter: float,
    sound_speed_gradient: float,
)

Characteristic seabed properties (Ainslie Table 9.1, printed p. 454).

Attributes

Name	Description
name	Sediment name.
grain_size	Grain size Mz (phi units).
sound_speed_ratio	$c_{\mathrm{sed}}/c_{\mathrm{w}}$.
density_ratio	${\rho }_{\mathrm{sed}}/{\rho }_{\mathrm{w}}$.
attenuation_db_per_wavelength	${\beta }_{\mathrm{sed}}$, in dB per wavelength.
loss_parameter	$\epsilon ={\beta }_{\mathrm{sed}}/(40\pi {\log }_{10}e)$ (Equation 9.23).
sound_speed_gradient	c', the sediment sound-speed gradient, in s⁻¹ (0 for sand, 1 for mud).