Reflection Module

The reflection module contains functions for calculating reflection factors and impedances.

Attributes

SPECIFIC_HEAT_RATIO module-attribute

SPECIFIC_HEAT_RATIO = 1.4

Specific heat ratio of air \(\gamma\).

POROSITY_DECREASE module-attribute

POROSITY_DECREASE = 120.0

Rate of exponential decrease of porosity with depth \(\alpha\).

SOUNDSPEED module-attribute

SOUNDSPEED = 343.0

Speed of sound in air \(c\).

DENSITY module-attribute

DENSITY = 1.296

Density of air \(\rho\).

Classes

Boundary

Boundary(
    frequency,
    flow_resistivity,
    density=DENSITY,
    soundspeed=SOUNDSPEED,
    porosity_decrease=POROSITY_DECREASE,
    specific_heat_ratio=SPECIFIC_HEAT_RATIO,
    angle=None,
    distance=None,
    impedance_model="db",
    reflection_model="plane",
)

An object describing a boundary.

ATTRIBUTE	DESCRIPTION
frequency	Frequency. Single value or vector for a frequency range.
flow_resistivity	Flow resistivity \(\sigma\).
density	Density of air \(\rho\). Note: This value is only required for when calculating the impedance according to Attenborough's model. See impedance_attenborough.
soundspeed	Speed of sound in air \(c\). Note: This value is required when calculating the impedance according to Attenborough's model or when calculating the spherical wave reflection factor. See respectively impedance_attenborough and reflection_factor_spherical_wave.
porosity_decrease	Rate of exponential decrease of porosity with depth \(\alpha\). Note: This value is only required for when calculating the impedance according to Attenborough's model. See impedance_attenborough.
specific_heat_ratio	Ratio of specific heats \(\gamma\) for air. Note: This value is only required for when calculating the impedance according to Attenborough's model. See impedance_attenborough.
angle	Angle of incidence \(\theta\).
distance	Path length of the reflected ray \(r\). Note: This value is only required for when calculating the spherical wave reflection factor. See reflection_factor_spherical_wave.
impedance_model	Impedance model. Note: Possibilities are db and att for respectively impedance_delany_and_bazley and impedance_attenborough.
reflection_model	Reflection factor model. Note: Possibilities are plane and spherical for respectively reflection_factor_plane_wave and reflection_factor_spherical_wave.

Source code in acoustic_toolbox/reflection.py

def __init__(  # pylint: disable=too-many-instance-attributes
    self,
    frequency,
    flow_resistivity,
    density=DENSITY,
    soundspeed=SOUNDSPEED,
    porosity_decrease=POROSITY_DECREASE,
    specific_heat_ratio=SPECIFIC_HEAT_RATIO,
    angle=None,
    distance=None,
    impedance_model="db",
    reflection_model="plane",
):
    """Initialize the boundary."""
    self.frequency = frequency
    self.flow_resistivity = flow_resistivity
    self.density = density
    self.soundspeed = soundspeed

    self.porosity_decrease = porosity_decrease

    self.specific_heat_ratio = specific_heat_ratio

    self.angle = angle
    self.distance = distance

    self.impedance_model = impedance_model
    self.reflection_model = reflection_model

Attributes

wavenumber property

wavenumber: float

Wavenumber \(k\).

RETURNS	DESCRIPTION
float	Wavenumber calculated as:
float	$$
float	k = \frac{2 \pi f}{c}
float	$$

impedance property

impedance: ndarray

Impedance according to chosen impedance model defined using impedance_model.

RAISES	DESCRIPTION
ValueError	If the impedance model is incorrect.

reflection_factor property

reflection_factor: ndarray

Reflection factor according to chosen reflection factor model defined using reflection_model.

RAISES	DESCRIPTION
AttributeError	If the angle is not specified.
AttributeError	If the distance is not specified.
ValueError	..shrug..

Functions

plot_impedance

plot_impedance(filename=None) -> Figure

Plot magnitude and phase of the impedance as function of frequency.

PARAMETER	DESCRIPTION
filename	File name to save the plot to. DEFAULT: None

RETURNS	DESCRIPTION
Figure	The figure. TYPE: Figure

Source code in acoustic_toolbox/reflection.py

def plot_impedance(self, filename=None) -> Figure:
    """Plot magnitude and phase of the impedance as function of frequency.

    Args:
        filename: File name to save the plot to.

    Returns:
        Figure: The figure.
    """
    fig = plt.figure()

    ax0 = fig.add_subplot(211)
    ax0.set_title("Magnitude of impedance")
    ax0.semilogx(self.frequency, np.abs(self.impedance))
    ax0.set_xlabel(r"$f$ in Hz")
    ax0.set_ylabel(r"$\left|Z\right|$")
    ax0.grid()

    ax0 = fig.add_subplot(212)
    ax0.set_title("Angle of impedance")
    ax0.semilogx(self.frequency, np.angle(self.impedance))
    ax0.set_xlabel(r"$f$ in Hz")
    ax0.set_ylabel(r"$\angle Z$")
    ax0.grid()

    plt.tight_layout()

    if filename:
        fig.savefig(filename, transparant=True)
    return fig

plot_reflection_factor

plot_reflection_factor(filename=None) -> Figure

Plot reflection factor.

PARAMETER	DESCRIPTION
filename	File name to save the plot to. DEFAULT: None

RETURNS	DESCRIPTION
Figure	The figure. TYPE: Figure

Source code in acoustic_toolbox/reflection.py

def plot_reflection_factor(self, filename=None) -> Figure:
    """Plot reflection factor.

    Args:
        filename: File name to save the plot to.

    Returns:
        Figure: The figure.
    """
    if self.frequency is None:
        raise ValueError("No frequency specified.")
    if self.angle is None:
        raise ValueError("No angle specified.")

    try:
        n_f = len(self.frequency)
    except TypeError:
        n_f = 1
    try:
        n_a = len(self.angle)
    except TypeError:
        n_a = 1

    if n_f == 1 and n_a == 1:
        raise ValueError("Either frequency or angle needs to be a vector.")

    elif n_f == 1 or n_a == 1:
        if (
            n_f == 1 and n_a > 1
        ):  # Show R as function of angle for a single frequency.
            xlabel = r"$\theta$ in degrees"
        elif (
            n_f > 1 and n_a == 1
        ):  # Show R as function of frequency for a single angle.
            xlabel = r"$f$ in Hz"
        R = self.reflection_factor
        fig = plt.figure()

        ax0 = fig.add_subplot(211)
        ax0.set_title("Magnitude of reflection factor")
        ax0.semilogx(self.frequency, np.abs(R))
        ax0.set_xlabel(xlabel)
        ax0.set_ylabel(r"$\left|R\right|$")
        ax0.grid()

        ax1 = fig.add_subplot(212)
        ax1.set_title("Phase of reflection factor")
        ax1.semilogx(self.frequency, np.angle(R))
        ax1.set_xlabel(xlabel)
        ax1.set_ylabel(r"$\angle R$")
        ax1.grid()

    elif n_f > 1 and n_a > 1:  # Show 3D or pcolor
        R = self.reflection_factor
        fig = plt.figure()

        # grid = AxesGrid(fig, 111, nrows_ncols=(2, 2), axes_pad=0.1, cbar_mode='each', cbar_location='right')
        ax0 = fig.add_subplot(211)
        # ax0 = grid[0]
        ax0.set_title("Magnitude of reflection factor")
        ax0.pcolormesh(self.frequency, self.angle * 180.0 / np.pi, np.abs(R))
        # ax0.pcolor(self.angle, self.frequency, np.abs(R))
        # ax0.set_xlabel(xlabel)
        # ax0.set_ylabel(r'$\left|R\right|$')
        ax0.grid()

        ax1 = fig.add_subplot(212)
        # ax1 = grid[1]
        ax1.set_title("Phase of reflection factor")
        ax1.pcolormesh(self.frequency, self.angle * 180.0 / np.pi, np.angle(R))
        # ax1.pcolor(self.angle, self.frequency, np.angle(R))
        # ax0.set_xlabel(xlabel)
        # ax0.set_ylabel(r'$\angle R$')
        ax1.grid()

    else:
        raise RuntimeError("Oops...")

    # plt.tight_layout()

    if filename:
        fig.savefig(filename, transparant=True)
    else:
        return fig

Functions

reflection_factor_plane_wave

reflection_factor_plane_wave(impedance, angle)

Plane wave reflection factor \(R\).

PARAMETER	DESCRIPTION
impedance	Normalized impedance \(Z\).
angle	Angle of incidence \(\theta\).

RETURNS	DESCRIPTION
float	Plane wave reflection factor calculated as: $$ R = \frac{Z \cos{\theta} - 1}{Z \cos{\theta} + 1} $$

Source code in acoustic_toolbox/reflection.py

def reflection_factor_plane_wave(impedance, angle):
    r"""Plane wave reflection factor $R$.

    Args:
      impedance: Normalized impedance $Z$.
      angle: Angle of incidence $\theta$.

    Returns:
        float: Plane wave reflection factor calculated as:
            $$
            R = \frac{Z \cos{\theta} - 1}{Z \cos{\theta} + 1}
            $$
    """
    return (impedance * np.cos(angle) - 1.0) / (impedance * np.cos(angle) + 1.0)

numerical_distance

numerical_distance(impedance, angle, distance, wavenumber)

Numerical distance \(w\).

PARAMETER	DESCRIPTION
impedance	Normalized impedance \(Z\).
angle	Angle of incidence \(\theta\).
distance	Path length of the reflected ray \(r\).
wavenumber	Wavenumber \(k\).

RETURNS	DESCRIPTION
complex	Numerical distance calculated as: $$ w = \sqrt{-j k r \left( 1 + \frac{1}{Z} \cos{\theta} - \sqrt{1 - \left( \frac{1}{Z} \right)^2} \sin{\theta} \right) } $$

Source code in acoustic_toolbox/reflection.py

def numerical_distance(impedance, angle, distance, wavenumber):
    r"""Numerical distance $w$.

    Args:
      impedance: Normalized impedance $Z$.
      angle: Angle of incidence $\theta$.
      distance: Path length of the reflected ray $r$.
      wavenumber: Wavenumber $k$.

    Returns:
        complex: Numerical distance calculated as:
            $$
            w = \sqrt{-j k r  \left( 1 + \frac{1}{Z} \cos{\theta} - \sqrt{1 - \left( \frac{1}{Z} \right)^2} \sin{\theta} \right) }
            $$
    """
    return np.sqrt(
        -1j
        * wavenumber
        * distance
        * (
            1.0
            + 1.0 / impedance * np.cos(angle)
            - np.sqrt(1.0 - (1.0 / impedance) ** 2.0) * np.sin(angle)
        )
    )

reflection_factor_spherical_wave

reflection_factor_spherical_wave(
    impedance, angle, distance, wavenumber
)

Spherical wave reflection factor \(Q\).

PARAMETER	DESCRIPTION
impedance	Normalized impedance \(Z\).
angle	Angle of incidence \(\theta\).
distance	Path length of the reflected ray \(r\).
wavenumber	Wavenumber \(k\).

RETURNS	DESCRIPTION
complex	Spherical wave reflection factor calculated as: $$ Q = R \left(1 - R \right) F $$ where \(R\) is the plane wave reflection factor as calculated in reflection_factor_plane_wave and \(F\) is given by $$ F = 1 - j \sqrt{\pi} w e^{-w^2} \mathrm{erfc} \left( j w \right) $$

Source code in acoustic_toolbox/reflection.py

def reflection_factor_spherical_wave(impedance, angle, distance, wavenumber):
    r"""Spherical wave reflection factor $Q$.

    Args:
      impedance: Normalized impedance $Z$.
      angle: Angle of incidence $\theta$.
      distance: Path length of the reflected ray $r$.
      wavenumber: Wavenumber $k$.

    Returns:
        complex: Spherical wave reflection factor calculated as:
            $$
            Q = R \left(1 - R \right) F
            $$

            where $R$ is the plane wave reflection factor as calculated in [reflection_factor_plane_wave][acoustic_toolbox.reflection.reflection_factor_plane_wave] and $F$ is given by
            $$
            F = 1 - j \sqrt{\pi} w e^{-w^2} \mathrm{erfc} \left( j w \right)
            $$
    """
    w = numerical_distance(impedance, angle, distance, wavenumber)

    F = 1.0 - 1j * np.sqrt(np.pi) * w * np.exp(-(w**2.0)) * erfc(1j * w)

    plane_factor = reflection_factor_plane_wave(impedance, angle)
    return plane_factor * (1.0 - plane_factor) * F

impedance_delany_and_bazley

impedance_delany_and_bazley(frequency, flow_resistivity)

Normalised specific acoustic impedance according to the empirical one-parameter model by Delany and Bazley.

PARAMETER	DESCRIPTION
frequency	Frequency \(f\).
flow_resistivity	Flow resistivity \(\sigma\).

RETURNS	DESCRIPTION
complex	Impedance calculated as: $$ Z = 1 + 9.08 \left( \frac{1000f}{\sigma} \right)^{-0.75} - 11.9 j \left( \frac{1000f}{\sigma} \right)^{-0.73} $$

Source code in acoustic_toolbox/reflection.py

def impedance_delany_and_bazley(frequency, flow_resistivity):
    r"""Normalised specific acoustic impedance according to the empirical one-parameter model by Delany and Bazley.

    Args:
      frequency: Frequency $f$.
      flow_resistivity: Flow resistivity $\sigma$.

    Returns:
        complex: Impedance calculated as:
            $$
            Z = 1 + 9.08 \left( \frac{1000f}{\sigma} \right)^{-0.75} - 11.9 j \left( \frac{1000f}{\sigma} \right)^{-0.73}
            $$
    """
    return (
        1.0
        + 9.08 * (1000.0 * frequency / flow_resistivity) ** (-0.75)
        - 1j * 11.9 * (1000.0 * frequency / flow_resistivity) ** (-0.73)
    )

impedance_attenborough

impedance_attenborough(
    frequency,
    flow_resistivity,
    density=DENSITY,
    soundspeed=SOUNDSPEED,
    porosity_decrease=POROSITY_DECREASE,
    specific_heat_ratio=SPECIFIC_HEAT_RATIO,
)

Normalised specific acoustics impedance according to the two-parameter model by Attenborough.

PARAMETER	DESCRIPTION
frequency	Frequency \(f\).
flow_resistivity	Flow resistivity \(\sigma\).
soundspeed	Speed of sound in air \(c\). DEFAULT: SOUNDSPEED
density	Density of air \(\rho\). DEFAULT: DENSITY
porosity_decrease	Rate of exponential decrease of porosity with depth \(\alpha\). DEFAULT: POROSITY_DECREASE
specific_heat_ratio	Ratio of specific heats \(\gamma\) for air. DEFAULT: SPECIFIC_HEAT_RATIO

RETURNS	DESCRIPTION
complex	Impedance calculated as: $$ Z = \frac{\left( 1-j \right) \sqrt{\sigma/f}}{\sqrt{\pi \gamma_0 \rho_0}} - \frac{jc\alpha}{8 \pi \gamma_0 f} $$

Source code in acoustic_toolbox/reflection.py

def impedance_attenborough(
    frequency,
    flow_resistivity,
    density=DENSITY,
    soundspeed=SOUNDSPEED,
    porosity_decrease=POROSITY_DECREASE,
    specific_heat_ratio=SPECIFIC_HEAT_RATIO,
):
    r"""Normalised specific acoustics impedance according to the two-parameter model by Attenborough.

    Args:
      frequency: Frequency $f$.
      flow_resistivity: Flow resistivity $\sigma$.
      soundspeed: Speed of sound in air $c$.
      density: Density of air $\rho$.
      porosity_decrease: Rate of exponential decrease of porosity with depth $\alpha$.
      specific_heat_ratio: Ratio of specific heats $\gamma$ for air.

    Returns:
        complex: Impedance calculated as:
            $$
            Z = \frac{\left( 1-j \right) \sqrt{\sigma/f}}{\sqrt{\pi \gamma_0 \rho_0}} - \frac{jc\alpha}{8 \pi \gamma_0 f}
            $$
    """
    return (1.0 - 1j) * np.sqrt(flow_resistivity / frequency) / np.sqrt(
        np.pi * specific_heat_ratio * density
    ) - 1j * soundspeed * porosity_decrease / (
        8.0 * np.pi * specific_heat_ratio * frequency
    )

:::