electroacoustics.piston

Radiation of a rigid circular piston set in an infinite baffle.

The baffled circular piston is the canonical acoustic radiator: a flat rigid disc of radius a vibrating with a uniform normal velocity in an otherwise rigid infinite plane. It is the model behind a loudspeaker cone in a large cabinet, the open end of a duct, and the reference source for the radiation efficiency of any finite vibrating surface, so its two results — the radiation impedance the air presents to the piston and the directivity of the far field — are the base of the electroacoustics domain (Beranek & Mellow, Acoustics: Sound Fields, Transducers and Vibration 2nd ed., §4.4; Bies, Hansen & Howard, Engineering Noise Control 5th ed.).

Radiation impedance. The reaction force of the air on the piston is $F = Z_{r} u$ with the mechanical radiation impedance

Z_{r} = ρ c S (R_{1} (2 k a) + j X_{1} (2 k a)), S = π a^{2}

where $k = ω / c$ is the wavenumber, $ρ c$ the characteristic impedance of air and S the piston area. The dimensionless piston resistance and reactance functions are (Beranek & Mellow Eq. (4.30))

R_{1} (x) = 1 - \frac{2 J _{1} ( x )}{x}, X_{1} (x) = \frac{2 H _{1} ( x )}{x}

with J1 the Bessel function of the first kind and H1 the Struve function, both of order one, evaluated at $x = 2 k a$ .

Low frequency ( $k a ≪ 1$ ): $R_{1} \to (k a)^{2} /2$ so the radiated power rises as $f^{2}$ , and $X_{1} \to (8/3 π) k a$ . The reactance is mass-like, $X_{r} = ρ c S X_{1} = ω M_{r}$ with the radiation (accreted) mass $M_{r} = 8 ρ a^{3} /3$ (Beranek & Mellow Eq. (4.32)): the piston drags an extra $8 ρ a^{3} /3$ of air, equivalent to a layer $8 a /3 π$ thick over its face.
High frequency ( $k a ≫ 1$ ): $R_{1} \to 1$ and $X_{1} \to 0$ , so $Z_{r} \to ρ c S$ — the piston radiates as if into an infinite tube and the air loads it purely resistively.

Directivity. The far-field pressure of the baffled piston varies with the polar angle theta from the axis as (Beranek & Mellow Eq. (4.42))

D (θ) = \frac{2 J _{1} ( k a sin θ )}{k a sin θ}, D (0) = 1

The main lobe narrows as ka grows; its first null is at $k a sin θ = 3.8317$ (the first zero of J1), which exists only once $k a > 3.8317$ . The directivity factor Q (on-axis intensity over the intensity of a point source of equal power radiating into the full sphere) and the directivity index $D I = 10 lo g_{10} Q$ follow from integrating $∣ D ∣^{2}$ over the radiating hemisphere,

Q = \frac{2}{\int _{0}^{π /2} ∣ D ( θ ) ∣ ^{2} sin θ d θ}

which tends to $Q = 2$ ( $D I = 3.01$ dB, the half-space baffle gain) at low ka and to $Q \sim (k a)^{2}$ ( $D I \sim 20 lo g_{10} k a$ ) at high ka.

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piston_directivity

piston_directivity(ka: ArrayLike, theta: ArrayLike) -> np.ndarray | float

Far-field directivity $D = 2 J_{1} (k a sin θ) / (k a sin θ)$ .

The pressure amplitude of a baffled circular piston relative to its on-axis value (Beranek & Mellow Eq. (4.42)), normalized so $D (0) = 1$ .

Parameters

Name	Description
`ka`	Wavenumber-radius product `ka` (scalar or array).
`theta`	Polar angle from the axis, rad (scalar or array). Broadcast against `ka`.

Returns: D (float for scalar inputs, else an array).

piston_directivity_pattern

piston_directivity_pattern(
    ka: ArrayLike,
    angles: ArrayLike | None = None,
) -> PistonDirectivity

Far-field directivity pattern of one or more baffled circular pistons.

Samples the directivity $D (θ) = 2 J_{1} (k a sin θ) / (k a sin θ)$ (Beranek & Mellow Eq. (4.42)) at every ka over a polar-angle grid and bundles it into a PistonDirectivity that exposes .plot(). The main lobe narrows as ka grows; its first null appears once ka passes the first zero of J1 ( $k a sin θ = 3.8317$ ).

Parameters

Name	Description
`ka`	Wavenumber-radius product(s) `ka` (scalar or 1-D array), each non-negative.
`angles`	Polar angles `theta` from the axis, rad (1-D). `None` (default) uses 361 points spanning the front hemisphere `-90 deg` to `+90 deg`, 0.5 deg apart.

Returns: A PistonDirectivity.

piston_reactance

piston_reactance(x: ArrayLike) -> np.ndarray | float

Piston reactance function $X_{1} (x) = 2 H_{1} (x) / x$ (H1 Struve order 1).

The imaginary part of the normalized radiation impedance of a baffled circular piston (Beranek & Mellow Eq. (4.30)). It rises as $(8/3 π) k a$ (mass-like) at low $x = 2 k a$ and decays to 0 at high x.

Parameters

Name	Description
`x`	Argument $x = 2 k a$ (scalar or array), dimensionless.

Returns: X1(x) (float for scalar input, else an array).

piston_resistance

piston_resistance(x: ArrayLike) -> np.ndarray | float

Piston resistance function $R_{1} (x) = 1 - 2 J_{1} (x) / x$ .

The real part of the normalized radiation impedance of a baffled circular piston, as a function of $x = 2 k a$ (Beranek & Mellow Eq. (4.30)). It rises as $x^{2} /8 = (k a)^{2} /2$ at low x and tends to 1 at high x.

Parameters

Name	Description
`x`	Argument $x = 2 k a$ (scalar or array), dimensionless.

Returns: R1(x) (float for scalar input, else an array).

PistonDirectivity

PistonDirectivity(
    angles: np.ndarray,
    ka: np.ndarray,
    directivity: np.ndarray,
    directivity_db: np.ndarray,
)

Far-field directivity pattern of a baffled circular piston.

Bundles the far-field directivity $D (θ) = 2 J_{1} (k a sin θ) / (k a sin θ)$ (Beranek & Mellow Eq. (4.42)) of one or more baffled circular pistons over a shared polar-angle grid, so the classic beam pattern can be drawn with plot. The maths is piston_directivity; this is a thin, plottable bundle around it.

Attributes

Name	Description
`angles`	Polar angles `theta` from the axis, rad.
`ka`	Wavenumber-radius products `ka`, one per pattern (a 1-D array).
`directivity`	Linear directivity $D (θ)$ , normalized so $D (0) = 1$ , as a `(len(ka), len(angles))` array; row `i` is the pattern for `ka[i]`.
`directivity_db`	Directivity in dB, $20 lo g_{10} ∣ D ∣$ , same shape as `directivity` (the side-lobe nulls floor at a large negative value rather than `-inf`).

PistonDirectivity.plot()

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

Plot the far-field directivity (beam) pattern on a polar axes.

Draws the directivity in dB against the polar angle: one curve per ka value as a single family (still one concept, the directivity pattern). A polar axes is created when ax is None. Requires matplotlib (pip install phonometry[plot]).

Parameters

Name	Description
`ax`	Existing (polar) axes, or `None` to create a figure.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the `Axes.plot` call when the result holds a single `ka`; ignored for a multi-`ka` family so one user color or label cannot collapse the per-curve styling.

Returns: The axes.

plot_piston_geometry

plot_piston_geometry(
    radius: float,
    ax: Axes | None = None,
    *,
    angles: ArrayLike | None = None,
    directivity: ArrayLike | None = None,
    lobe_label: str | None = None,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw a baffled piston to scale, optionally with a directivity lobe.

The rigid baffle is the vertical wall, the piston the plate of radius a set into it; when angles/directivity are given the normalised far-field lobe is overlaid on the radiation side.

Parameters

Name	Description
`radius`	Piston radius `a`, in metres.
`ax`	Existing axes, or `None` to create a figure.
`angles`	Far-field angles, in radians (0 on axis), matching `directivity`.
`directivity`	Linear directivity values in `[0, 1]`.
`lobe_label`	Optional legend label for the lobe (e.g. the `ka`).
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the piston rectangle.

Returns: The axes.

radiating_piston

radiating_piston(
    radius: float,
    frequencies: ArrayLike,
    *,
    speed_of_sound: float = 343.0,
    density: float = 1.206,
    angles: ArrayLike | None = None,
) -> RadiatingPistonResult

Radiation impedance and directivity of a rigid baffled circular piston.

Evaluates the piston resistance $R_{1} (2 k a)$ and reactance $X_{1} (2 k a)$ , the mechanical radiation impedance $ρ c S (R_{1} + j X_{1})$ , the low-frequency radiation mass $8 ρ a^{3} /3$ and the directivity index over the given frequencies (Beranek & Mellow §4.4). Pass angles to also sample the far-field directivity pattern $D (θ)$ .

Parameters

Name	Description
`radius`	Piston radius `a`, m.
`frequencies`	Frequencies `f`, Hz (scalar or 1-D array), all > 0.
`speed_of_sound`	Speed of sound `c`, m/s (default 343).
`density`	Air density $ρ$ , kg/m3 (default 1.206).
`angles`	Optional polar angles `theta` from the axis, rad, at which to sample the directivity pattern.

Returns: A RadiatingPistonResult.

RadiatingPistonResult

RadiatingPistonResult(
    frequencies: np.ndarray,
    ka: np.ndarray,
    resistance: np.ndarray,
    reactance: np.ndarray,
    radiation_resistance: np.ndarray,
    radiation_reactance: np.ndarray,
    radiation_mass: float,
    directivity_index: np.ndarray,
    angles: np.ndarray | None,
    directivity: np.ndarray | None,
    radius: float,
    speed_of_sound: float,
    density: float,
)

Radiation impedance and directivity of a baffled circular piston.

Attributes

Name	Description
`frequencies`	Frequencies `f`, Hz.
`ka`	Wavenumber-radius product `ka` at each frequency.
`resistance`	Normalized piston resistance $R_{1} (2 k a)$ (real part of $Z_{r} / (ρ c S)$ ).
`reactance`	Normalized piston reactance $X_{1} (2 k a)$ (imaginary part of $Z_{r} / (ρ c S)$ ).
`radiation_resistance`	Mechanical radiation resistance $ρ c S R_{1}$ , N s/m.
`radiation_reactance`	Mechanical radiation reactance $ρ c S X_{1}$ , N s/m.
`radiation_mass`	Low-frequency accreted air mass $M_{r} = 8 ρ a^{3} /3$ , kg (a single value; the mass limit of `radiation_reactance / omega`).
`directivity_index`	Directivity index $D I = 10 lo g_{10} Q$ , dB.
`angles`	Polar angles of `directivity`, rad, or `None` if not requested.
`directivity`	Far-field directivity $D (θ)$ as a `(n_freq, n_angle)` array, or `None` if `angles` was not given.
`radius`	Piston radius `a`, m.
`speed_of_sound`	Speed of sound `c`, m/s.
`density`	Air density $ρ$ , kg/m3.

RadiatingPistonResult.plot()

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

Plot the normalized piston resistance and reactance against ka.

Reproduces the classic Beranek & Mellow figure: R1 rising to 1 and X1 peaking then decaying, over the ka range of the result. Requires matplotlib (pip install phonometry[plot]).

Parameters

Name	Description
`language`	Label language, `"en"` (default) or `"es"`.

RadiatingPistonResult.plot_geometry()

RadiatingPistonResult.plot_geometry(
    ax: Axes | None = None,
    *,
    frequency_index: int = -1,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Draw the baffled piston to scale, with its directivity lobe.

Requires matplotlib (pip install phonometry[plot]); returns the Axes.

Parameters

Name	Description
`ax`	Existing axes, or `None` to create a figure.
`frequency_index`	Index of the frequency whose far-field lobe is overlaid (default: the highest); ignored when the result carries no directivity.
`language`	Label language, `"en"` (default) or `"es"`.
`kwargs`	Forwarded to the piston rectangle.

Created and maintained by José Manuel Requena Plens ORCID All projects