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 = rho c S ( R1(2ka) + j X1(2ka) ),      S = pi a^2,

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

R1(x) = 1 - 2 J1(x) / x,        X1(x) = 2 H1(x) / x,

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

• Low frequency (ka << 1): R1 -> (ka)^2 / 2 so the radiated power rises as f^2, and X1 -> (8 / 3 pi) ka. The reactance is mass-like, X_r = rho c S X1 = omega M_r with the radiation (accreted) mass

  M_r = 8 rho a^3 / 3

  (Beranek & Mellow Eq. (4.32)): the piston drags an extra 8 rho a^3 / 3 of air, equivalent to a layer 8a / 3 pi thick over its face.

• High frequency (ka >> 1): R1 -> 1 and X1 -> 0, so Z_r -> rho 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(theta) = 2 J1(ka sin theta) / (ka sin theta),      D(0) = 1.

The main lobe narrows as ka grows; its first null is at ka sin theta = 3.8317 (the first zero of J1), which exists only once ka > 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 DI = 10 log10 Q follow from integrating |D|^2 over the radiating hemisphere,

Q = 2 / integral_0^(pi/2) |D(theta)|^2 sin theta d theta,

which tends to Q = 2 (DI = 3.01 dB, the half-space baffle gain) at low ka and to Q ~ (ka)^2 (DI ~ 20 log10 ka) at high ka.

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piston_directivity

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

Far-field directivity D = 2 J1(ka sin theta) / (ka sin theta).

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(theta) = 2 J1(ka sin theta) / (ka sin theta) (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 (ka sin theta = 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 X1(x) = 2 H1(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 pi) ka (mass-like) at low x = 2ka and decays to 0 at high x.

Parameters

Name	Description
x	Argument x = 2ka (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 R1(x) = 1 - 2 J1(x) / x.

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

Parameters

Name	Description
x	Argument x = 2ka (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(theta) = 2 J1(ka sin theta) / (ka sin theta) (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(theta), 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 log10 |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.

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 R1(2ka) and reactance X1(2ka), the mechanical radiation impedance rho c S (R1 + j X1), the low-frequency radiation mass 8 rho 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(theta).

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 rho, 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 R1(2ka) (real part of Z_r / (rho c S)).
reactance	Normalized piston reactance X1(2ka) (imaginary part of Z_r / (rho c S)).
radiation_resistance	Mechanical radiation resistance rho c S R1, N s/m.
radiation_reactance	Mechanical radiation reactance rho c S X1, N s/m.
radiation_mass	Low-frequency accreted air mass M_r = 8 rho a^3/3, kg (a single value; the mass limit of radiation_reactance / omega).
directivity_index	Directivity index DI = 10 log10 Q, dB.
angles	Polar angles of directivity, rad, or None if not requested.
directivity	Far-field directivity D(theta) 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 rho, 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".