ARF#

Examples using this class are:

Frontiers: Copper Particle in Viscous Oil

Doinikov Rigid (1994): Sandstone in Glycerin

Doinikov 1994 Models

Possible Numerical Problems

class osaft.solutions.doinikov1994rigid.arf.ARF(f, R_0, rho_s, rho_f, c_f, eta_f, zeta_f, p_0, wave_type, position=None, long_wavelength=False, small_boundary_layer=False, large_boundary_layer=False, fastened_sphere=False, background_streaming=True, N_max=5)[source]#

Bases: ARFArbitraryRadius

ARF according to Doinikov’s theory (viscous fluid-rigid sphere; 1994)

There is a large number of options for the computation of the ARF in different limiting cases. For a more detailed explanation see this example.

Parameters:

f (Frequency | float | int) – Frequency [Hz]
R_0 (Sphere | float | int) – Radius of the sphere [m]
rho_s (float) – Density of the sphere [kg/m^3]
rho_f (float) – Density of the fluid [kg/m^3]
c_f (float) – Speed of sound of the fluid [m/s]
eta_f (float) – shear viscosity [Pa s]
zeta_f (float) – bulk viscosity [Pa s]
p_0 (float) – Pressure amplitude of the field [Pa]
wave_type (WaveType) – Type of wave, traveling or standing
position (None | float, optional) – Position within the standing wave field [m]
Default: None
long_wavelength (bool, optional) – using \(x \ll 1\)
Default: False
small_boundary_layer (bool, optional) – \(x \ll x_v \ll 1\)
Default: False
large_boundary_layer (bool, optional) – :math`x ll 1 ll x_v`
Default: False
fastened_sphere (bool, optional) – use theory of fastened sphere
Default: False
background_streaming (bool, optional) – background streaming contribution
Default: True
N_max (int, optional) – Highest order mode
Default: 5

Public Data Attributes:

Inherited from ARFArbitraryBL

`D_0`	Approximation for the coefficient D_0 from Eq (6.1)
`D_1`	Approximation for the coefficient D_1 from Eq (6.2)
`G_0`	Approximation to value G_0 from Eq (6.3)
`G_1`	Approximation to value G_1 from Eq (6.4)
`G_2`	Approximation to value G_2 from Eq (6.5)
`G_3`	Approximation to value G_3 from Eq (6.6)
`G_4`	Approximation to value G_4 from Eq (6.7)

Inherited from ARFLimiting

`long_wavelength`	Use limiting case for ARF calculation
`fastened_sphere`	Use limiting case of fastened sphere

Inherited from ScatteringField

`mu_1`	\(\mu_1\) according to (3.16)
`mu_2`	\(\mu_2\) according to (3.17)
`mu_3`	\(\mu_3\) according to (3.18)
`mu_4`	\(\mu_4\) according to (3.19)

Inherited from BaseDoinikov1994Rigid

`supported_wavetypes`
`rho_s`	Wraps to `osaft.core.solids.RigidSolid.rho_s`

Inherited from BaseDoinikov1994

`x`	Product of `k_f` and `R_0` \(\hat{x}=k_f R_0\)
`x_v`	Product of `k_v` and `R_0` \(x_v=k_v R_0\)
`x_0`	Real part of \(x\)
`norm_delta`	normalized viscous boundary thickness \(\tilde{\delta}=\frac{\delta}{R_0}\)
`position`	Wraps to `osaft.core.backgroundfields.BackgroundField.position`
`p_0`	Wraps to `osaft.core.backgroundfields.BackgroundField.p_0`
`wave_type`	Wraps to `osaft.core.backgroundfields.BackgroundField.wave_type`
`rho_f`	Wraps to `osaft.core.fluids.ViscousFluid.rho_f`
`c_f`	Wraps to `osaft.core.fluids.ViscousFluid.c_f`
`eta_f`	Wraps to `osaft.core.fluids.ViscousFluid.eta_f`
`zeta_f`	Wraps to `osaft.core.fluids.ViscousFluid.zeta_f`
`rho_t`	Returns the ratio of the densities \(\tilde{\rho}=\frac{\rho_f}{\rho_s}\)
`kappa_f`	Wraps to `osaft.core.fluids.ViscousFluid.kappa_f`
`k_f`	Wraps to `osaft.core.fluids.ViscousFluid.k_f`
`k_v`	Wraps to `osaft.core.fluids.ViscousFluid.k_v`
`delta`	Wraps to `osaft.core.fluids.ViscousFluid.delta`
`abs_pos`	Wraps to `osaft.core.backgroundfields.BackgroundField.abs_pos`

Inherited from BaseSphereFrequencyComposite

`R_0`	Wrapper for `osaft.core.geometries.Sphere.R_0`
`area`	Wrapper for `osaft.core.geometries.Sphere.area`
`volume`	Wrapper for `osaft.core.geometries.Sphere.volume`

Inherited from BaseFrequencyComposite

`f`	wrapper for `osaft.core.frequency.Frequency.f`
`omega`	wrapper for `osaft.core.frequency.Frequency.omega`

Inherited from BaseSolution

`supported_wavetypes`
`wave_type`	returns the wave type of the solution

Inherited from BaseScatteringRigidParticle

R_0

Inherited from BaseScatteringDoinikov1994

`k_f`	Returns the wave number in the fluid \(k_f\) [1/m]
`k_v`	Returns the viscous wave number in the fluid \(k_v\) [1/m]

Inherited from BaseScattering

`N_max`	Cutoff mode number for infinite sums
`field`
`omega`
`R_0`
`rho_f`
`k_f`

Inherited from BaseARFLimiting

`small_boundary_layer`	Use limiting case of a small viscous boundary layer \(\delta\)
`large_boundary_layer`	Use limiting case of a large viscous boundary layer \(\delta\)
`background_streaming`	Background streaming contribution to the ARF
`field`
`fluid`
`sphere`
`abs_pos`
`delta`
`k_f`
`R_0`
`rho_f`
`x`
`x_v`

Inherited from BaseGeneralARF

`field`
`abs_pos`
`background_streaming`
`k_f`
`N_max`
`rho_f`
`wave_type`
`x`
`x_v`

Public Methods:

compute_arf()

Acoustic radiation fore in [N]

Inherited from ARFArbitraryBL

`D_n_limit`(n)	Approximation for the coefficient D_0, D_1 from Eq (6.1), (6.2)
`S_9n_contribution`()	S_9n contribution to the background streaming

Inherited from ScatteringField

`xi_n`(n)	coefficient \(\xi_n\) (3.22)
`gamma_n`(n)	coefficient \(\gamma_n\) (3.22)
`alpha_n`(n)	coefficient \(\alpha_n\) (3.13) and (3.20)
`beta_n`(n)	coefficient \(\beta_n\) (3.13) and (3.20)
`particle_velocity`(t)	Particle velocity

Inherited from BaseDoinikov1994

A_in(n)

Wraps to osaft.core.backgroundfields.BackgroundField.A_in

Inherited from BaseFrequencyComposite

input_variables()

Returns all properties that are settable.

Inherited from BaseSolution

`copy`()	Returns a copy of the object
`check_wave_type`()	Checks if `wave_type` is in `supported_wavetypes`

Inherited from BaseScatteringRigidParticle

`particle_velocity`(t)	Particle velocity
`radial_particle_velocity`(r, theta, t[, mode])	Particle velocity in radial direction
`tangential_particle_velocity`(r, theta, t[, mode])	Particle velocity in tangential direction

Inherited from BaseScatteringDoinikov1994

`potential_coefficient`(n)	Wrapper to the fluid scattering coefficients for an inviscid fluid
`V_r_sc`(n, r)	Radial scattering field velocity term of mode n without Legendre coefficients
`V_theta_sc`(n, r)	Tangential scattering field velocity term of mode n without Legendre coefficients
`A_in`(n)	Incoming wave amplitude
`alpha_n`(n)	\(\alpha_n\) coefficient
`beta_n`(n)	\(\beta_n\) coefficient

Inherited from BaseScattering

`radial_acoustic_fluid_velocity`(r, theta, t, ...)	Returns the value for the radial acoustic velocity in [m/s].
`tangential_acoustic_fluid_velocity`(r, theta, ...)	Returns the value for the tangential acoustic velocity in [m/s].
`pressure`(r, theta, t, scattered, incident[, ...])	Returns the acoustic pressure [Pa].
`potential_coefficient`(n)	Wrapper to the fluid scattering coefficients for an inviscid fluid
`velocity_potential`(r, theta, t, scattered, ...)	Returns the velocity potential of the fluid in [m^2/s].
`radial_particle_velocity`(r, theta, t[, mode])	Returns the value for the radial particle velocity in [m/s].
`tangential_particle_velocity`(r, theta, t[, mode])	Returns the value for the tangential particle velocity in [m/s].
`radial_particle_displacement`(r, theta, t[, mode])	Particle displacement in radial direction
`tangential_particle_displacement`(r, theta, t)	Particle displacement in tangential direction
`radial_mode_superposition`(radial_func, r, ...)	Returns either a single mode (`mode=int`) or a the sum until `N_max` (`mode=None`).
`tangential_mode_superposition`(...)	Returns either a single mode (`mode=int`) or a the sum until `N_max` (`mode=None`).
`V_r_i`(n, r)	Radial incident field velocity term of mode n without Legendre coefficients
`V_theta_i`(n, r)	Tangential incident field velocity term of mode n without Legendre coefficients
`V_r_sc`(n, r)	Radial scattering field velocity term of mode n without Legendre coefficients
`V_theta_sc`(n, r)	Tangential scattering field velocity term of mode n without Legendre coefficients
`V_r`(n, r, scattered, incident)	Superposition of `V_r_sc()` and `V_r_i()` depending on `scattered` and `incident`
`V_theta`(n, r, scattered, incident)	Superposition of `V_theta_sc()` and `V_theta_i()` depending on `scattered` and `incident`

Inherited from BaseARF

compute_arf()

Returns the value for the ARF in Newton [N].

Inherited from BaseGeneralARF

`alpha_n`(n)	param n:
`beta_n`(n)	param n:
`D_n`(n)	coefficient \(D_{n}\)
`G_n_l`(n, l, x)	Coefficient \(G_{n}^{(l)}(x)\) from the appendix
`G_sum_1`(n)	Numerically stable evaluation of \((G_{n}^{(1)}(x) + G_{n}^{(2)}(x)) / 2\)
`G_sum_2`(n)	Numerically stable evaluation of \((G_{n}^{(1)}(x^) + G_{n}^{(2)}(x)) / 2\)
`K_n_l`(n, l)	Coefficient \(K_{n}^{(l)}(x)\) from the appendix
`K_sum`(n)	Numerically stable evaluation of \((K_{n}^{(1)}(x) + K_{n}^{(2)}(x)) / 2\)
`L_n_l`(n, l)	Numerically stable evaluation of \((K_{n}^{(1)}(x) + K_{n}^{(2)}(x)) / 2\)
`L_sum`(n)	Numerically stable evaluation of \((L_{n}^{(1)}(x) + L_{n}^{(2)}(x)) / 2\)
`B_kq`(k, q, m, n)	Coefficient \(B_{kq}\) from the appendix
`J_nm_j`(x1, x2, n, m, j)	Integral \(J_{nm}^{(j)}\)
`H_nm_j`(x1, x2, n, m, j)	Integral \(H_{nm}^{(j)}\)
`H_nm_j_sum`(x1, x2, n, m, j)	Coefficient \(H_{nm}^{(j)}\)
`S_1n`(n)	coefficient \(S_{1n}\)
`S_2n`(n)	coefficient \(S_{2n}\)
`S_3n`(n)	coefficient \(S_{3n}\)
`S_4n`(n)	coefficient \(S_{4n}\)
`S_5n`(n)	coefficient \(S_{5n}\)
`S_6n`(n)	coefficient \(S_{6n}\)
`S_7n`(n)	coefficient \(S_{7n}\)
`S_8n`(n)	coefficient \(S_{8n}\)
`S_9n`(n)	coefficient \(S_{9n}\)

A_in(n)#

Wraps to osaft.core.backgroundfields.BackgroundField.A_in

Parameters:: n (int) – mode number
Return type:: complex

static B_kq(k, q, m, n)#

Coefficient \(B_{kq}\) from the appendix

Parameters:

k (int) – \(k\)
q (int) – \(q\)
m (int) – \(m\)
n (int) – \(n\)

Return type:

complex

D_n(n)#

coefficient \(D_{n}\)

Parameters:: n (int) – order
Return type:: complex

D_n_limit(n)#

Approximation for the coefficient D_0, D_1 from Eq (6.1), (6.2)

Parameters:: n (int) –
Return type:: complex

static G_n_l(n, l, x)#

Coefficient \(G_{n}^{(l)}(x)\) from the appendix

Parameters:

n (int) – order
l (int) – kind of Hankel function
x (complex) –

Return type:

complex

G_sum_1(n)#

Numerically stable evaluation of \((G_{n}^{(1)}(x) + G_{n}^{(2)}(x)) / 2\)

Parameters:: n – order

G_sum_2(n)#

Numerically stable evaluation of \((G_{n}^{(1)*}(x^*) + G_{n}^{(2)}(x)) / 2\)

Parameters:: n – order

H_nm_j(x1, x2, n, m, j)#

Integral \(H_{nm}^{(j)}\)

Parameters:

x1 (complex) – \(x_1\)
x2 (complex) – \(x_2\)
n (int) – \(n\)
m (int) – \(m\)
j (int) – \(j\)

Return type:

complex

H_nm_j_sum(x1, x2, n, m, j)#

Coefficient \(H_{nm}^{(j)}\)

Parameters:

x1 (complex) – argument x1
x2 (complex) – argument x2
n (int) – order n
m (int) – order m
j (int) – exponent

Return type:

complex

J_nm_j(x1, x2, n, m, j)#

Integral \(J_{nm}^{(j)}\)

Parameters:

x1 (complex) – \(x_1\)
x2 (complex) – \(x_2\)
n (int) – \(n\)
m (int) – \(m\)
j (int) – \(j\)

Return type:

complex

K_n_l(n, l)#

Coefficient \(K_{n}^{(l)}(x)\) from the appendix

Parameters:

n (int) – order
l (int) – kind of Hankel function

Return type:

complex

K_sum(n)#

Numerically stable evaluation of \((K_{n}^{(1)}(x) + K_{n}^{(2)}(x)) / 2\)

Parameters:: n (int) – order

L_n_l(n, l)#

Numerically stable evaluation of \((K_{n}^{(1)}(x) + K_{n}^{(2)}(x)) / 2\)

Parameters:

n (int) – order
l (int) – kind of Hankel function

Return type:

complex

L_sum(n)#

Numerically stable evaluation of \((L_{n}^{(1)}(x) + L_{n}^{(2)}(x)) / 2\)

Parameters:: n (int) – order
Return type:: complex

S_1n(n)#

coefficient \(S_{1n}\)

Parameters:: n (int) – order
Return type:: complex

S_2n(n)#

coefficient \(S_{2n}\)

Parameters:: n (int) – order
Return type:: complex

S_3n(n)#

coefficient \(S_{3n}\)

Parameters:: n (int) – order
Return type:: complex

S_4n(n)#

coefficient \(S_{4n}\)

Parameters:: n (int) – order
Return type:: complex

S_5n(n)#

coefficient \(S_{5n}\)

Parameters:: n (int) – order
Return type:: complex

S_6n(n)#

coefficient \(S_{6n}\)

Parameters:: n (int) – order
Return type:: complex

S_7n(n)#

coefficient \(S_{7n}\)

Parameters:: n (int) – order
Return type:: complex

S_8n(n)#

coefficient \(S_{8n}\)

Parameters:: n (int) – order
Return type:: complex

S_9n(n)#

coefficient \(S_{9n}\)

Parameters:: n (int) – order
Return type:: complex

S_9n_contribution()#

S_9n contribution to the background streaming

Return type:: complex

V_r(n, r, scattered, incident)#

Superposition of V_r_sc() and V_r_i() depending on scattered and incident

At least one of the two must be True.

Parameters:

n (int) – mode
r (float) – radial coordinate [m]
scattered (bool) – add scattered field
incident (bool) – add incident

Return type:

complex

V_r_i(n, r)#

Radial incident field velocity term of mode n without Legendre coefficients

Returns radial incident field velocity in [m/s]

Parameters:

n (int) – mode
r (float | Sequence) – radial coordinate [m]

Return type:

complex

V_r_sc(n, r)#

Radial scattering field velocity term of mode n without Legendre coefficients

Returns radial scattering field velocity in [m/s]

Parameters:

n (int) – mode
r (float) – radial coordinate [m]

Return type:

complex

V_theta(n, r, scattered, incident)#

Superposition of V_theta_sc() and V_theta_i() depending on scattered and incident

At least one of the two must be True.

Parameters:

n (int) – mode
r (float) – radial coordinate [m]
scattered (bool) – add scattered field
incident (bool) – add incident

Return type:

complex

V_theta_i(n, r)#

Tangential incident field velocity term of mode n without Legendre coefficients

Returns tangential incident field velocity in [m/s]

Parameters:

n (int) – mode
r (float | Sequence) – radial coordinate [m]

Return type:

complex

V_theta_sc(n, r)#

Tangential scattering field velocity term of mode n without Legendre coefficients

Returns tangential scattering field velocity in [m/s]

Parameters:

n (int) – mode
r (float) – radial coordinate [m]

Return type:

complex

alpha_n(n)#

coefficient \(\alpha_n\) (3.13) and (3.20)

Parameters:: n (int) – order
Return type:: complex

beta_n(n)#

coefficient \(\beta_n\) (3.13) and (3.20)

Parameters:: n (int) – order
Return type:: complex

check_wave_type()#

Checks if wave_type is in supported_wavetypes

Raises:: WrongWaveTypeError – If wave_type is not supported
Return type:: None

compute_arf()[source]#

Acoustic radiation fore in [N]

It logs the current values of x and x_v.

Raises:

WrongWaveTypeError – if wrong wave_type
AssumptionWarning – if used solution might not be valid

Return type:

float

copy()#

Returns a copy of the object

Return type:: BaseSolution

gamma_n(n)#

coefficient \(\gamma_n\) (3.22)

Parameters:: n (int) – order
Return type:: complex

classmethod input_variables()#

Returns all properties that are settable.

Returns a list of the names of all properties that are settable, i.e. all properties that wrap a PassiveVariable.

Return type:: list[str]

particle_velocity(t)#

Particle velocity

Returns the value of the particle velocity in the direction of the axis of rotational symmetry of the radiation field in [m/s]

Parameters:: t (float) – time [s]
Return type:: complex

potential_coefficient(n)#

Wrapper to the fluid scattering coefficients for an inviscid fluid

This method must be implemented by every theory to have a common interface for other modules.

Parameters:: n (int) – mode
Return type:: complex

pressure(r, theta, t, scattered, incident, mode=None)#

Returns the acoustic pressure [Pa].

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
scattered (bool) – add scattered field
incident (bool) – add incident
mode (None | int, optional) – specific mode number of interest; if None then all modes until N_max
Default: None

Return type:

complex

radial_acoustic_fluid_velocity(r, theta, t, scattered, incident, mode=None)#

Returns the value for the radial acoustic velocity in [m/s].

This method must be implemented by every theory to have a common interface for other modules.

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
scattered (bool) – scattered field contribution
incident (bool) – incident field contribution
mode (None | int, optional) – specific mode number of interest; if None then all modes until N_max
Default: None

Return type:

complex | NDArray

radial_mode_superposition(radial_func, r, theta, t, mode=None)#

Returns either a single mode (mode=int) or a the sum until N_max (mode=None).

If mode=int the formula is

\[e^{-i\omega t}\, f_{\text{mode}}(r) \,P_{\text{mode}}(\cos\theta)\]

If mode=None the formula is

\[e^{-i\omega t} \sum_{n=0}^{\text{N}_{\text{max}}} \,f_n(r) \,P_n(\cos\theta)\]

where \(f_\text{n}(r)\) is the radial_func(n, r) passed to the method.

Parameters:

radial_func (Callable[[int, float], complex]) – radial function dependent on r
r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (int, optional) – specific mode number of interest; if None then all modes until N_max
Default: None

Return type:

complex | NDArray

radial_particle_displacement(r, theta, t, mode=None)#

Particle displacement in radial direction

Returns the value of the particle displacement in radial direction in [m]

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (None | int, optional) – specific mode number of interest; if None that all modes until N_max
Default: None

Return type:

complex | NDArray

radial_particle_velocity(r, theta, t, mode=None)#

Particle velocity in radial direction

Returns the value of the particle velocity in radial direction in [m/s]

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (None | int, optional) – specific mode number of interest; if None that all modes until N_max
Default: None

Return type:

complex | NDArray

tangential_acoustic_fluid_velocity(r, theta, t, scattered, incident, mode=None)#

Returns the value for the tangential acoustic velocity in [m/s].

This method must be implemented by every theory to have a common interface for other modules.

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
scattered (bool) – scattered field contribution
incident (bool) – incident field contribution
mode (None | int, optional) – specific mode number of interest; if None then all modes until N_max
Default: None

Return type:

complex | NDArray

tangential_mode_superposition(tangential_func, r, theta, t, mode)#

Returns either a single mode (mode=int) or a the sum until N_max (mode=None).

If mode=int the formula is

\[e^{-i\omega t}\, f_\text{mode}(r) \,P^1_{\text{mode}}(\cos\theta)\]

If mode=None the formula is

\[e^{-i\omega t}\sum_{n=0}^{\text{N}_{\text{max}}} \,f_n(r) \,P^1_n(\cos\theta)\]

where \(f_n(r)\) is the tangential_func(n, r) passed to the method.

Parameters:

tangential_func (Callable[[int, float], complex]) – tangential function dependent on r
r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (int) – specific mode number of interest; if None then all modes until N_max

Return type:

complex | NDArray

tangential_particle_displacement(r, theta, t, mode=None)#

Particle displacement in tangential direction

Returns the value of the particle displacement in tangential direction in [m]

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (None | int, optional) – specific mode number of interest; if None that all modes until N_max
Default: None

Return type:

complex | NDArray

tangential_particle_velocity(r, theta, t, mode=None)#

Particle velocity in tangential direction

Returns the value of the particle velocity in tangential direction in [m/s]

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
mode (None | int, optional) – specific mode number of interest; if None that all modes until N_max
Default: None

Return type:

complex | NDArray

velocity_potential(r, theta, t, scattered, incident, mode=None)#

Returns the velocity potential of the fluid in [m^2/s].

Parameters:

r (float | Sequence) – radial coordinate [m]
theta (float | Sequence) – tangential coordinate [rad]
t (float | Sequence) – time [s]
scattered (bool) – add scattered field
incident (bool) – add incident
mode (None | int, optional) – specific mode number of interest; if None then all modes until N_max
Default: None

Return type: