<filename>beamshapes/point_source_on_a_sphere.py<gh_stars>1-10
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Point source on a sphere
========================
An infinitesimally oscillating on the surface of a sphere. Useful where the 
source is very small in comparison to the sphere.


References 
----------
Chp 12, secn. 12.5, <NAME>., & <NAME>. (2012). Acoustics: sound fields and transducers.
Academic Press.

"""
import numpy as np 
from sympy import symbols, legendre, lambdify, I, Sum, cos
from beamshapes.special_functions import sph_hankel2
from beamshapes.utilities import dB

k, R, n, theta, NN, kR = symbols('k R n theta NN kR')
costheta = cos(theta)
kR = k*R
# eqn. 12.46

numerator_sumterm = I**(n+1)*(2*n+1)**2*legendre(n, costheta)
denominator_sumterm = n*sph_hankel2.subs({'n':n-1,'z':kR}) - (n+1)*sph_hankel2.subs({'n':n+1,'z':kR})
sum_term = Sum(numerator_sumterm/denominator_sumterm, (n, 0, NN))
d_theta = (1/(kR**2))*sum_term

d_theta_func = lambdify([theta,k,R,NN],d_theta)

docs_for_dtheta = '''On-axis level for point on a sphere. 
                 Lambdified version of :code:`d_theta` SymPy expression.
                 
                 Parameters
                 ----------
                 theta : np.float>0
                    Elevation/azimuth off-axis angle in radians. 
                 k : np.float>0
                    Wavenumber
                 R : np.float>0
                    Radius of sphere 
                 NN : int
                    Number of summation steps that must be calculated.
                 
                 Returns 
                 -------
                 _ : np.float, Off-axis level at :math:`\\theta` for point-source on a sphere.
                 '''
d_theta_func.__doc__ = docs_for_dtheta

# eqn. 12.47 
onaxis_num_sumterm = I**n*(2*n+1)**2
onaxissum_term = Sum(onaxis_num_sumterm/denominator_sumterm, (n,0,NN))
d_zero = (1/(kR**2))*onaxissum_term

d_zero_func = lambdify([k,R,NN],d_zero)

docs_for_dzero = '''
                 On-axis level for point on a sphere. 
                 Lambdified version of :code:`d_zero` SymPy expression.
                 
                 Parameters
                 ----------
                 k : np.float>0
                    Wavenumber
                 R : np.float>0
                    Radius of sphere 
                 NN : int 
                    Number of summation steps that must be calculated.
                 
                 Returns 
                 -------
                 _ : np.float,  On-axis level for point-source on a sphere.

                 '''

d_zero_func.__doc__ = docs_for_dzero

def point_source_on_a_sphere_directivity(angles, param, parallel=False):
    '''Directivity for point source on a sphere.
        
    Parameters
    ----------
        angles : list/np.array-like
            The angles for which the directivity is to be calculated. 
            All angles in radians.

        param: dictionary 
            With the following keys and entries:
                k : np.float > 0 
                    Wavenumber
                R : np.float > 0
                    Radius of sphere
    
    Returns 
    -------
    _ : None
    dtheta_by_d0 : np.array
        20log10(D_theta/D_0)
             
    '''
    kv, Rv = [param[each] for each in ['k','R']]
    NNv = param.get('NN', int(10+2*kv*Rv)) # trial formula - need to see if it matches w textbook
    
    onaxis_value = abs(d_zero_func(kv,Rv,NNv))
    offaxis_values = []
    for angle in angles:
        offaxis_values.append(abs(d_theta_func(angle, kv, Rv, NNv)))
    dtheta_by_d0 = dB(np.abs(np.array(offaxis_values))/np.abs(onaxis_value))
    return None , dtheta_by_d0

<filename>beamshapes/point_source_on_a_sphere.py<gh_stars>1-10
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Point source on a sphere
========================
An infinitesimally oscillating on the surface of a sphere. Useful where the 
source is very small in comparison to the sphere.


References 
----------
Chp 12, secn. 12.5, <NAME>., & <NAME>. (2012). Acoustics: sound fields and transducers.
Academic Press.

"""
import numpy as np 
from sympy import symbols, legendre, lambdify, I, Sum, cos
from beamshapes.special_functions import sph_hankel2
from beamshapes.utilities import dB

k, R, n, theta, NN, kR = symbols('k R n theta NN kR')
costheta = cos(theta)
kR = k*R
# eqn. 12.46

numerator_sumterm = I**(n+1)*(2*n+1)**2*legendre(n, costheta)
denominator_sumterm = n*sph_hankel2.subs({'n':n-1,'z':kR}) - (n+1)*sph_hankel2.subs({'n':n+1,'z':kR})
sum_term = Sum(numerator_sumterm/denominator_sumterm, (n, 0, NN))
d_theta = (1/(kR**2))*sum_term

d_theta_func = lambdify([theta,k,R,NN],d_theta)

docs_for_dtheta = '''On-axis level for point on a sphere. 
                 Lambdified version of :code:`d_theta` SymPy expression.
                 
                 Parameters
                 ----------
                 theta : np.float>0
                    Elevation/azimuth off-axis angle in radians. 
                 k : np.float>0
                    Wavenumber
                 R : np.float>0
                    Radius of sphere 
                 NN : int
                    Number of summation steps that must be calculated.
                 
                 Returns 
                 -------
                 _ : np.float, Off-axis level at :math:`\\theta` for point-source on a sphere.
                 '''
d_theta_func.__doc__ = docs_for_dtheta

# eqn. 12.47 
onaxis_num_sumterm = I**n*(2*n+1)**2
onaxissum_term = Sum(onaxis_num_sumterm/denominator_sumterm, (n,0,NN))
d_zero = (1/(kR**2))*onaxissum_term

d_zero_func = lambdify([k,R,NN],d_zero)

docs_for_dzero = '''
                 On-axis level for point on a sphere. 
                 Lambdified version of :code:`d_zero` SymPy expression.
                 
                 Parameters
                 ----------
                 k : np.float>0
                    Wavenumber
                 R : np.float>0
                    Radius of sphere 
                 NN : int 
                    Number of summation steps that must be calculated.
                 
                 Returns 
                 -------
                 _ : np.float,  On-axis level for point-source on a sphere.

                 '''

d_zero_func.__doc__ = docs_for_dzero

def point_source_on_a_sphere_directivity(angles, param, parallel=False):
    '''Directivity for point source on a sphere.
        
    Parameters
    ----------
        angles : list/np.array-like
            The angles for which the directivity is to be calculated. 
            All angles in radians.

        param: dictionary 
            With the following keys and entries:
                k : np.float > 0 
                    Wavenumber
                R : np.float > 0
                    Radius of sphere
    
    Returns 
    -------
    _ : None
    dtheta_by_d0 : np.array
        20log10(D_theta/D_0)
             
    '''
    kv, Rv = [param[each] for each in ['k','R']]
    NNv = param.get('NN', int(10+2*kv*Rv)) # trial formula - need to see if it matches w textbook
    
    onaxis_value = abs(d_zero_func(kv,Rv,NNv))
    offaxis_values = []
    for angle in angles:
        offaxis_values.append(abs(d_theta_func(angle, kv, Rv, NNv)))
    dtheta_by_d0 = dB(np.abs(np.array(offaxis_values))/np.abs(onaxis_value))
    return None , dtheta_by_d0

#!/usr/bin/env python3 # -*- coding: utf-8 -*- Point source on a sphere ======================== An infinitesimally oscillating on the surface of a sphere. Useful where the  source is very small in comparison to the sphere.   References  ---------- Chp 12, secn. 12.5, <NAME>., & <NAME>. (2012). Acoustics: sound fields and transducers. Academic Press. # eqn. 12.46 On-axis level for point on a sphere.                   Lambdified version of :code:`d_theta` SymPy expression.                                    Parameters                  ----------                  theta : np.float>0                     Elevation/azimuth off-axis angle in radians.                   k : np.float>0                     Wavenumber                  R : np.float>0                     Radius of sphere                   NN : int                     Number of summation steps that must be calculated.                                    Returns                   -------                  _ : np.float, Off-axis level at :math:`\\theta` for point-source on a sphere. # eqn. 12.47 On-axis level for point on a sphere.                   Lambdified version of :code:`d_zero` SymPy expression.                                    Parameters                  ----------                  k : np.float>0                     Wavenumber                  R : np.float>0                     Radius of sphere                   NN : int                      Number of summation steps that must be calculated.                                    Returns                   -------                  _ : np.float,  On-axis level for point-source on a sphere. Directivity for point source on a sphere.              Parameters     ----------         angles : list/np.array-like             The angles for which the directivity is to be calculated.              All angles in radians.          param: dictionary              With the following keys and entries:                 k : np.float > 0                      Wavenumber                 R : np.float > 0                     Radius of sphere          Returns      -------     _ : None     dtheta_by_d0 : np.array         20log10(D_theta/D_0) # trial formula - need to see if it matches w textbook