Transmission-reflection coefficients#
Fluid-solid interface#
Fluid to solid:
arim.ut.fluid_solid()Solid to fluid:
arim.ut.solid_l_fluid()andarim.ut.solid_t_fluid()
References: [KK] and [Schmerr].
Example:
1"""
2Compute and plot the transmission and reflection coefficients.
3
4"""
5
6import warnings
7
8import matplotlib.pyplot as plt
9import numpy as np
10
11from arim.model import snell_angles, solid_l_fluid
12
13# %% Parameters
14
15SAVEFIG = False
16
17# water:
18c_fluid = 1480.0
19rho_fluid = 1000.0
20
21# aluminium :
22c_l = 6320.0
23c_t = 3130.0
24rho_solid = 2700.0
25
26# %% Computation of reflection and transmission coefficients
27
28# Remark: by using complex angles, complex reflection and transmission coefficients
29# are computed.
30alpha_l = np.asarray(np.linspace(0, np.pi / 2, 50000), dtype=float)
31
32alpha_fluid = snell_angles(alpha_l, c_l, c_fluid)
33alpha_t = snell_angles(alpha_l, c_l, c_t)
34
35reflection_l, reflection_t, transmission = solid_l_fluid(
36 alpha_l, rho_fluid, rho_solid, c_fluid, c_l, c_t, alpha_fluid, alpha_t
37)
38
39alpha_fluid_deg = np.rad2deg(alpha_fluid.real)
40alpha_l_deg = np.rad2deg(alpha_l.real)
41alpha_t_deg = np.rad2deg(alpha_t.real)
42
43phase = lambda x: np.angle(x, deg=True)
44
45# %% Plot reflection and transmission coefficients (alpha_fluid axis)
46
47fig, axes = plt.subplots(nrows=2, sharex=True)
48ax = axes[0]
49ax.plot(alpha_l_deg, np.abs(transmission), label="transmission")
50ax.plot(alpha_l_deg, np.abs(reflection_l), label="reflection L")
51ax.plot(alpha_l_deg, np.abs(reflection_t), label="reflection T")
52ax.set_title("Tranmission and reflection coefficients from solid L to liquid")
53ax.set_ylabel("absolute value")
54ax.legend(loc="best")
55
56ax = axes[1]
57ax.plot(alpha_l_deg, phase(transmission), label="transmission", linewidth=2)
58ax.plot(alpha_l_deg, phase(reflection_l), label="reflection L")
59ax.plot(alpha_l_deg, phase(reflection_t), label="reflection T")
60
61ax.set_ylabel("phase (deg)")
62ax.set_xlabel("angle of the incident L wave in solid (deg)")
63
64if SAVEFIG:
65 fig.savefig("solid_l_to_liquid_alpha_fluid")
66
67# %% Computation of the repartition of energy
68
69# incident pressure (no effect because of normalisation, we keep it for clarity of the formulas)
70pres_i = 1.0
71
72# cross section areas:
73area_fluid = np.cos(alpha_fluid).real
74area_l = np.cos(alpha_l).real
75area_t = np.cos(alpha_t).real
76
77# Compute the energy incoming and outcoming: the principle of conservation of energy must be respected.
78# Reference: Schmerr, §6.3.2
79
80# Incoming energy
81inc_energy = 0.5 * pres_i**2 / (rho_solid * c_l) * area_l
82
83# Outgoing energy
84energy_trans = (
85 0.5 * (np.abs(transmission) * pres_i) ** 2 / (rho_fluid * c_fluid) * area_fluid
86)
87energy_refl_l = 0.5 * (np.abs(reflection_l) * pres_i) ** 2 / (rho_solid * c_l) * area_l
88energy_refl_t = 0.5 * (np.abs(reflection_t) * pres_i) ** 2 / (rho_solid * c_t) * area_t
89out_energy = energy_trans + energy_refl_l + energy_refl_t
90
91ratio_trans = energy_trans / inc_energy
92ratio_refl_l = energy_refl_l / inc_energy
93ratio_refl_t = energy_refl_t / inc_energy
94
95# Verify the conservation of energy:
96if not (np.allclose(ratio_trans + ratio_refl_l + ratio_refl_t, 1.0)):
97 warnings.warn("The conservation of energy is not respected.")
98
99# %% Plot energy
100
101fig, ax = plt.subplots()
102ax.plot(alpha_l_deg, ratio_trans, label="transmission")
103ax.plot(alpha_l_deg, ratio_refl_l, label="reflection L")
104ax.plot(alpha_l_deg, ratio_refl_t, label="reflection T")
105ax.plot(alpha_l_deg, ratio_trans + ratio_refl_l + ratio_refl_t, "--", label="total")
106ax.set_title("Repartition of energy: solid to liquid interface")
107ax.set_xlabel("angle of the incident wave in liquid (deg)")
108ax.set_ylabel("normalised energy (1)")
109ax.set_xlabel("angle of the incident L wave in solid (deg)")
110ax.set_ylim([0, 1.05])
111ax.legend(loc="best")
112if SAVEFIG:
113 fig.savefig("solid_l_to_liquid_energy")
1"""
2Compute and plot the transmission and reflection coefficients.
3
4Case: an incident longitudinal in the fluid hits the wall of an solid part.
5"""
6
7import warnings
8
9import matplotlib.pyplot as plt
10import numpy as np
11
12from arim.model import fluid_solid, snell_angles
13
14# %% Parameters
15
16# Transmission coefficients above the critical angles corresponds to
17# inhomogeneous surface waves instead of bulk waves
18# Set to True to force the coefficients to zero.
19NULL_TRANSMISSION_ABOVE_CRIT_ANGLES = False
20SAVEFIG = False
21
22# water:
23c_fluid = 1480.0
24rho_fluid = 1000.0
25
26# aluminium :
27c_l = 6320.0
28c_t = 3130.0
29rho_solid = 2700.0
30
31# %% Computation of reflection and transmission coefficients
32
33# Critical angles
34critical_l = np.arcsin(c_fluid / c_l)
35critical_t = np.arcsin(c_fluid / c_t)
36print(f"Critical angle L: {np.rad2deg(critical_l):.3f}°")
37print(f"Critical angle T: {np.rad2deg(critical_t):.3f}°")
38
39# Remark: by using complex angles, complex reflection and transmission coefficients
40# are computed.
41alpha_fluid = np.asarray(np.linspace(0, np.pi / 2, 50000), dtype=complex)
42
43alpha_l = snell_angles(alpha_fluid, c_fluid, c_l)
44alpha_t = snell_angles(alpha_fluid, c_fluid, c_t)
45
46reflection, transmission_l, transmission_t = fluid_solid(
47 alpha_fluid, rho_fluid, rho_solid, c_fluid, c_l, c_t, alpha_l, alpha_t
48)
49
50alpha_fluid_deg = np.rad2deg(alpha_fluid.real)
51alpha_l_deg = np.rad2deg(alpha_l.real)
52alpha_t_deg = np.rad2deg(alpha_t.real)
53
54if NULL_TRANSMISSION_ABOVE_CRIT_ANGLES:
55 transmission_l[alpha_fluid > critical_l] = 0
56 transmission_t[alpha_fluid > critical_t] = 0
57
58# %% Plot reflection and transmission coefficients (alpha_fluid axis)
59
60fig, axes = plt.subplots(nrows=2, sharex=True)
61ax = axes[0]
62ax.plot(alpha_fluid_deg, np.abs(reflection), label="reflection")
63ax.plot(alpha_fluid_deg, np.abs(transmission_l), label="transmission L")
64ax.plot(alpha_fluid_deg, np.abs(transmission_t), label="transmission T")
65ax.set_title("Tranmission and reflection coefficients from liquid to solid")
66ax.axvline(
67 x=np.rad2deg(critical_l), color="k", linestyle="--", label="critical angle L"
68)
69ax.axvline(x=np.rad2deg(critical_t), color="k", linestyle="-", label="critical angle T")
70ax.set_ylabel("absolute value")
71ax.legend(loc="best")
72
73ax = axes[1]
74ax.plot(alpha_fluid_deg, np.angle(reflection, True), label="reflection")
75ax.plot(alpha_fluid_deg, np.angle(transmission_l, True), label="transmission L")
76ax.plot(alpha_fluid_deg, np.angle(transmission_t, True), label="transmission T")
77ax.axvline(
78 x=np.rad2deg(critical_l), color="k", linestyle="--", label="critical angle L"
79)
80ax.axvline(x=np.rad2deg(critical_t), color="k", linestyle="-", label="critical angle T")
81
82ax.set_ylabel("phase (deg)")
83ax.set_xlabel("angle of the incident wave in liquid (deg)")
84
85if SAVEFIG:
86 fig.savefig("liquid_to_solid_alpha_fluid")
87
88# %%Plot reflection and transmission coefficients (alpha_t axis)
89
90fig, axes = plt.subplots(nrows=2, sharex=True)
91ax = axes[0]
92ax.plot(alpha_t_deg, np.abs(reflection), label="reflection")
93ax.plot(alpha_t_deg, np.abs(transmission_l), label="transmission L")
94ax.plot(alpha_t_deg, np.abs(transmission_t), label="transmission T")
95ax.set_title("Tranmission and reflection coefficients from liquid to solid")
96ax.set_ylabel("absolute value")
97ax.legend(loc="best")
98
99ax = axes[1]
100ax.plot(alpha_t_deg, np.angle(reflection, True), label="reflection")
101ax.plot(alpha_t_deg, np.angle(transmission_l, True), label="transmission L")
102ax.plot(alpha_t_deg, np.angle(transmission_t, True), label="transmission T")
103
104ax.set_ylabel("phase (deg)")
105ax.set_xlabel("angle of the transmitted wave T (deg)")
106
107if SAVEFIG:
108 fig.savefig("liquid_to_solid_alpha_t")
109
110# %%Plot reflection and transmission coefficients (alpha_l axis)
111
112fig, axes = plt.subplots(nrows=2, sharex=True)
113ax = axes[0]
114ax.plot(alpha_l_deg, np.abs(reflection), label="reflection")
115ax.plot(alpha_l_deg, np.abs(transmission_l), label="transmission L")
116ax.plot(alpha_l_deg, np.abs(transmission_t), label="transmission T")
117ax.set_title("Tranmission and reflection coefficients from liquid to solid")
118ax.set_ylabel("absolute value")
119ax.legend(loc="best")
120
121ax = axes[1]
122ax.plot(alpha_l_deg, np.angle(reflection, True), label="reflection")
123ax.plot(alpha_l_deg, np.angle(transmission_l, True), label="transmission L")
124ax.plot(alpha_l_deg, np.angle(transmission_t, True), label="transmission T")
125
126ax.set_ylabel("phase (deg)")
127ax.set_xlabel("angle of the transmitted wave L (deg)")
128
129if SAVEFIG:
130 fig.savefig("liquid_to_solid_alpha_l")
131
132# %% Computation of the repartition of energy
133
134# incident pressure (no effect because of normalisation, we keep it for clarity of the formulas)
135pres_i = 10000.0
136
137# cross section areas:
138area_r = np.cos(alpha_fluid).real
139area_l = np.cos(alpha_l).real
140area_t = np.cos(alpha_t).real
141
142# Compute the energy incoming and outcoming: the principle of conservation of
143# energy must be respected.
144# Reference: Schmerr, §6.3.2
145# See also Schmerr, §6.2.5 for the considerations on the inhomogeneous waves
146# which propagate after the critical angle.
147
148# Incoming energy
149inc_energy = 0.5 * pres_i**2 / (rho_fluid * c_fluid) * area_r
150
151# Outgoing energy
152energy_refl = 0.5 * (np.abs(reflection) * pres_i) ** 2 / (rho_fluid * c_fluid) * area_r
153energy_l = 0.5 * (np.abs(transmission_l) * pres_i) ** 2 / (rho_solid * c_l) * area_l
154energy_t = 0.5 * (np.abs(transmission_t) * pres_i) ** 2 / (rho_solid * c_t) * area_t
155out_energy = energy_refl + energy_l + energy_t
156
157ratio_refl = energy_refl / inc_energy
158ratio_l = energy_l / inc_energy
159ratio_t = energy_t / inc_energy
160
161# Verify the conservation of energy:
162if not (np.allclose(ratio_refl + ratio_l + ratio_t, 1.0)):
163 warnings.warn("The conservation of energy is not respected.")
164
165# %% Plot energy
166
167fig, ax = plt.subplots()
168ax.plot(alpha_fluid_deg, ratio_refl, label="reflection")
169ax.plot(alpha_fluid_deg, ratio_l, label="transmission L")
170ax.plot(alpha_fluid_deg, ratio_t, label="transmission T")
171ax.plot(alpha_fluid_deg, ratio_refl + ratio_l + ratio_t, "--", label="total")
172ax.axvline(
173 x=np.rad2deg(critical_l), color="k", linestyle="--", label="critical angle L"
174)
175ax.axvline(x=np.rad2deg(critical_t), color="k", linestyle="-", label="critical angle T")
176ax.set_title("Repartition of energy of the bulk waves: liquid to solid interface")
177ax.set_xlabel("angle of the incident wave in liquid (deg)")
178ax.set_ylabel("normalised energy (1)")
179# ylim([0, 1.05])
180ax.legend(loc="best")
181if SAVEFIG:
182 fig.savefig("liquid_to_solid_energy")
See also
Full reference: arim.model, arim.ut