Reflection and transmission

Once we have calculated the effective wavenumber, we can calculate the average reflection and transmission.

Here we will use an incident plane wave

\[u_\text{in} = U e^{i \mathbf k \cdot \mathbf r},\]

where $U$ is the amplitude, which is usually set to $U=1$ for acoustics, $k = \|\mathbf k\|$ is the incident wavenumber, and $\mathbf k$ can be a two or three dimensional vector.

We will assume that $u_\text{in}$ is arriving from $z<0$ ($x<0$ for 2D). If the material occupies the region $\mathcal R = \{ z>0 : \mathbf r \in \mathbb R^3\}$, then the reflected wave will be given by

\[u_\text{R} = R e^{i (k_x x + k_y y - k_z z)}.\]

The code below calculates $R$, which is called the reflection coefficient. Both reflection and transmission are simpler to calculate when there exists only one effective wavenumber. Currently, we have only implemented the reflection coefficient for multiple effective wavenumbers for 2D acoustics.

Many tests for 3D reflection and transmission are in test/acoustics-3D/planar-symmetry.jl, while tests for 2D are in the files in the folder test/acoustics-2D.

The formulas used to calculate the below can mostly be found in Gower & Kristensson 2020 and Gower et al. 2018

2D acoustics

Low frequency reflection

The simplest case is for low frequency, where the average reflection coefficient $R$ reduces to the reflection coefficient from a homogeneous material:

\[R = \frac{q_* \cos \theta_\text{in} - \cos \theta_*}{q_* \cos \theta_\text{in} + \cos \theta_*},\]

where

\[\mathcal R = \{ x>0 : \mathbf r \in \mathbb R^2\} \;\; \text{and} \;\; \mathbf k = k (\cos \theta_\text{in}, \sin \theta_\text{in}).\]

In code this becomes

spatial_dim = 2
medium = Acoustic(spatial_dim; ρ=1.2, c=1.5)

# Choose the species
radius1 = 0.1
s1 = Specie(
    Acoustic(spatial_dim; ρ=10.2, c=10.1), radius1;
    volume_fraction=0.2
);
radius2 = 0.2
s2 = Specie(
    Acoustic(spatial_dim; ρ=0.2, c=4.1), radius2;
    volume_fraction=0.15

);
species = [s1,s2]

# Choose the frequency
ω = 1e-4
k = ω / medium.c

# Calculate the equivalent effective medium in the asymptotic low frequency limit
eff_medium = effective_medium(medium, species)

normal = [-1.0,0.0] # an outward normal to the surface

# Define the material region
material = Material(medium,Halfspace(normal),species)

# define a plane wave source travelling at a 45 degree angle in relation to the material
source = PlaneSource(medium, [cos(pi/4.0),sin(pi/4.0)])

R = reflection_coefficient(ω,source, eff_medium, material.shape);

round(R * 100) / 100
# output

0.14 + 0.0im

One plane wave mode

Note there are formulas for low volume fraction expansions of the reflection coefficient, see reflection_coefficient_low_volumefraction. However, it is better to use the exact expression, as the different in computational cost is minimal.

As an example, we will use the same material defined for the low frequency case

k_effs = wavenumbers(ω, medium, species; tol = 1e-6, num_wavenumbers = 1, basis_order = 1)

# Calculate the wavemode for the first wavenumber
wave1 = WaveMode(ω, k_effs[1], source, material; tol = 1e-6, basis_order = 1)

R = reflection_coefficient(ω, wave1, source, material)

round(R * 100) / 100

# output
0.14 + 0.0im

Multiple effective modes

See A numerical matching method for an example that uses multiple effective wave modes to calculate the reflection coefficient.

3D acoustics

Low frequency reflection

The simplest case is for low frequency, where the average reflection coefficient $R$ reduces to the reflection coefficient from a homogeneous material:

\[R = \frac{q_* \cos \theta_\text{in} - \cos \theta_*}{q_* \cos \theta_\text{in} + \cos \theta_*},\]

where

\[\mathcal R = \{ z>0 : \mathbf r \in \mathbb R^3\} \;\; \text{and} \;\; \mathbf k = k (\cos \phi_\text{in} \sin \theta_\text{in}, \sin \phi_\text{in} \sin \theta_\text{in}, \cos \theta_\text{in}).\]

In code this becomes

spatial_dim = 3
medium = Acoustic(spatial_dim; ρ=0.3, c=0.5)

# Choose the species
radius1 = 0.1
s1 = Specie(
    Acoustic(spatial_dim; ρ=10.2, c=10.1), radius1;
    volume_fraction=0.2
);
species = [s1]

# Choose the frequency
ω = 1e-5
k = ω / medium.c

# For the limit of low frequencies we can define
eff_medium = effective_medium(medium, species)

# Define a plate
r = maximum(outer_radius.(species))
normal = [0.0,0.0,-1.0] # an outward normal to both surfaces of the
width = 150.0 # plate width
origin = [0.0,0.0,width/2] # the centre of the plate


# the size of the effective low frequency limit material is one particle radius smaller
plate_low = Plate(normal,width - 2r,origin)
halfspace_low = Halfspace(normal,[0.0,0.0,r])

# define a plane wave source travelling at a 45 degree angle in relation to the material
source = PlaneSource(medium, [cos(pi/4.0),0.0,sin(pi/4.0)])

Ramp1 = reflection_coefficient(ω, source, eff_medium, halfspace_low)

# planewave_amplitudes returns the
amps = planewave_amplitudes(ω, source, eff_medium, plate_low)
Ramp = amps[1]
Tamp = amps[2]

round(1000 * Tamp) / 1000

# output

1.0 + 0.002im

The function planewave_amplitudes returns [R, T, P1, P2] where R is the reflection coefficient, T is the coefficient of the transmitted wave, and P1 (P2) are the amplitudes of the wave travling forward (backward) inside the plate.

One plane wave mode

Using only one plane wave mode we can calculate both reflection and transmission from a plate.

k_effs = wavenumbers(ω, medium, species;
    tol = 1e-6,
    num_wavenumbers = 1,
    basis_order = 1
)
k_eff = k_effs[1]

abs(k_eff - ω / eff_medium.c) < 1e-10

halfspace = Halfspace(normal)
plate = Plate(normal,width,origin)

material = Material(medium,halfspace,species)

# Calculate the wavemode for the first wavenumber
# the WaveMode function calculates the types of waves and solves the needed boundary conditions
wavemode = WaveMode(ω, k_eff, source, material; tol = 1e-6, basis_order = 1);

Reff = reflection_coefficient(wavemode, source, material)

material = Material(medium,plate,species)
wavemodes = WaveMode(ω, k_eff, source, material; tol = 1e-6, basis_order = 1);
RTeff = reflection_transmission_coefficients(wavemodes, source, material);

abs(Ramp1 - Reff) < 1e-6

abs.(RTeff - [Ramp; Tamp]) .< [1e-4, 5e-4]

# Note that summing the reflection and transmission from the homogeneous low frequency medium gives 1
sum(abs.([Ramp,Tamp]).^2) ≈ 1.0
sum(abs.(RTeff).^2)

Formulas from Gower & Kristensson 2020.