# Equivalent symmetries

Here we show how to formulate dispersion equations for wave mode which are not just plane waves. See Theoretical background for an overview.

The dispersion equations become simpler the more symmetries that are shared between the incident wave and the material geometry. The simplest case is `PlanarAzimuthalSymmetry`

, and includes the case where a plane wave is directly incident upon a flat surface. If no symmetries are present, then the type `WithoutSymmetry`

is used and leads to a general dispersion equation for materials which occupy a simple connected domain. This general dispersion equation has spurious roots and is computationally heavier to solve, see Gower & Kristensson 2020.

## Choose the microstructure

```
spatial_dim = 3
medium = Acoustic(spatial_dim; ρ=1.2, c=1.5)
# Below we explicitly define the shape of the particles as being spheres
s1 = Specie(
Acoustic(spatial_dim; ρ=1.0, c=0.5), Sphere(spatial_dim, 0.4);
volume_fraction=0.2
);
# We use just one specie to speed up the calculations.
species = [s1]
```

## Calculate the wavenumbers

```
ω = 0.9
tol = 1e-7
# Calculate the wavenumbers for PlanarAzimuthalSymmetry()
AP_kps = wavenumbers(ω, medium, species;
num_wavenumbers = 4, tol = tol,
symmetry = PlanarAzimuthalSymmetry())
# Calculate the wavenumbers for just PlanarSymmetry()
P_kps = wavenumbers(ω, medium, species;
num_wavenumbers = 4, tol = tol,
symmetry = PlanarSymmetry{3}())
# Select a subset to test. The dispersion equation WithoutSymmetry can be unstable for higher order effective wavenumbers.
AP_kps = AP_kps[1:min(length(AP_kps),14)]
P_kps = P_kps[1:min(length(P_kps),14)]
```

## Test different dispersion equations

```
AP_det = dispersion_equation(ω, medium, species, PlanarAzimuthalSymmetry{spatial_dim}())
P_det = dispersion_equation(ω, medium, species, PlanarSymmetry{spatial_dim}())
AR_det = dispersion_equation(ω, medium, species, AzimuthalSymmetry{spatial_dim}())
R_det = dispersion_equation(ω, medium, species, WithoutSymmetry{spatial_dim}())
# As plane waves with azimuthal symmetry is a sub-case of plane-waves, and all materials allow for the effective wavenumbers of plane waves, all the below determinant equations should be satisfied
maximum(AP_det.(AP_kps)) < tol
maximum(P_det.(AP_kps)) < tol
maximum(AR_det.(AP_kps)) < tol^2
maximum(R_det.(AP_kps)) < tol^3
# However, there do exist effective wavenumbers for plane-waves which have eigen-vectors that do not satisfy azimuthal symmetry. This is why maximum(AP_det.(P_kps)) != 0.0
maximum(AP_det.(P_kps)) > tol
maximum(P_det.(P_kps)) < tol
maximum(AR_det.(P_kps)) < tol^2
maximum(R_det.(P_kps)) < tol^3
```