Example: Slowness Profiles
Slowness profiles are a plot of the inverse-phase velocity of each bulk wave component. They are found by solving the Christoffel equation, and only depend on the material properties of each material. The slowness profiles given by this software are plotted in terms of 
vs 
. For an isotropic material the slowness profiles for each bulk wave are spherical, however, this is not true for an anisotropic material. Slowness profiles can be used to determine the angles of refraction between multi-layered media as well as the direction of energy propagation and skew angle [1]. An example of the slowness profiles for isotropic-glass and transverse-isotropic beryl material is shown here.
vs . For an isotropic material the slowness profiles for each bulk wave are spherical, however, this is not true for an anisotropic material. Slowness profiles can be used to determine the angles of refraction between multi-layered media as well as the direction of energy propagation and skew angle [1]. An example of the slowness profiles for isotropic-glass and transverse-isotropic beryl material is shown here.[1] Rose, Joseph L. "Ultrasonic guided waves in solid media.", Cambridge university press, (2014).
Initialize a Medium Object
Firstly create a Medium object with each material needed to generate a plot. Here, the slowness profiles of water, glass, beryl will be generated.
% Initialize the Medium:
my_medium = Medium('water', Inf, 'glass', 0.001,...
'beryl2', Inf);
Calculating Slowness Profiles
The slowness profiles can be calculated using the .calculateSlowness method.
% Calculate the slowness profiles:
my_medium.calculateSlowness;
Plotting the Slowness Profiles
The slowness profiles can be plotted using the .plotSlowness method. Since these are symmetric about the 
and 
axes, they are only plotted for a single quadrant. The slowness profiles of the (quasi-)longitudinal, (quasi-)shear-vertical and (quasi-)shear-horizontal bulk waves are shown.
and axes, they are only plotted for a single quadrant. The slowness profiles of the (quasi-)longitudinal, (quasi-)shear-vertical and (quasi-)shear-horizontal bulk waves are shown.% Plot slowness profiles:
my_medium.plotSlowness;
Manually Plotting the Slowness Profiles
The slowness data is stored in obj.slowness in the form of a structure.
% Slowness data for the second material:
slowness_data_2 = my_medium(2).slowness;
% The .calcualteSlowness function will return complex values and only the
% real part should be plotted.
figure;
% plot the (q)-L component (kz_qL1)
plot(real(slowness_data_2.kx), abs(real(slowness_data_2.kz_qL1)), 'k' )
hold on
% plot the (q)-SV component (kzt_qSV1)
plot(real(slowness_data_2.kx), abs(real(slowness_data_2.kz_qSV1)), 'b' )
% plot the (q)-SH component (kz_qSH)
plot(real(slowness_data_2.kx), abs(real(slowness_data_2.kz_qSH)), 'r--')
hold off
% labels
axis equal
xlabel('k_x /\omega')
ylabel('k_z /\omega')
legend('L', 'SV', 'SH')
title('Manual slowness plot')
Calculating Errors
As glass is an isotropic material, the slowness profiles are spherical and the magnitudes of L, 
and 
when 
or 
are equal to the reciprocal of the compressional- and shear-speeds of glass. For the transverse-isotropic case, when , the value of 
is equal to 
and 
is equal to 
. When , the value of is equal to 
and the value of is equal to .
and when or are equal to the reciprocal of the compressional- and shear-speeds of glass. For the transverse-isotropic case, when , the value of is equal to and is equal to . When , the value of is equal to and the value of is equal to . Errors for Glass
% Predefined value compressional/longitudinal wave:
cL_glass = 5570;
% From the slowness calculation:
cL_glass_slowness = 1 / sqrt(real(slowness_data_2.kx(1)).^2 ...
+ real(slowness_data_2.kz_qL1(1)).^2);
% Normalised error:
difference_error_CL = abs((cL_glass - cL_glass_slowness) / 5570);
% Predefined value shear wave:
cS_Glass = 3430;
% From the slowness calculation, shear-vertical:
cS_glass_slowness_1 = 1 / sqrt(real(slowness_data_2.kx(1)).^2 +...
real(slowness_data_2.kz_qSV1(1)).^2);
% From the slowness calculation, shear-horizontal:
cS_glass_slowness_2 = 1 / sqrt(real(slowness_data_2.kx(1)).^2 + ...
real(slowness_data_2.kz_qSH(1)).^2);
% Normalized Error:
difference_error_CSV = abs((cS_Glass - cS_glass_slowness_1) / 3430);
difference_error_CSH = abs((cS_Glass - cS_glass_slowness_2) / 3430);
% Display errors for glass:
disp(['Error in cL :', num2str(difference_error_CL) ])
disp(['Error in cSV :', num2str(difference_error_CSV)])
disp(['Error in cSH :', num2str(difference_error_CSH)])
Errors for Beryl
% Get the stiffness matrix for beryl:
stiffness_matrix_beryl = my_medium(3).stiffness_matrix;
density_beryl = my_medium(3).density;
% Compare qL at kx = 0:
diff_error_qL_kx0 = (sqrt(density_beryl/stiffness_matrix_beryl(3,3)) -...
abs(my_medium(3).slowness.kz_qL1(1))) / ...
sqrt(density_beryl / stiffness_matrix_beryl(3,3));
% Compare qL at kz = 0:
[idx] = find(real(my_medium(3).slowness.kz_qL1) == 0);
diff_error_qL_kz0 = (my_medium(3).slowness.kx(idx(1)) - ...
sqrt(density_beryl / stiffness_matrix_beryl(1, 1))) / ...
sqrt(density_beryl / stiffness_matrix_beryl(1, 1));
% Compare qSV at kx = 0:
diff_error_qSV_kx0 = (sqrt(density_beryl / stiffness_matrix_beryl(5, 5)) -...
abs(my_medium(3).slowness.kz_qSV1(1))) / ...
sqrt(density_beryl / stiffness_matrix_beryl(5, 5));
% Compare qSV at kz = 0:
[idx] = find(real(my_medium(3).slowness.kz_qSV1) == 0);
diff_error_qSV_kz0 = (my_medium(3).slowness.kx(idx(1)) - ...
sqrt(density_beryl / stiffness_matrix_beryl(5, 5))) / ...
sqrt(density_beryl / stiffness_matrix_beryl(5, 5));
Errors for beryl are within numerical precision however, the profile is not calculated exactly when 
: this has been indicated with an *.
: this has been indicated with an *.disp(['Error in qL at kx = 0 :', num2str(diff_error_qL_kx0)])
disp(['Error in qL at kz = 0 * :', num2str(diff_error_qL_kz0)])
disp(['Error in qSV at kx = 0 :', num2str(diff_error_qSV_kx0)])
disp(['Error in qSV at kz = 0 * :', num2str(diff_error_qSV_kz0)])