Update test_christoffel_stepbystep.ipynb

src/test_christoffel_stepbystep.ipynb

@@ -27,9 +27,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "The goal of PyRockWave's Christoffel module is to estimate various seismic properties as a function of direction using the Christoffel equation (Christoffel, 1877). These properties include the $V_p$ (compressional), $V_{s1}$, $V_{s2}$ (shear) wave velocities (phase and group) in km/s, the polarisation directions of the sound waves, the shear wave splitting, the coordinates or the ray surface, the power flow angle and the enhancement factor. We will define all these parameters throughout this notebook.\n",
+    "\n",
     "## Input data\n",
     "\n",
-    "The neccesary input data to estimate seismic velocities through any direction of a particular material using the Christoffel equation is the density and the stiffness tensor of the material. The stiffness tensor $C$, a fundamental property of a material that generalizes Hooke's law in three dimensions relating strains stresses in the elastic regime. This is usually provided (or abbreviated) in a 6x6 matrix ($C_{ij}$) using Voigt's notation. "
+    "The necessary input data to estimate the seismic properties in any direction of a given material using the Christoffel equation are the density and the stiffness tensor of the material.\n",
+    "\n",
+    "The stiffness tensor $C$ is a fundamental property of a material that generalises Hooke's law in three dimensions, relating strains to stresses in the elastic regime. This is usually given (or abbreviated) in a 6x6 matrix ($C_{ij}$) using Voigt's notation, and values are in $GPa$. Density is usually given in...TODO "
    ]
   },
   {
@@ -243,13 +247,21 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Step 4: Estimate the eigenvalues and eigenvectors\n",
+    "## Step 4: Estimate the eigenvalues $\\lambda$ and eigenvectors\n",
     "\n",
-    "TODO\n",
+    "The eigenvalues $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$ of the Christoffel tensor $M_{ij}$ are always positive and related to the wave velocities $V_p$, $V_{s1}$, $V_{s2}$ propagating in the direction $\\vec{n}$ by the formulae\n",
     "\n",
     "$$\n",
-    "\\lambda = v^2_p\n",
-    "$$"
+    "V_p = \\sqrt{ \\frac{\\lambda_1}{\\rho} }, \\quad V_{s1} = \\sqrt{ \\frac{\\lambda_2}{\\rho} }, \\quad V_{s2} = \\sqrt{ \\frac{\\lambda_3}{\\rho} }\n",
+    "$$\n",
+    "\n",
+    "where $\\rho$ denotes the material density. Note that if the Christoffel tensor is already normalized to the density this is just\n",
+    "\n",
+    "$$\n",
+    "v_p = \\sqrt{\\lambda}, \\quad \\lambda = v^2_p\n",
+    "$$\n",
+    "\n",
+    "where $v_p$ denotes phase velocities."
    ]
   },
   {