add missing changes

microphones.ipynb

@@ -59,46 +59,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font color='blue'>\n",
-    "*\n",
-    "For the gradient, we need the derivative with respect to each component of $\\mathbf x$. Using the chain rule of differentiation, we can write for $x$:\n",
-    "\n",
-    "$$\n",
-    "  \\frac{\\partial}{\\partial x} P(\\mathbf x | \\mathbf x_s, \\omega) = \n",
-    "      \\frac{\\partial}{\\partial x}\n",
-    "      \\frac{\\mathrm{exp}\\left(-j\\frac{\\omega}{c} |\\mathbf x-\\mathbf x_{\\mathrm s}| \\right)}{4\\pi|\\mathbf x-\\mathbf x_{\\mathrm s}|} =     \n",
-    "  \\left(\\frac{\\partial (|\\mathbf x - \\mathbf x_s|)}{\\partial x}\\right) \\cdot\n",
-    "  \\left(\\frac{\\partial }{\\partial (|\\mathbf x -\\mathbf  x_s|)} \n",
-    "  \\frac{\\mathrm{exp}\\left(-j\\frac{\\omega}{c} |\\mathbf x-\\mathbf x_{\\mathrm s}| \\right)}{4\\pi|\\mathbf x-\\mathbf x_{\\mathrm s}|}\n",
-    "  \\right)\n",
-    "$$\n",
-    "\n",
-    "The first derivative yields:\n",
-    "$$ \n",
-    "    \\left(\\frac{\\partial (|\\mathbf x - \\mathbf x_s|}{\\partial x}\\right) = \n",
-    "    \\left(\\frac{\\partial (\\sqrt{(x-x_s)^2 + (y-y_s)^2 + (z-z_s)^2}}{\\partial x}\\right) =\n",
-    "    \\frac{x-x_s}{\\sqrt{(x-x_s)^2 + (y-y_s)^2 + (z-z_s)^2}} =\n",
-    "    \\frac{x-x_s}{|\\mathbf x - \\mathbf x_s|}\n",
-    "$$\n",
-    "\n",
-    "The second derivative with $u = |\\mathbf x - \\mathbf x_s|$ yields:\n",
-    "\n",
-    "$$ \n",
-    "  \\frac{\\partial }{\\partial u} \n",
-    "  \\frac{\\mathrm{exp}\\left(-j\\frac{\\omega}{c} u \\right)}{4\\pi u} =\n",
-    "  \\left(-j\\frac{\\omega}{c} - \\frac{1}{u} \\right) \\frac{\\mathrm{exp}\\left(-j\\frac{\\omega}{c} u \\right)}{4\\pi u}\n",
-    "$$\n",
-    "\n",
-    "Repeating the procedure for $y$ and $z$ and evaulating the scalar product with $\\mathbf n_m$ results in\n",
-    "\n",
-    "$$\n",
-    "\\frac{\\partial}{\\partial \\mathbf n_m} P_{ps}(\\mathbf x | \\mathbf x_s, \\omega) =\n",
-    "    \\left(j\\frac{\\omega}{c} + \\frac{1}{|\\mathbf x- \\mathbf x_s|} \\right) \n",
-    "    \\frac{(\\mathbf x_s - \\mathbf x) \\cdot \\mathbf n_m}{|\\mathbf x- \\mathbf x_s|} \n",
-    "    \\frac{\\mathrm{exp}\\left(-j\\frac{\\omega}{c} |\\mathbf x_{\\mathrm s} - \\mathbf x| \\right)}{4\\pi|\\mathbf x-\\mathbf x_{\\mathrm s}|}\n",
-    "$$\n",
-    "*\n",
-    "</font>"
+    "  "
    ]
   },
   {
@@ -152,13 +113,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font color='blue'>\n",
-    "\n",
-    "$$\n",
-    "M_0(\\Theta) = 1\n",
-    "$$\n",
-    "\n",
-    "</font>"
+    "  "
    ]
   },
   {
@@ -172,17 +127,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font color='blue'>\n",
-    "\n",
-    "$$\n",
-    " M_0(\\Theta) = \n",
-    "    \\lim_{|\\mathbf x_s|\\rightarrow \\infty} \n",
-    "    \\left(j\\frac{\\omega}{c} + \\frac{1}{|\\mathbf x - \\mathbf x_s|} \\right) \n",
-    "    \\frac{(\\mathbf x_s - \\mathbf x) \\cdot \\mathbf n_m}{|\\mathbf x- \\mathbf x_s|} = \n",
-    "    j\\frac{\\omega}{c} \\frac{\\mathbf x_s}{|\\mathbf x_s|} \\cdot \\mathbf n_m =\n",
-    "    j\\frac{\\omega}{c} \\cos( \\Theta )\n",
-    "$$\n",
-    "</font>"
+    "  "
    ]
   },
   {
@@ -347,27 +292,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font color='blue'>\n",
-    "*\n",
-    "For the directivity-index, we need the integral in the denominator\n",
-    "$$\n",
-    "\\int_0^\\pi (a+(a-1)\\cos(\\Theta))^2 \\sin(\\theta)\\, \\mathrm d \\Theta =\n",
-    "\\int_0^\\pi \\left( a^2 + 2a(a-1)\\cos(\\Theta) + (a-1)^2\\cos^2(\\Theta)\\right) \\sin(\\Theta) \\, \\mathrm d \\Theta\n",
-    "$$\n",
-    "\n",
-    "Substitute $u = \\cos(\\Theta)$ which yields $\\mathrm du = - \\mathrm \\sin(\\Theta) \\mathrm d \\Theta$. This results in the equivalent integral\n",
-    "\n",
-    "$$\n",
-    "\\int_{-1}^{1} \\left( a^2 + 2a(a-1)u + (a-1)^2 u^2\\right) \\, \\mathrm d u = 2\\left[a^2 + \\frac{1}{3}(a-1)^2 \\right]\n",
-    "$$\n",
-    "\n",
-    "The final directivity-factor reads\n",
-    "\n",
-    "$$\n",
-    "  \\gamma(a, \\omega) = \\frac{1}{a^2 + \\frac{1}{3}(a-1)^2}\n",
-    "$$\n",
-    "*\n",
-    "</font>"
+    "  "
    ]
   },
   {
@@ -381,19 +306,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font color='blue'>\n",
-    "*\n",
-    "A local extremum is possibly reached if the first derivative is zero .\n",
-    "$$\n",
-    "  \\frac{\\partial}{\\partial a} \\gamma(a, \\omega) = \\frac{2a+\\frac{2}{3}(a-1)}{(a^2 + \\frac{1}{3}(a-1)^2)^2} = 0\n",
-    "$$\n",
-    "Multiplying both side with the denominator yields\n",
-    "$$ \n",
-    "  2a+\\frac{2}{3}(a-1) = 0 \\Rightarrow a = \\frac{1}{4}\n",
-    "$$\n",
-    "This value corresponds to a hyper-cardiod pattern\n",
-    "*\n",
-    "</font>"
+    "  "
    ]
   },
   {