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   "source": [
    "# Exercise 2: Scattering from a pressure-release sphere"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The problem\n",
    "\n",
    "In the [first](../tutorials/1_sphere_scatterer_null_field.ipynb) and [second](../tutorials/2_sphere_scatterer_direct.ipynb) tutorials, we looked at two formulations for a rigid scattering problem.\n",
    "\n",
    "In this exercise, you will write your own code to solve a pressure-release scattering problem. As in the tutorials, we will use a unit sphere for $\\Omega$, and we define the incident wave by\n",
    "\n",
    "$$\n",
    "p_{\\text{inc}}(\\mathbf x) = \\mathrm{e}^{\\mathrm{i} k x_0}.\n",
    "$$\n",
    "\n",
    "where $\\mathbf x = (x_0, x_1, x_2)$.\n",
    "\n",
    "Acoustic waves are governed by the Helmholtz equation:\n",
    "\n",
    "$$\n",
    "\\Delta p_\\text{total} + k^2 p_\\text{total} = 0, \\quad \\text{ in } \\mathbb{R}^3 \\backslash \\Omega,\n",
    "$$\n",
    "\n",
    "where $p_\\text{total}$ is the total pressure. We can split $p_\\text{total}$ into incident and scattered pressures by writing $p_\\text{s}+p_\\text{inc}$. The scattered pressure ($p_\\text{s}$) satisfies the Sommerfeld radiation condition\n",
    "\n",
    "$$\n",
    "\\frac{\\partial p_\\text{s}}{\\partial r}-\\mathrm{i}kp_\\text{s}=o(r^{-1})\n",
    "$$\n",
    "\n",
    "when $r:=|\\mathbf{x}|\\rightarrow\\infty$.\n",
    "\n",
    "For our problem, we impose a Dirichlet boundary condition:\n",
    "\n",
    "$$\n",
    "p_\\text{total}=0, \\quad \\text{ on } \\Gamma,\n",
    "$$\n",
    "\n",
    "where $\\Gamma$ is the surface of the sphere $\\Omega$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The formulation\n",
    "\n",
    "We use a formulation based on the [direct formulation in the second tutorial](../tutorials/2_sphere_scatterer_direct.ipynb).\n",
    "\n",
    "### Representation formula\n",
    "\n",
    "For this problem, we use the following representation formula:\n",
    "\n",
    "$$\n",
    "p_\\text{s} = \\mathcal{D}u -\\mathcal{S}\\lambda,\n",
    "$$\n",
    "\n",
    "where $\\mathcal{S}$ is the single layer potential operator; $\\mathcal{D}$ is the double layer potential operator; $u$ is the value (or trace) of $p_\\text{s}$ on the surface $\\Gamma$; and $\\lambda$ is the normal derivative of $p_\\text{s}$ on the surface $\\Gamma$.\n",
    "\n",
    "For this problem, our boundary condition tells us that $u=-p_\\text{inc}$ on $\\Gamma$.\n",
    "\n",
    "### Boundary integral equation\n",
    "\n",
    "For this problem, we want to solve the following boundary integral equation:\n",
    "\n",
    "$$\n",
    "\\mathsf{S}\\lambda = -(\\mathsf{D} - \\tfrac{1}{2}\\mathsf{I})p_\\text{inc},\n",
    "$$\n",
    "\n",
    "where $\\mathsf{S}$ is the single layer boundary operator; $\\mathsf{D}$ is the double layer boundary operator, and $\\mathsf{I}$ is the identity operator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving with Bempp\n",
    "\n",
    "Your task is to adapt and combine the two example codes in [first](../tutorials/1_sphere_scatterer_null_field.ipynb) and [second](../tutorials/2_sphere_scatterer_direct.ipynb) tutorials to solve this problem and plot a slice of the solution at $z=0$\n",
    "\n",
    "To get you started, I've copies the first few lines (which were the same in both examples) into the cell below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import bempp.api\n",
    "from bempp.api.operators.boundary import helmholtz, sparse\n",
    "from bempp.api.operators.potential import helmholtz as helmholtz_potential\n",
    "from bempp.api.linalg import gmres\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "k = 8.\n",
    "\n",
    "grid = bempp.api.shapes.regular_sphere(3)\n",
    "\n",
    "space = bempp.api.function_space(grid, \"DP\", 0)\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What next?\n",
    "\n",
    "After attempting this exercises, you should read [tutorial 3](../tutorials/3_convergence.ipynb)."
   ]
  }
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