From f20ed66ef5066e6b55d06e2d4e428e0d50fb8a7d Mon Sep 17 00:00:00 2001
From: ChristiaanDHoogervorst
 <145754988+ChristiaanDHoogervorst@users.noreply.github.com>
Date: Thu, 29 Feb 2024 14:28:43 +0100
Subject: [PATCH] Add files via upload

---
 notebooks/3b_wave_transformation.ipynb        | 148 ++++++
 ...wave_transformation_in_the_nearshore.ipynb | 474 ++++++++++++++++++
 2 files changed, 622 insertions(+)
 create mode 100644 notebooks/3b_wave_transformation.ipynb
 create mode 100644 notebooks/3c_wave_transformation_in_the_nearshore.ipynb

diff --git a/notebooks/3b_wave_transformation.ipynb b/notebooks/3b_wave_transformation.ipynb
new file mode 100644
index 0000000..4243a8e
--- /dev/null
+++ b/notebooks/3b_wave_transformation.ipynb
@@ -0,0 +1,148 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a57196e5-eb53-4eab-8bb2-b11c2a313e56",
+   "metadata": {},
+   "source": [
+    "<figure>\n",
+    "  <IMG SRC=\"https://raw.githubusercontent.com/mbakker7/exploratory_computing_with_python/master/tudelft_logo.png\" WIDTH=250 ALIGN=\"right\">\n",
+    "</figure>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fcad68ab-a006-40c5-93ad-5eb0a10e41e8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%run Initialize/init_3b.ipynb"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a3b7540-f1c8-4188-9321-c162619892e7",
+   "metadata": {},
+   "source": [
+    "# 1) Wave asymmetry and skewness"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9c61dd36-8df2-48f7-b54e-a4f2f5af06a2",
+   "metadata": {},
+   "source": [
+    "Wave asymmetry and skewness can be demonstrated by using a second order wave, as in equation 1. The phase difference between the wave componets influences the shape of the combined wave. You can assess the impact of phase differences and the amplitudes on the assymetry on skewness in the interactive graph below. The figures 5.13 and 5.16 of the book can be reproduced with this figure, without scaling the y-axis. What are the amplitudes and the phases of wave component 1 and 2 to reproduce these figures? You can check it below the graph.\n",
+    "\n",
+    "\\begin{equation}\n",
+    "\\tag{1}\n",
+    "\\eta = \\eta_1 cos(wt-kx-\\phi_1) + \\eta_2 cos(2(wt-kx)-\\phi_2)\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d0ce79ef-1e0f-4a72-b152-4b71b46f5ab8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "second_order_waves()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "67ead777-88c0-4389-9998-f117368fae50",
+   "metadata": {},
+   "source": [
+    "Can you make the left panel of figure 5.15, without scaling to (dividing by)  $\\eta_1$? You can rewrite equation 1 such that the surface elevation is a function of x/L. You may assume that t=0 to simplify the process. You can put this equation in the formula below to check your outcome, which will automatically be plotted for 3 wave lenghts."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c1f2816b-95f2-4e6a-8606-5fb329f8b81f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x_L = 1 # The ratio of x/L (an arbitrary value can be usefull to check your code)\n",
+    "eta1 = 1\n",
+    "eta2 = ...\n",
+    "phi1 = ...\n",
+    "phi2 = ...\n",
+    "\n",
+    "\n",
+    "def second_order_waves_x_L(x_L, eta1, eta2, phi1, phi2):\n",
+    "    ...\n",
+    "    return eta\n",
+    "\n",
+    "show_second_order_waves_x_L()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cdff687-6362-43c1-834b-ab569f4d017c",
+   "metadata": {},
+   "source": [
+    "Can you rewrite equation 1 now so that the surface elavation is a function of t/T. You may assume that x=0 to simplify the process. You can put this equation in the formula below to check your outcome.\n",
+    "figuur 5.15 right panel, x-axis etc."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ee83ee3c-a7fa-4f7a-80cb-cfb6fbdc3758",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "t_T = 1# t/T, an arbitrary value for testing\n",
+    "\n",
+    "def second_order_waves_t_T(t_T, more_arguments):\n",
+    "        ...\n",
+    "        return eta\n",
+    "\n",
+    "show_second_order_waves_t_T()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8a4143ae-5b9c-4011-86de-4f9a4ed9d646",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Skewed_assymetric_waves()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d34dd514-32ab-4a9b-9932-d39c47547dfb",
+   "metadata": {},
+   "source": [
+    "A final thought provocing question on this section, what is the influence on the time/space averaged value of $\\eta^3$ when the amplitudes and phases of the wave components change?"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/notebooks/3c_wave_transformation_in_the_nearshore.ipynb b/notebooks/3c_wave_transformation_in_the_nearshore.ipynb
new file mode 100644
index 0000000..ed2f1bc
--- /dev/null
+++ b/notebooks/3c_wave_transformation_in_the_nearshore.ipynb
@@ -0,0 +1,474 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a57196e5-eb53-4eab-8bb2-b11c2a313e56",
+   "metadata": {},
+   "source": [
+    "<figure>\n",
+    "  <IMG SRC=\"https://raw.githubusercontent.com/mbakker7/exploratory_computing_with_python/master/tudelft_logo.png\" WIDTH=250 ALIGN=\"right\">\n",
+    "</figure>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "59f51bad-f362-48d4-b5c5-9dd22f07579a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "run Initialize/init_3c.ipynb"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "55200145-c32c-444e-b925-9e73e173cf2a",
+   "metadata": {},
+   "source": [
+    "# week 3c: Wave transformation in the near-shore "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "75e92d3c-f6b9-40c4-9fe1-3082483a6ff7",
+   "metadata": {},
+   "source": [
+    "We continue on the application of the dispersion relationship in the cross-shore, which was discussed in week 3 and the Wave course. You can use Forward Euler to calculate a derivative, an example in the cell below to calculate dh/dx. Note that the last value of dh_dx remains 0, since there is no forward location. This can be ignored in this notebook."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0558a7a1-7c82-4f55-ae25-9d9f0467079c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x = np.arange(0,10+1,1)\n",
+    "h = -100 + x**2\n",
+    "\n",
+    "dh_dx = np.zeros(len(h)) # dh/dx\n",
+    "for i in range(len(dh_dx)-1):\n",
+    "    dh_dx[i] = (h[i+1] - h[i])/(x[i+1]-x[i])\n",
+    "\n",
+    "plt.figure(figsize = (3,1))\n",
+    "plt.plot(x,h, label = 'h')\n",
+    "plt.plot(x,dh_dx, label = 'dh_dx')\n",
+    "plt.legend();"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "17205ca8-dfd6-4622-8482-c64cfd445857",
+   "metadata": {},
+   "source": [
+    "## 1) Wave height (H), radiation stress (Sxx), mean water level by waves ($\\eta$) for normal incident waves"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e5dd8576-5be2-40dd-916d-32d9d8453cfc",
+   "metadata": {},
+   "source": [
+    "In Waves you should have programmed a function to compute the wave height (H), radiation stress (Sxx), and mean water level ($\\eta$) along the cross-shore for normal incident waves. Can you reproduce/copy these functions below? If not, you can uncomment the content in the cell below and use this to answer the upcoming questions. However, it is highly recommended to make those functions. As inputs (arguments) you should consider the offshore deep-water wave height (H0), the wave period (T), the offshore water depth (h0) and the bed slope (slope). Below is a setup of the environmental conditions, for a a spatial grid that is defined with x_range. The function check_W3_normal_wave_setup will check your answers. You can also ignore RuntimeWarnings, for example, divide by zero encountered in scalar power."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a903c1f3-af10-46e5-bfc3-8d086a18923b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "H0 = 1.5                         # The offshore wave height [m]\n",
+    "T = 6                            # The wave period [s]\n",
+    "h0 = 20                          # The largest water depth\n",
+    "slope = 1/30                     # The constant bed slope\n",
+    "x_range = np.arange(0,900,1)     # The cross-shore directed grid, the horizontal axis\n",
+    "\n",
+    "def W3_normal_wave(x_range, H0, T, h0, slope):\n",
+    "\n",
+    "    # The environmental conditions\n",
+    "    x = x_range               # the horizontal axis\n",
+    "    zbed = -h0 + slope*x      # bed elevation [m]\n",
+    "    h = -zbed                 # still water depth [m]\n",
+    "    h[h < 0] = 0              # no negative depths (A depth of 0 can cause warnings, these can be ignored to keep the code straightforward)\n",
+    "    \n",
+    "    # given conditions\n",
+    "    gamma = 0.8               # wave breaking ratio\n",
+    "\n",
+    "    '''Complete the code here'''\n",
+    "\n",
+    "    return H, n, Ksh, E, Sxx, eta\n",
+    "\n",
+    "\n",
+    "check_W3_normal_wave()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5d0aff2b-605e-4205-9f19-86ba318cf08f",
+   "metadata": {},
+   "source": [
+    "If you are not able to make the functions, you can set Load_answers = True (rather than False) to get the values for the offshore wave angle (angle), the wave height (H), the values for n, the shoaling parameter (Ksh), the wave energy (E), the radiant stress (Sxx), and the surface elevation (eta), with mean-sea-level at 0 m."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "afa50f94-2678-4f8d-998f-2b571a0e48b9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Load_answers = False\n",
+    "\n",
+    "if Load_answers == True:\n",
+    "    angle = 0\n",
+    "    H = [1.5, 1.49975468745948, 1.4995086464001774, 1.499261878337649, 1.4990143848294484, 1.498766167475419, 1.4985172279179815, 1.4982675678424247, 1.4980171889771967, 1.497766093094194, 1.497514282009055, 1.497261757581446, 1.4970085217153555, 1.4967545763593837, 1.496499923507032, 1.4962445651969951, 1.4959885035134517, 1.4957317405863537, 1.495474278591717, 1.4952161197519125, 1.494957266335957, 1.4946977206598027, 1.4944374850866269, 1.4941765620271235, 1.4939149539397911, 1.4936526633312261, 1.493389692756409, 1.4931260448189956, 1.4928617221716067, 1.4925967275161178, 1.4923310636039466, 1.492064733236343, 1.491797739264678, 1.4915300845907318, 1.491261772166984, 1.4909928049968981, 1.4907231861352133, 1.4904529186882303, 1.4901820058140973, 1.4899104507231011, 1.4896382566779522, 1.4893654269940688, 1.4890919650398684, 1.48881787423705, 1.4885431580608817, 1.4882678200404862, 1.487991863759126, 1.4877152928544888, 1.4874381110189718, 1.4871603219999656, 1.48688192960014, 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+    "    "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d72471e7-9c77-4ee6-a6db-de93a048e75b",
+   "metadata": {},
+   "source": [
+    "Below is the cross-shore distribution of several wave characteristics shown for the wave conditions described on top. Feel free to study the effects of changing these parameters on the wave characteristics. You can also use these to answer the questions below."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8e563df0-685e-47e8-a859-d2cfa904ab9c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_Q1()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "95345d24-4ade-4679-9502-9c440107d6b5",
+   "metadata": {},
+   "source": [
+    "## 2) Wave height (H), radiation stress (Sxx), mean water level by waves ($\\eta$) for oblique waves"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2920520f-db13-40de-9de8-3d20f7d16d08",
+   "metadata": {},
+   "source": [
+    "The influence of refraction should also be considered for waves that approach the shore under an angle. Can you calculate the wave heigth (H), radiation stress (Sxx), and mean water level (eta) for oblique waves by completing the function below? The function W3_plot_oblique_waves(angle1, angle2) will plot these characteristics along the cross-shore for the two given angles. This function will give a runtime warning when the water depth (h) is 0, you may ignore this to keep the code as straightforward as possible. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6b4b0fb3-dbac-40cd-b494-57a143b711b4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "H0 = 1.5                         # The offshore wave height [m]\n",
+    "T = 6                            # The wave period [s]\n",
+    "h0 = 20                          # The largest water depth\n",
+    "slope = 1/30                     # The constant bed slope\n",
+    "x_range = np.arange(0,900,1)     # The cross-shore directed grid, the horizontal axis\n",
+    "angle = 30                       # The offshore wave angle\n",
+    "\n",
+    "def W3_wave_setup(x_range, H0, T, h0, slope, angle):\n",
+    "    \n",
+    "    # The environmental conditions\n",
+    "    x = x_range               # the horizontal axis\n",
+    "    zbed = -h0 + slope*x      # bed elevation [m]\n",
+    "    h = -zbed                 # still water depth [m]\n",
+    "    h[h < 0] = 0              # no negative depths\n",
+    "\n",
+    "    # given conditions\n",
+    "    gamma = 0.8               # wave breaking ratio\n",
+    "\n",
+    "    ''' You can build on the results from normal-indicent waves'''\n",
+    "    \n",
+    "    ...\n",
+    "    \n",
+    "    return H, Sxx, eta, c, n  # c and n are useful later on\n",
+    "\n",
+    "check_W3_wave_setup()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51e2110a-d085-4d52-a7b1-3d8882a321e7",
+   "metadata": {},
+   "source": [
+    "You can use the interactive widget below to assess the influence of changing environmental conditions on the wave characteristics"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d045aad5-ad82-44b2-bb52-a5d0fe6bc027",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_show_oblique_waves()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb378c1f-7267-4ab0-a15b-eb1cbee22bf3",
+   "metadata": {},
+   "source": [
+    "## 3) The shorewards directed force (Fx)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "32cd7ecc-7ee1-4883-9125-735f3cfae4e9",
+   "metadata": {},
+   "source": [
+    "We dive a bit deeper into the force in the cross-shore direction that drives the wave setup (Fx). Can you calculate this force along the cross-shore for an alongshore uniform coast? For this question, you can consider using a similar approach as the setup is calculated, by using forward euler."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0bf515d7-a7e6-4bab-b108-4f258ce9f90c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def W3_Fx(x_range, H0, T, h0, slope, angle):\n",
+    "\n",
+    "    '''It is recommended to use the earlier defined function W3_wave_setup()'''\n",
+    "    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)\n",
+    "\n",
+    "    ...\n",
+    "    \n",
+    "    return Fx\n",
+    "\n",
+    "W3_plot_Fx()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ca954ae3-7057-4f0f-a227-3ad6ae95497f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_show_Fx()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40a688e5-8544-4fc1-9637-edf792b6c5d6",
+   "metadata": {},
+   "source": [
+    "## 4) Alongshore shear stresses (Syx) and alongshore force (Fy)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "428d376f-71d5-4b4c-8fa2-6c8448cfc898",
+   "metadata": {},
+   "source": [
+    "Can you compute the radiation shear stress in y direction (Syx)?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f1f3b159-8f30-43e2-9c6b-d958f4297d01",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def W3_Syx(x_range, H0, T, d0, slope, angle):\n",
+    "\n",
+    "    '''It is recommended to use the earlier defined function W3_wave_setup()'''\n",
+    "    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)\n",
+    "\n",
+    "    ...\n",
+    "    \n",
+    "    return Syx\n",
+    "\n",
+    "W3_plot_Syx()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c448338b-3a5e-42a1-9f1e-8e586d0ee600",
+   "metadata": {},
+   "source": [
+    "And can you compute the alongshore force (Fy) for an alongshore uniform coast?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c555c3e7-ac1e-414d-91af-0dcd8ac4b0d5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def W3_Fy(x_range, H0, T, d0, slope, angle):\n",
+    "\n",
+    "    '''It is recommended to use the earlier defined function(s)'''\n",
+    "    x = x_range\n",
+    "    ...\n",
+    "    \n",
+    "    return Fy\n",
+    "\n",
+    "W3_plot_Fy()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d30d5759-7970-47a9-98bf-80e958b36f72",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_show_Syx_Fy()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "262420b5-5d7e-4d45-b4b8-43275aa1cb97",
+   "metadata": {},
+   "source": [
+    "## 5) Longshore current (V)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af022363-dffb-48da-acb8-40f5f3007d70",
+   "metadata": {},
+   "source": [
+    "Can you calculate the distribution of the longshore current, when a bottom roughness (r) of 6 cm is used in the friction factor $c_f$?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0be6cbd7-0b05-44f5-a847-b459c5b316b9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def W3_V(x_range, H0, T, d0, slope, angle, r):\n",
+    "\n",
+    "    # The environmental conditions\n",
+    "    x = x_range               # the horizontal axis\n",
+    "    zbed = -h0 + slope*x      # bed elevation [m]\n",
+    "    h = -zbed                 # still water depth [m]\n",
+    "    h[h < 0] = 0              # no negative depths\n",
+    "\n",
+    "    # With\n",
+    "    gamma = 0.8\n",
+    "    \n",
+    "    '''It is recommended to use the earlier defined function W3_wave_setup()'''\n",
+    "    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)\n",
+    "\n",
+    "    ...\n",
+    "    \n",
+    "    return V\n",
+    "\n",
+    "W3_plot_V()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d2c9d554-bc54-4efa-9d60-056125b9635b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_show_V()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "69c8fab3-bdad-47ab-8ec4-ed637fcca545",
+   "metadata": {},
+   "source": [
+    "## 6) Wave induced local current, orbital velocity"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "47795cf9-506e-4717-91a5-dc5ee1102ef9",
+   "metadata": {},
+   "source": [
+    "Can you calculate the amplitude of the velocity (u0) near the bed through the linear wave theory?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c5550679-5d39-405f-b8e4-9e0ba12d46b8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def W3_u0(x_range, H0, T, d0, slope, angle):\n",
+    "\n",
+    "    '''It is recommended to use the earlier defined function W3_wave_setup()'''\n",
+    "    H, Sxx, eta, c, n = W3_wave_setup(x_range, H0, T, h0, slope, angle)\n",
+    "\n",
+    "    L = c*T\n",
+    "    k = 2*np.pi/L                                 # The wave number\n",
+    "    \n",
+    "    snell_constant = np.sin(np.deg2rad(angle))/c[0]  # apply snell's law\n",
+    "    theta_radians = np.arcsin(snell_constant * c)\n",
+    "    \n",
+    "    # page 199 of the book\n",
+    "    omega = 2*np.pi/T\n",
+    "    u0 = 0.5* omega * H / (np.sin(theta_radians)*k*h)\n",
+    "    \n",
+    "    return u0\n",
+    "\n",
+    "W3_plot_u0()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e37b13cf-4310-41c2-8291-d358505980c3",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "W3_show_U0()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f5ea94ec-da81-4eed-807c-37d639c1257a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}