Add formulas to README (#15)

README.md

@@ -1,11 +1,13 @@
 
-# FrequenPy
+<h1 align="center" style="border-bottom: none;"> FrequenPy </h1>
 
-![coverage_badge](https://codecov.io/gh/tomaslink/frequenpy/branch/master/graph/badge.svg) 
-
-_FrequenPy_ is a high-precision physics engine dedicated to the study of standing waves and visualization of its normal modes.
+<p align="center">
+  <a>
+    <img alt="Coverage" src="https://codecov.io/gh/tomaslink/frequenpy/branch/master/graph/badge.svg">
+  </a>
+</p>
 
-This package has educational purposes. 
+**frequenpy** is a high-precision physics engine dedicated to the study and visualization of standing waves.
 
 ## Wave theory
 
@@ -21,21 +23,38 @@ This results have been (and can be) demonstrated experimentally.
 
 <div align="justify">
 
-  According to wave theory, any arbitrary movement of the string
-  can be decomposed into a superposition of natural modes of oscillation.
-  In each natural mode **m**,
-  all masses in the system oscilate at the same frequency ***f(m)***
-  and pass through the equilibrium position at the same time.
-  This natural modes of oscillation are called **normal modes**.
-
+  A flexible elastic string with tension **T** is loaded with **N** identical particles,
+  each of mass **m**, equally spaced a distance **a** apart.
+  Let us hold the string fixed at two points,
+  one at a distance **a** to the left of the first particle
+  and the other at a distance **a** to the right of the Nth particle.
+
+  According to the theory, the movement of each of the masses in the vertical direction
+  can be decomposed into a superposition of **N** **normal modes** modes of oscillation.
+  That way, the $y$ position of the particle **n** as a function of time is
+  ```math
+    y_n(t) = \sum_{p=1}^N A_p \sin(k_p n a) \cos(\omega_p t + \theta_p).
+  ```
+  Where $A_p$ and $\theta_p$ depend on the initial conditions,
+  $k_p$ will depend on the boundary conditions and $\omega_p$ will have the form
+  ```math
+    \omega_p = 2 \omega_0 \sin\left(\frac{p \pi}{2(N + 1)}\right)
+    \qquad,
+    \qquad
+    \omega_0 = \sqrt{\frac{T}{ma}}.
+  ```
+
   There are as many normal modes as there are degrees of freedom (masses) in the system.
-  So, a string loaded with **N** masses, will have **N** **normal modes**.
-  The first will corresponde to the lowest frequency (called fundamental)
-  and each next will be higher than the previous one, until we reach the last and highest frequency.
-  Any movement, as strange as it may be, can be expressed as a superposition of those **N** normal modes
+  In each natural mode **p**,
+  all masses in the system oscilate at the same frequency $\omega_p$
+  and pass through the equilibrium position at the same time.
+  The first mode, **p=1**, corresponds to the lowest frequency (called fundamental)
+  and each subsequent mode will have a frequency higher than the previous one.
+  Any movement of the string, as strange as it may be,
+  can be expressed as a superposition of those **N** normal modes
   (some will contribute more than others to the final movement). 
 
-  As the number of masses gets higher and highter (***N*** ---> ***∞***),
+  As the number of masses gets higher and highter ($N \rightarrow \infty$),
   we approximate to the continuous system (a vibrating string - no discrete masses).
   In this simulation, you can use **N = 30** to see a continuous effect.
 
@@ -98,7 +117,7 @@ optional arguments:
 Example: frequenpy loaded_string --masses 3 --modes 1 2 3 --speed 0.1 --boundary 0
 ```
 
-Remember that for system of **N** masses there are N normal modes.
+Remember that for system of **N** masses there are **N** normal modes.
 You can pass only one of them or a combination of several, e.g. "2 6 3".
 The order doesn't matter. 
 

frequenpy/loaded_string/animation.py

@@ -10,10 +10,10 @@
 logger = logging.getLogger(__name__)
 
 
-LINE_WIDTH = 1
+LINE_WIDTH = 0.5
 LINE_MARKERTYPE = 'o'
 LINE_MARKERSIZE = 8
-LINE_MARKERFACECOLOR = 'None'
+LINE_MARKERFACECOLOR = 'gray'
 LINE_COLOR = 'white'
 
 BACKGROUND_COLOR = 'black'
@@ -22,7 +22,7 @@
 WALL_WIDTH = 0.5
 WALL_COLOR = 'white'
 
-FIG_SIZE = (7, 2)
+FIG_SIZE = (7, 4)
 FIG_DPI = 100
 FIG_X_LIMIT = (-0.5, 0.5)
 FIG_Y_LIMIT = (-1, 1)
@@ -93,11 +93,10 @@ def _build_line_without_markers(self):
 
     def _build_frames(self):
         self._loaded_string.apply_speed(self._speed)
-        frames = range(0, self._n_frames)
 
         return [
             self._loaded_string.position_at_time_t(t)
-            for t in frames
+            for t in range(0, self._n_frames)
         ]
 
     def _build_figure(self):
@@ -117,6 +116,8 @@ def _build_figure(self):
         ax = plt.axes(xlim=FIG_X_LIMIT, ylim=FIG_Y_LIMIT, frameon=False)
         ax.set_yticks([])
         ax.set_xticks([])
+        ax.set_xlim(-0.5, 0.51)
+        ax.set_ylim(-0.5, 0.51)
 
         ax.add_line(self._left_support(support_distance_from_origin))
         ax.add_line(self._right_support(support_distance_from_origin))
@@ -151,7 +152,7 @@ def _build_animation(self):
             self._figure,
             self._update,
             frames=self._n_frames,
-            interval=5,
+            interval=10,
             blit=True,
             repeat=True)
 

frequenpy/loaded_string/loaded_string.py

@@ -3,10 +3,10 @@
 
 
 LONGITUDE = 1
-TENSION = 0.5
+TENSION = 1
 MASS = 0.1
 
-AMPLITUDE = 0.5
+AMPLITUDE = 0.2
 THETA = 0
 
 
@@ -18,6 +18,8 @@ def __init__(self, N, modes):
         self._modes = modes
         self._masses = range(0, N + 2)
         self._a = LONGITUDE / (N + 1)
+
+        self._omega0 = np.sqrt(TENSION / (MASS * self._a))
         self._omega = self._omega()
 
         self.rest_position = self._get_rest_position()
@@ -57,7 +59,7 @@ def modes(self):
 
     def _omega(self):
         return {
-            p: 2 * np.sqrt(TENSION / MASS * self._a) * np.sin(self._wavenumber(p) * self._a / 2)
+            p: 2 * self._omega0 * np.sin((self._wavenumber(p) * self._a) / 2)
             for p in self._modes
         }