Fourier sine series coefficients

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Toby Wood 1 year, 4 months ago

Published this.

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Toby Wood 1 year, 4 months ago

Created this as a copy of Toby JSXGraph one slider, two plots - 2023-11-15.

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Question statement

The plot below shows the first 10 coefficients of the Fourier sine series for the function
$f(x) = \displaystyle\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{l} = \begin{cases}hx/d & \mbox{if}\ 0 \leqslant x \leqslant d \\ h(l-x)/(l-d) & \mbox{if}\ d \leqslant x \leqslant l\end{cases}$ .

These coefficients are given by $b_n = \frac{2h}{d(l-d)}\left(\frac{l}{\pi n}\right)^2\sin\frac{\pi nd}{l}$ .

{diagram}

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			Name	Type	Generated Value

			diagram	jsxgraphboard	HTML node
			python_code	string	%matplotlib qt import numpy a

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​x
 
jessiecode(700,200,[-5.6,1.5,1.4,-0.5],safe("""
  
  sinc = function(x) {
    if(abs(x)<0.001) {
      return 1;
    }
    else {
    return sin(x)/x;
    };
  };
  fourier_coefficient = function(i, d) {
    return 2/(PI*i)*(sinc(PI*i*d)-(-1)^i*sinc(PI*i*(1-d)));
  };
    
  // A slider for the value of d
  d = slider([-2,0.5],[-1,0.5],[0,0.5,1]) <<suffixlabel: "$d=$", postlabel: "$l$">>;
    
  // Plot of the graph of f(x)/h
  functiongraph(function(x) {
      if(x - d < 0) {
        return x/d;
      } else {
        return (1-x)/(1-d); 
      };
  }, 0, 1);
  
  // Axes and labels for the graph
  xaxis = arrow([0,0], [1.2,0]) <<fixed: true, highlight: false, strokeColor: 'black', strokeWidth: 1>>;
  point(1.2,0) <<size: 0, name: "$x/l$", label: <<autoPosition: true, offset:[-5, -10]>> >>;
  ticks(xaxis) <<drawLabels: true, ticksPerLabel: 5, ticksDistance: 0.5, label: <<anchorY: 'top', anchorX: 'middle', offset: [0,-10]>>, fixed: true, highlight: false>>;
  yaxis = arrow([0,0], [0,1.2]) <<fixed: true, highlight: false, strokeColor: 'black', strokeWidth: 1>>;
  point(0,1.2) <<size: 0, name: "$f(x)/h$", label: <<autoPosition: true, offset:[-30, 0]>> >>;
  ticks(yaxis) <<drawLabels: true, ticksPerLabel: 5, ticksDistance: 1, label: <<anchorX: 'right', offset: [-10,0]>>, fixed: true, highlight: false>>;
    
  xaxis2 = arrow([-5,0.2],[-2,0.2]) <<fixed: true, highlight: false, strokeColor: 'black', strokeWidth: 1>>;
  point(-2,0.2) <<size: 0, name: "$n$", label: <<autoPosition: true, offset:[-5, -10]>> >>;
  ticks(xaxis2) <<scale: 3/11, ticksDistance: 1, drawLabels: true, ticksPerLabel: 1, label: <<anchorY: 'top', anchorX: 'middle', offset: [0,-10]>>, fixed: true, highlight: false, minorTicks: 0>>;
    
  yaxis2 = arrow([-5,-0.3],[-5,1.2]) <<fixed: true, highlight: false, strokeColor: 'black', strokeWidth: 1>>;
  point(-5,1.2) <<size: 0, name: "$b_n/h$", label: <<autoPosition: true, offset:[-40, 0]>> >>;
  ticks(yaxis2) <<scale: 0.8, ticksDistance: 0.5, anchor: 1/3, drawLabels: true, ticksPerLabel: 1, label: <<anchorX: 'right', offset: [-10,0]>>, fixed: true, highlight: false, minorTicks: 0>>;
​
  // Functions to transform coordinates for the plot of Fourier series
  x_series = function(x) {
    return x * 3*10/11 - 5;
  };
  y_series = function(y) {
    return y*0.8 + 0.2;
  };
    
  // Points representing the coefficients of the Fourier series
  make_point = function (i) {
    return point(x_series(i/10),function() { return y_series(fourier_coefficient(i,d.Value())); }) <<name: ''>>; 
  };
    
  for(i=1;i<=10;i=i+1) {
    make_point(i);
  };
  
"""),
  ["axis": false]
)

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Generated value: `jsxgraphboard`

0,0

$d=$ 0.50

$l$

$x/l$

0.5

$f(x)/h$

$n$

$b_n/h$

0.5

−0.5

This variable doesn't seem to be used anywhere.

Parts

Reproducing the plots with Python Code

Code

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You can make this figure yourself in Python by running the code below. But you'll need to code the function fourier_coefficient yourself first.

{code_block(python_code,"python")}

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xxxxxxxxxx
 
d_init = 0.5
​
def fourier_coefficient(n,d):
# Add your code here to calculate the value of b_n/h, for given values of n and d/l
    return
​
# Plot the first 10 coefficients, assuming that d/l = d_init initially
vals = np.arange(10)+1
points, = coeffs.plot(vals,fourier_coefficient(vals,d_init),'o')
coeffs.set_ylim((-0.5,1))
coeffs.set_xlabel("$n$",loc='right')
coeffs.set_ylabel("$b_n/h$",loc='top',rotation='horizontal')

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xxxxxxxxxx
 
import sys
import matplotlib
​
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np
fig = plt.figure()
fig.savefig(sys.stdout, format='svg')
coeffs = plt.axes()

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fig.savefig(sys.stdout, format='svg')

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  ["Test 1", 1, "True"]
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  ["arithmetic works", "1+1 == 2"]
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Taxonomy: mathcentre

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Toby Wood 1 year, 4 months ago

Toby Wood 1 year, 4 months ago

Error in variable testing condition

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Generated value: `jsxgraphboard`

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Fourier sine series coefficients

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Toby Wood 1 year, 4 months ago

Toby Wood 1 year, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: jsxgraphboard

Parts

Code

Generated value: `jsxgraphboard`