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waves.py
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waves.py
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# -*- coding: utf-8 -*-
from constants import *
import scipy.integrate
from manimlib.imports import *
USE_ALMOST_FOURIER_BY_DEFAULT = True
NUM_SAMPLES_FOR_FFT = 1000
DEFAULT_COMPLEX_TO_REAL_FUNC = lambda z : z.real
class Block(Rectangle):
CONFIG = {
"mass": 1,
"velocity": 0,
"width": 0.2,
"height": 6.0,
"fill_opacity": 1,
"stroke_width": 1,
"stroke_color": WHITE,
"fill_color": None,
}
def __init__(self, **kwargs):
digest_config(self, kwargs)
Rectangle.__init__(self, side_length=self.width, **kwargs)
class AirMolecule(Circle):
CONFIG = {
"velocity": 0,
"radius": 0.08,
"fill_opacity": 1,
"stroke_width": 0.1,
"stroke_color": BLUE,
"fill_color": BLUE,
}
def __init__(self, **kwargs):
digest_config(self, kwargs)
Circle.__init__(self, **kwargs)
class SlidingBlocks(VGroup):
CONFIG = {
"block_config": {
"distance": 1,
"mass": 1,
"velocity": 0,
},
"membrane":-4.5,
"floor_pos":-3,
"amplitude": 0.4, #0.4
"frequency":3, #1
"timer":0,
}
def __init__(self, scene, **kwargs):
VGroup.__init__(self, **kwargs)
self.scene = scene
self.block = self.get_block(**self.block_config)
self.point_field = self.get_points()
self.add(
self.block,
)
self.add_updater(self.__class__.update_positions)
def get_block(self, distance, **kwargs):
block = Block(**kwargs)
block.move_to(
self.floor_pos * DOWN +
(self.membrane + distance) * LEFT,
DL,
)
return block
def get_points(self,**kwargs):
points = [x*RIGHT+y*UP
for x in np.arange(-4.1,8,0.25) #-4
for y in np.arange(-3,3,0.25)
]
point_field=[]
for point in points:
circle=AirMolecule().shift(point)
circle.x = point[0]
circle.y = point[1]
point_field.append(circle)
self.add(circle)
return point_field
def update_positions(self, dt):
block= self.block
floor_y = self.floor_pos
#point_field = self.point_fieldmju765
self.timer +=dt
ps_block= self.amplitude* np.sin(self.frequency*self.timer )
block.move_to(
(self.membrane+ ps_block) * RIGHT +
floor_y * UP,
DL,
)
for point in self.point_field:
ps_point = self.amplitude*np.sin(self.frequency * (self.timer - point.x)) #cos
point.move_to(
(point.x+ ps_point) * RIGHT +
point.y * UP,
DL,
)
class MovingString(VGroup):
CONFIG = {
"amplitude": 0.5,
"frequency":0.25,
"timer":0,
}
def __init__(self, scene, **kwargs):
VGroup.__init__(self, **kwargs)
self.scene = scene
self.point_field = self.get_points()
self.add_updater(self.__class__.update_positions)
def get_block(self, distance, **kwargs):
block = Block(**kwargs)
block.move_to(
self.floor_pos * DOWN +
(self.membrane + distance) * LEFT,
DL,
)
return block
def get_points(self,**kwargs):
points = [x*RIGHT
for x in np.arange(-2*np.pi,2*np.pi,np.pi/20)
for y in np.arange(-3,3,0.3)
]
point_field=[]
for point in points:
circle=AirMolecule().shift(point)
circle.x = point[0]
circle.y = 0
point_field.append(circle)
self.add(circle)
return point_field
def update_positions_working(self, dt):
self.timer +=dt
for point in self.point_field:
ps_point = (self.amplitude*np.sin(self.timer*5))*np.cos(self.frequency*point.x)
#ps_point += ((self.amplitude*np.sin(self.timer*5))*np.sin(self.frequency*4*point.x))/4
ps_point += (self.amplitude*np.sin(self.timer*5)*np.sin(self.frequency*6*point.x))/4
ps_point += (self.amplitude*np.sin(self.timer*5)*np.sin(self.frequency*8*point.x))/4
ps_point += (self.amplitude*np.sin(self.timer*5)*np.sin(self.frequency*10*point.x))/4
point.move_to(
point.x*RIGHT+
(ps_point) * UP,
DL
)
class WhatIsATone(Scene):
CONFIG = {
"A_frequency" : 2.1,
"A_color" : YELLOW,
"D_color" : PINK,
"F_color" : TEAL,
"C_color" : RED,
"sum_color" : GREEN,
"equilibrium_height" : 0,
}
def construct(self):
self.question=TextMobject("Was ist ein Ton?")
self.question.scale_in_place(1.3)
self.play(FadeIn(self.question))
self.headline = deepcopy(self.question)
self.headline.shift(3*UP)
self.play(Transform(self.question,self.headline))
self.show_sine()
def show_sine(self):
axes = Axes(
y_min = -2, y_max = 2,
x_min = 0, x_max = 12,
number_line_config = {"include_tip" : False},
)
axes.stretch_to_fit_height(2)
axes.to_corner(2*LEFT)
#axes.shift(UP)
frequency = self.A_frequency
graph = self.get_wave_graph(frequency, axes)
func = graph.underlying_function
graph.set_color(self.A_color)
frequency = self.A_frequency
graph = self.get_wave_graph(frequency, axes)
func = graph.underlying_function
graph.set_color(self.A_color)
pressure = TextMobject("Luftdruck")
time = TextMobject("Zeit")
for label in pressure, time:
label.scale_in_place(0.8)
pressure.next_to(axes.y_axis, UP)
pressure.to_edge(LEFT, buff = MED_SMALL_BUFF)
time.next_to(axes.x_axis.get_right(), DOWN+LEFT)
axes.labels = VGroup(pressure, time)
n = 10
brace = Brace(Line(
axes.coords_to_point(n/frequency, func(n/frequency)),
axes.coords_to_point((n+1)/frequency, func((n+1)/frequency)),
), UP)
words = brace.get_text("440 pro Sekunde", buff = SMALL_BUFF)
words.scale(0.8, about_point = words.get_bottom())
self.play(
ShowCreation(graph, run_time = 6, rate_func=linear),
#ShowCreation(equilibrium_line),
)
self.wait(3)
self.play(
FadeIn(pressure),
ShowCreation(axes.y_axis)
)
self.play(
Write(time),
ShowCreation(axes.x_axis)
)
self.wait(10)
def get_wave_graph(self, frequency, axes):
tail_len = 3.0
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = 0.7*np.cos(2*np.pi*frequency*x)
if x - x_min < tail_len:
value *= smooth((x-x_min)/tail_len)
if x_max - x < tail_len:
value *= smooth((x_max - x )/tail_len)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class AirPressureFast(Scene):
CONFIG = {
"sliding_blocks_config": {
"block_config": {
"mass": 1e0,
"velocity": -2,
}
},
"wait_time": 10,
"frequency" : 1,
"amplitude":2,
"A_color" : YELLOW,
"D_color" : PINK,
"F_color" : TEAL,
"C_color" : RED,
"sum_color" : GREEN,
"equilibrium_height" : 1.5,
"plane_kwargs" : {},
}
def setup(self):
plane = NumberPlane(**self.plane_kwargs)
self.track_time()
self.add_blocks_and_points()
def add_blocks_and_points(self):
self.blocks = SlidingBlocks(self, **self.sliding_blocks_config)
self.add(self.blocks)
def track_time(self):
time_tracker = ValueTracker()
time_tracker.add_updater(lambda m, dt: m.increment_value(dt))
self.add(time_tracker)
self.get_time = time_tracker.get_value
def construct(self):
self.wait(10)
def get_wave_graph(self, axes):
tail_len = 1.0
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = self.amplitude*np.cos(2*np.pi*self.frequency*x)
if x - x_min < tail_len:
value *= smooth((x-x_min)/tail_len)
if x_max - x < tail_len:
value *= smooth((x_max - x )/tail_len)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*self.frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class FrequencyPlot(Scene):
CONFIG = {
"A_frequency" : 2.1,
"A_color" : YELLOW,
"D_color" : PINK,
"F_color" : TEAL,
"C_color" : RED,
"sum_color" : GREEN,
"equilibrium_height" : 1.5,
}
def construct(self):
self.play_A440()
self.play_lower_pitch()
self.wait(10)
def play_A440(self):
self.axes = Axes(
y_min = -2, y_max = 2,
x_min = 0, x_max = 12,
number_line_config = {"include_tip" : False},
)
axes = self.axes
A_label = TextMobject("A440")
A_label.set_color(self.A_color)
#A_label.to_edge(DOWN)
#A_label.shif(UP)
self.set_variables_as_attrs(A_label)
axes.stretch_to_fit_height(2)
axes.to_corner(UP+LEFT)
axes.shift(1.5*DOWN)
eh = self.equilibrium_height
equilibrium_line = DashedLine(
axes.coords_to_point(0, eh),
axes.coords_to_point(axes.x_max, eh),
stroke_width = 2,
stroke_color = LIGHT_GREY
)
frequency = self.A_frequency
graph = self.get_wave_graph(frequency, axes)
func = graph.underlying_function
graph.set_color(self.A_color)
pressure = TextMobject("Luftdruck")
time = TextMobject("Zeit")
for label in pressure, time:
label.scale_in_place(0.8)
pressure.next_to(axes.y_axis, UP)
pressure.to_edge(LEFT, buff = MED_SMALL_BUFF)
time.next_to(axes.x_axis.get_right(), DOWN+LEFT)
axes.labels = VGroup(pressure, time)
n = 10
brace = Brace(Line(
axes.coords_to_point(n/frequency, func(n/frequency)),
axes.coords_to_point((n+1)/frequency, func((n+1)/frequency)),
), UP)
words = brace.get_text("440 pro Sekunde", buff = SMALL_BUFF)
words.scale(0.8, about_point = words.get_bottom())
self.play(
FadeIn(pressure),
ShowCreation(axes.y_axis)
)
self.play(
Write(time),
ShowCreation(axes.x_axis)
)
self.play(
FadeIn(A_label),
)
self.play(
ShowCreation(graph, run_time = 4, rate_func=linear),
ShowCreation(equilibrium_line),
)
axes.add(equilibrium_line)
self.play(
GrowFromCenter(brace),
Write(words)
)
self.wait(2)
graph.save_state()
self.play(
FadeOut(brace),
FadeOut(words),
)
self.set_variables_as_attrs(axes, A_graph = graph)
def play_lower_pitch(self):
axes = self.axes
D_axes= deepcopy(axes)
D_axes.shift(3*DOWN)
frequency = self.A_frequency*(2.0/3.0)
graph = self.get_wave_graph(frequency, D_axes)
graph.set_color(self.D_color)
D_label = TextMobject("D294")
D_label.set_color(self.D_color)
D_label.shift(3*DOWN)
pressure = TextMobject("Luftdruck")
time = TextMobject("Zeit")
for label in pressure, time:
label.scale_in_place(0.8)
pressure.next_to(D_axes.y_axis, UP)
pressure.to_edge(LEFT, buff = MED_SMALL_BUFF)
time.next_to(D_axes.x_axis.get_right(), DOWN+LEFT)
axes.labels = VGroup(pressure, time)
self.play(
FadeIn(pressure),
ShowCreation(D_axes.y_axis)
)
self.play(
Write(time),
ShowCreation(D_axes.x_axis)
)
self.play(
GrowFromCenter(D_label),
)
self.play(
ShowCreation(graph, run_time = 4, rate_func=linear),
)
self.set_variables_as_attrs(
D_label,
D_graph = graph
)
def get_wave_graph(self, frequency, axes):
tail_len = 3.0
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = 0.7*np.cos(2*np.pi*frequency*x)
if x - x_min < tail_len:
value *= smooth((x-x_min)/tail_len)
if x_max - x < tail_len:
value *= smooth((x_max - x )/tail_len)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class Instruments(Scene):
def construct(self):
self.show_both()
self.list_instruments()
self.forced_motion()
self.wait(5)
def show_both(self):
question = TextMobject("Wie produzieren Instrumente Töne?")
question.scale_in_place(1.3)
self.violin = SVGMobject("physik/violin.svg")
sax = SVGMobject("physik/sax.svg", stroke_width=0.1)
string = TextMobject("Saiteninstrumente")
self.string_heading = deepcopy(string)
self.string_heading.shift(3*UP)
self.string_heading.scale_in_place(1.3)
winds = TextMobject("Blasinstrumente")
string.scale_in_place(0.8)
winds.scale_in_place(0.8)
self.violin.shift(3*RIGHT+1.5*DOWN)
self.violin.scale_in_place(1.7)
sax.scale_in_place(1.5)
sax.shift(3*LEFT+1.5*DOWN)
string.shift(3*RIGHT+UP)
winds.shift(3*LEFT+UP)
self.play(GrowFromCenter(question))
self.wait(2)
headline = deepcopy(question)
headline.shift(2.5*UP)
self.play(Transform(question,headline))
self.wait(1)
self.play(FadeIn(winds))
self.play(FadeIn(sax))
self.play(FadeIn(string))
self.play(FadeIn(self.violin))
self.wait(2)
self.remove(winds)
self.remove(sax)
self.play(FadeOut(question))
self.play(Transform(string, self.string_heading))
def list_instruments(self):
piano = SVGMobject("physik/piano.svg",stroke_width=0.1)
guitar = SVGMobject("physik/guitar.svg",stroke_width=0.1)
cello = SVGMobject("physik/cello.svg",stroke_width=0.1)
guitar.scale_in_place(0.6)
cello.scale_in_place(1.6)
cello.shift(UP+5*LEFT)
piano.shift(1.5*DOWN+LEFT)
guitar.shift(5*RIGHT+2*UP)
self.play(FadeIn(cello))
self.play(FadeIn(guitar))
self.play(FadeIn(piano))
self.wait(5)
self.remove(self.violin,guitar,piano,cello)
def forced_motion(self):
cello = SVGMobject("physik/cello.svg",stroke_width=0.1)
cello.scale_in_place(2.5)
cello.shift(0.5*DOWN)
#big_violin.move_to(3*LEFT+1.5*UP)
#big_violin.scale_in_place(1.5)
bow = SVGMobject("physik/bow.svg",stroke_width=0.1)
bow.scale_in_place(0.1)
bow.shift(1.1*DOWN)
bow1 = bow.deepcopy()
bow2 = bow.deepcopy()
bow1.shift(RIGHT)
bow2.shift(LEFT)
self.play(FadeIn(cello))
self.wait(1)
self.play(FadeIn(bow))
self.play(Transform(bow,bow2),run_time=2)
self.play(Transform(bow,bow1),run_time=2)
self.play(Transform(bow,bow2),run_time=2)
self.play(Transform(bow,bow1),run_time=2)
class StringWithNodes(Scene):
CONFIG = {
"frequency" : 0.05,
"color" : YELLOW,
"equilibrium_height" : 0,
"amplitude": 1,
"plane_kwargs" : {},
}
def setup(self):
self.axes = Axes(
y_min = -2, y_max = 2,
x_min = -5, x_max = 5,
number_line_config = {"include_tip" : False},
)
# self.headline=TextMobject("Wie produzieren Instrumente Töne?")
# self.headline.scale_in_place(1.3)
# self.headline.shift(3*UP)
left_node= Circle(radius=0.1, fill_color=RED, fill_opacity= 1)
self.left_node = left_node
self.left_node.shift(self.axes.x_min*RIGHT)
right_node= Circle(radius=0.1, fill_color=RED, fill_opacity= 1)
self.right_node = right_node
self.right_node.shift(self.axes.x_max*RIGHT)
center_node= Circle(radius=0.1, fill_color=RED, fill_opacity= 1)
self.center_node = center_node
first_third_node= Circle(radius=0.1, fill_color=RED, fill_opacity= 1)
self.first_third_node = first_third_node
self.first_third_node.shift(self.axes.x_min/3*RIGHT)
second_third_node= Circle(radius=0.1, fill_color=RED, fill_opacity= 1)
self.second_third_node = second_third_node
self.second_third_node.shift(self.axes.x_max/3*RIGHT)
def construct(self):
#self.play(ShowCreation(self.headline))
self.pure_frequency()
self.first_ot()
self.second_ot()
self.wave_graph()
def pure_frequency(self):
string = self.get_wave_graph()
#self.string = string
state1 = self.get_wave_graph()
self.amplitude *= -1
state2 = self.get_wave_graph()
#self.state2 = string2
self.play(FadeIn(self.left_node))
self.play(FadeIn(self.right_node))
self.play(ShowCreation(string))
self.play(Transform(string,state2),run_time=(2))
self.play(Transform(string,state1),run_time=(2))
self.play(FadeOut(string))
def first_ot(self):
self.amplitude *= -1
self.frequency = self.frequency*2
string = self.get_wave_graph()
self.string = string
state1 = self.get_wave_graph()
self.amplitude *= -1
state2 = self.get_wave_graph()
#self.state2 = string2
self.play(FadeIn(self.center_node))
self.play(ShowCreation(string))
self.play(Transform(string,state2),run_time=(2))
self.play(Transform(string,state1),run_time=(2))
self.play(FadeOut(string))
self.play(FadeOut(self.center_node))
def second_ot(self):
self.amplitude *= -1
self.frequency = self.frequency/2*3
string = self.get_wave_graph()
state1 = self.get_wave_graph()
self.amplitude *= -1
state2 = self.get_wave_graph()
self.play(FadeIn(self.first_third_node))
self.play(FadeIn(self.second_third_node))
self.play(ShowCreation(string))
self.play(Transform(string,state2),run_time=(2))
self.play(Transform(string,state1),run_time=(2))
self.play(FadeOut(string))
self.play(FadeOut(self.left_node))
self.play(FadeOut(self.first_third_node))
self.play(FadeOut(self.second_third_node))
self.play(FadeOut(self.right_node))
def wave_graph(self):
graph = self.get_complex_graph()
self.play(ShowCreation(graph))
self.wait(5)
def get_wave_graph(self):
axes = self.axes
frequency = self.frequency
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = self.amplitude*np.sin(2*np.pi*frequency*(x-x_min))
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
def get_complex_graph(self):
axes = self.axes
frequency = 1.2
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = self.amplitude*np.sin(2*np.pi*frequency*(x-x_min))
value += (self.amplitude/1.5)*np.sin(2*np.pi*frequency*(x-x_min)*2)
value += (self.amplitude/1)*np.sin(2*np.pi*frequency*(x-x_min)*3)
value += (self.amplitude/2)*np.sin(2*np.pi*frequency*(x-x_min)*4)
value += (self.amplitude/1)*np.sin(2*np.pi*frequency*(x-x_min)*5)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class ComplexOscillation(Scene):
CONFIG = {
"sliding_blocks_config": {
"block_config": {
"mass": 1e0,
"velocity": -2,
}
},
"wait_time": 15,
"frequency" : 0.5,
"amplitude":2,
"A_color" : YELLOW,
"D_color" : PINK,
"F_color" : TEAL,
"C_color" : RED,
"sum_color" : GREEN,
"equilibrium_height" : 1.5,
"plane_kwargs" : {},
}
def setup(self):
plane = NumberPlane(**self.plane_kwargs)
self.track_time()
self.add_points()
def add_points(self):
self.blocks = MovingString(self, **self.sliding_blocks_config)
self.add(self.blocks)
def track_time(self):
time_tracker = ValueTracker()
time_tracker.add_updater(lambda m, dt: m.increment_value(dt))
self.add(time_tracker)
self.get_time = time_tracker.get_value
def construct(self):
self.wait(5)
class AddingFrequencies(Scene):
CONFIG = {
"equilibrium_height" : 1.5,
"amplitude" : 1,
"freq1" : 2,
"freq2" : 4,
"freq3" : 6,
}
def setup(self):
axes = Axes(
y_min = -2, y_max = 2,
x_min = 0, x_max = 12,
)
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = self.amplitude*np.cos(2*np.pi*self.freq1*x) + self.amplitude/2*np.cos(2*np.pi*self.freq2*x) + self.amplitude/3*np.cos(2*np.pi*self.freq3*x)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*self.freq1 + 1
self.graph = axes.get_graph(func, num_graph_points = int(ngp))
self.play(FadeIn(self.graph))
def get_wave_graph(self, frequency, axes):
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = 0.7*np.cos(2*np.pi*frequency*x)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class Intro(Scene):
def construct(self):
self.show_questions()
self.wait(5)
def show_questions(self):
tone = TextMobject("Was sind Töne?")
tone.scale_in_place(1.3)
tone.shift(2*UP)
instrument = TextMobject("Wie produzieren Instrumente Töne?")
instrument.scale_in_place(1.3)
computer = TextMobject("Wie können Computer Instrumente imitieren?")
computer.scale_in_place(1.3)
computer.shift(2*DOWN)
self.play(FadeIn(tone))
self.wait(0.5)
self.play(FadeIn(instrument))
self.wait(1)
self.play(FadeIn(computer))
class HarmonicSeries(Scene):
CONFIG = {
"fund_color" : YELLOW,
"first_color" : PINK,
"second_color" : TEAL,
"third_color" : RED,
"sum_color" : GREEN,
"equilibrium_height" : 1,
}
def setup(self):
self.axes = Axes(
y_min = -2, y_max = 2,
x_min = -4, x_max = 6,
number_line_config = {"include_tip" : False},
)
def construct(self):
self.fundamental()
self.first_ot()
self.second_ot()
self.third_ot()
self.combine()
self.wait(5)
def fundamental(self):
axes = self.axes
axes.shift(1.3*UP)
frequency = 0.25
self.graph0 = self.get_wave_graph(frequency,axes)
self.label0 = TextMobject("Grundton")
self.label0.next_to(self.graph0,LEFT)
self.play(FadeIn(self.label0))
self.play(ShowCreation(self.graph0))
def first_ot(self):
axes = self.axes
axes.shift(1.7*DOWN)
frequency = 0.5
self.graph1 = self.get_wave_graph(frequency,axes)
self.label1 = TextMobject("1. Oberton")
self.label1.next_to(self.graph1,LEFT)
self.play(FadeIn(self.label1))
self.play(ShowCreation(self.graph1))
def second_ot(self):
axes = self.axes
axes.shift(1.7*DOWN)
frequency = 0.75
self.graph2 = self.get_wave_graph(frequency,axes)
self.label2 = TextMobject("2. Oberton")
self.label2.next_to(self.graph2,LEFT)
self.play(FadeIn(self.label2))
self.play(ShowCreation(self.graph2))
def third_ot(self):
axes = self.axes
axes.shift(1.7*DOWN)
frequency = 1
self.graph3 = self.get_wave_graph(frequency,axes)
self.label3 = TextMobject("3. Oberton")
self.label3.next_to(self.graph3,LEFT)
self.play(FadeIn(self.label3))
self.play(ShowCreation(self.graph3))
def combine(self):
#self.add(self.graph1)
axes = Axes(
y_min = -2, y_max = 2,
x_min = -4, x_max = 6,
number_line_config = {"include_tip" : False},
)
axes.shift(1.3*UP)
def func1(x):
value = 0.4*np.cos(2*np.pi*0.25*(x-axes.x_min))
value += 0.4*np.cos(2*np.pi*0.5*(x-axes.x_min))
return value + self.equilibrium_height
ngp = 2*(axes.x_max - axes.x_min)*1 + 1
sum1 = axes.get_graph(func1, num_graph_points = int(ngp))
def func2(x):
value = 0.3*np.cos(2*np.pi*0.25*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.5*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.75*(x-axes.x_min))
return value + self.equilibrium_height
ngp = 2*(axes.x_max - axes.x_min)*1 + 1
sum2 = axes.get_graph(func2, num_graph_points = int(ngp))
def func3(x):
value = 0.3*np.cos(2*np.pi*0.25*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.5*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.75*(x-axes.x_min))
value += 0.25*np.cos(2*np.pi*0.1*(x-axes.x_min))
return value + self.equilibrium_height
ngp = 2*(axes.x_max - axes.x_min)*1 + 1
sum3 = axes.get_graph(func3, num_graph_points = int(ngp))
self.remove(self.label0, self.label1, self.label2, self.label3)
self.play(ReplacementTransform(self.graph1, self.graph0, run_time=0.25))
self.play(ReplacementTransform(self.graph0, sum1, run_time=0.4))
self.play(ReplacementTransform(self.graph2, sum1, run_time=0.6))
self.play(ReplacementTransform(sum1, sum2,run_time=0.3))
self.play(ReplacementTransform(self.graph3, sum2, run_time=0.8))
self.play(ReplacementTransform(sum2, sum3,run_time=0.3))
axes.x_min = -8
axes.x_max = 8
def func4(x):
value = 0.3*np.cos(2*np.pi*0.25*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.5*(x-axes.x_min))
value += 0.3*np.cos(2*np.pi*0.75*(x-axes.x_min))
value += 0.25*np.cos(2*np.pi*0.1*(x-axes.x_min))
return value + self.equilibrium_height
ngp = 2*(axes.x_max - axes.x_min)*1 + 1
centered_sum = axes.get_graph(func3, num_graph_points = int(ngp))
centered_sum.shift(2.5*DOWN)
self.play(Transform(sum3,centered_sum))
def get_wave_graph(self, frequency, axes):
tail_len = 0.5
x_min, x_max = axes.x_min, axes.x_max
def func(x):
value = 0.5*np.cos(2*np.pi*frequency*x)
if x - x_min < tail_len:
value *= smooth((x-x_min)/tail_len)
if x_max - x < tail_len:
value *= smooth((x_max - x )/tail_len)
return value + self.equilibrium_height
ngp = 2*(x_max - x_min)*frequency + 1
graph = axes.get_graph(func, num_graph_points = int(ngp))
return graph
class Fourier(GraphScene):
CONFIG={
"axes_config" : {
"x_min": 0,
"x_max" : 12,
"y_min" : -6,
"y_max": 6,
"y_axis_config": {
"unit_size" : 0.15,
"tick_frequency" : 16,
},
},
"frequency_axes_config" : {
"x_min":0,
"x_max":2,
"x_axis_config" : {
"unit_size" : 6,
"tick_frequency" : 0.25,
},
"y_min":0,
"y_max":1,
"y_axis_config" : {
"unit_size" : 2,
},
"x_labeled_nums" :range(0,2,1),
},
"equilibrium_height": 1,
}
def break_apart_and_add(self):
axes = Axes(**self.axes_config)
axes.to_corner(UP+LEFT)
func = lambda t : sum([
np.cos(TAU*f*t)
for f in (0.5, 1, 0.75, 1.5)
])
summed_graph = axes.get_graph(func)
summed_graph.set_color(RED)
sum_label = TextMobject ("f(x) = sin(1x)+sin(1.5x)+sin(2x)+sin(3x)")
sum_label.next_to(summed_graph,DOWN)
fund = self.get_wave_graph(0.5,axes)
fund.to_corner(DOWN,LEFT)
fund.shift(UP)
first_ot = self.get_wave_graph(0.75,axes)
first_ot.next_to(fund,3*UP)
second_ot = self.get_wave_graph(1,axes)
second_ot.next_to(first_ot,3*UP)
third_ot = self.get_wave_graph(1.5,axes)
third_ot.next_to(second_ot,3*UP)
self.play(FadeIn(axes))
self.play(ShowCreation(summed_graph))
self.wait(5)
self.play(TransformFromCopy(summed_graph, fund))
self.play(TransformFromCopy(summed_graph, first_ot))
self.play(TransformFromCopy(summed_graph, second_ot))
self.play(TransformFromCopy(summed_graph, third_ot))
self.wait(5)
first_ot_copy = first_ot.copy()
first_ot_copy.move_to(fund)
self.play(ReplacementTransform(first_ot,first_ot_copy))
second_ot_copy = second_ot.copy()
second_ot_copy.move_to(fund)
self.play(ReplacementTransform(second_ot,second_ot_copy))
third_ot_copy = third_ot.copy()
third_ot_copy.move_to(fund)
self.play(ReplacementTransform(third_ot,third_ot_copy))
func_copy = lambda t : sum([
2*np.cos(TAU*f*t)
for f in (0.5, 1, 0.75, 1.5)
])