One-Dimensional Superposition of Sine Waves

Modeling and visualization of linear superposition of sine waves and resultant interference patterns

• Python 3.11.4
• Matplotlib 3.7.2
• Seaborn 0.12.2
• NumPy 1.25.2

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Main Project Files

• Standalone Module
• IPython Notebook / Jupyter Notebook

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Content

Theoretical Background

• Wave equation
• Superposition principle

Parameters

• Wave model parameters
• Superposition plot parameters

Implementation

• Modeling of sine wave superposition phenomena and interference patterns

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Wave Equation

The sine wave equation in one-dimension can be represented as:

$y(x, t) = A \sin(kx \pm \omega t + \phi)$

where:

$y(x,t)$ is the wave displacement at position $x$ and time $t$
$A$ is the amplitude
$k = \frac{2\pi}{\lambda}$ is the wave number
$\omega = 2\pi f$ is the angular frequency
$\phi$ is the phase offset
$+$ or $−$ depends on the direction of propagation (right or left)

Superposition Principle

The superposition principle states that when two or more waves overlap in space, the resultant wave is the algebraic sum of their individual waves. When two waves $W_1$ and $W_2$ interfere, the resultant wave $W_R$ can be given by:

$W_R(x, t) = W_1(x, t) + W_2(x, t)$

Interference patterns are the result of the superposition of two or more waves. These patterns can be either constructive or destructive depending on the phase and amplitude of the interacting waves.

These patterns can display areas of both constructive and destructive interference, and they are commonly seen in phenomena like double-slit experiments and sound wave interference.

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Wave Model Parameters

model_sinewave() function

Parameter	Description	Type
x	Positions of the wave ($x$): Positions where the wave is evaluated (m)	numpy.ndarray
t	Time of evaluation ($t$): Time at which the wave is evaluated (s)	float
A	Amplitude ($A$): Maximum displacement from equilibrium (m)	float
wavelength	Wavelength ($\lambda$): Length of one complete wave cycle (m)	float
frequency	Frequency ($f$): Number of oscillations per second (Hz)	float
phi	Phase offset ($\phi$): Shifts the wave horizontally (radians)	float (optional)
propagation	Propagation direction (x-axis): 'right' for positive, 'left' for negative	string (optional)
phase_polarity	Phase polarity (y-axis): 'positive' retains form, 'negative' flips the wave vertically	string (optional)

These parameters can be used to represent the waves in the model:

$\large y(x, t) = A \sin\left( \frac{2\pi}{\text{wavelength}} \cdot x \pm 2\pi \cdot \text{frequency} \cdot t + \phi \right)$

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Superposition Plot Parameters

plot_wave_superposition() function

Parameter	Description	Type
wave_1_params	$W_1$: Wave 1 model parameters	dictionary
wave_2_params	$W_2$: Wave 2 model parameters	dictionary
dark_theme	Dark Theme: If True, uses a dark background for the plot	bool (optional)

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Implementation

Modeling of sine wave superposition phenomena and interference patterns

1. Standing waves

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	5.0	5.0
frequency	$f$: Frequency	90.0	90.0
wavelength	$\lambda$: Wavelength	10.0	10.0
phi	$\phi$: Phase offset	0.0	0.0
propagation	Propagation direction	right	left
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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2. Wave beats

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	5.0	5.0
frequency	$f$: Frequency	10.0	20.0
wavelength	$\lambda$: Wavelength	4.0	5.0
phi	$\phi$: Phase offset	0.0	0.0
propagation	Propagation direction	right	right
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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3. Constructive Interference

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	10.0	5.0
frequency	$f$: Frequency	90.0	90.0
wavelength	$\lambda$: Wavelength	10.0	10.0
phi	$\phi$: Phase offset	0.0	0.0
propagation	Propagation direction	right	right
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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4. Destructive Interference

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	10.0	5.0
frequency	$f$: Frequency	90.0	90.0
wavelength	$\lambda$: Wavelength	10.0	10.0
phi	$\phi$: Phase offset	0.0	$\pi$
propagation	Propagation direction	right	right
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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5. Perfect Destructive Interference (Cancellation)

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	10.0	10.0
frequency	$f$: Frequency	90.0	90.0
wavelength	$\lambda$: Wavelength	10.0	10.0
phi	$\phi$: Phase offset	0.0	$\pi$
propagation	Propagation direction	right	right
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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6. Fundamental Frequency + 2nd Harmonic

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	5.0	2.5
frequency	$f$: Frequency	90.0	180.0
wavelength	$\lambda$: Wavelength	4.0	2.0
phi	$\phi$: Phase offset	0.0	0.0
propagation	Propagation direction	right	right
phase_polarity	Phase polarity (y)	positive	positive

Parameter	Description	Value
dark_theme	Dark Theme	True

Output:

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7. General superposition

Parameter	Description	$W_1$	$W_2$
A	$A$: Amplitude	10.0	5.0
frequency	$f$: Frequency	90.0	110.0
wavelength	$\lambda$: Wavelength	6.0	10.0
phi	$\phi$: Phase offset	0.0	0.0
propagation	Propagation direction	right	left
phase_polarity	Phase polarity (y)	positive	negative

Parameter	Description	Value
dark_theme	Dark Theme	True

Output: