Standing Waves

📐 Definition

A standing wave is formed by adding two waves of the same frequency and amplitude traveling in opposite directions, yielding fixed spatial nodes with oscillating amplitude.

$$\sin(kx)\cos(\omega t) = \tfrac{1}{2}\bigl(\sin(kx-\omega t) + \sin(kx+\omega t)\bigr) = \operatorname{Im}\!\left(\tfrac{1}{2}\bigl(e^{i(kx-\omega t)} + e^{i(kx+\omega t)}\bigr)\right)$$

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Domain and Codomain

Defined over spatial coordinates and time where linear superposition applies. Outputs follow the underlying field type (often real-valued displacement or potential).

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⚙️ Key Properties

• Nodes remain fixed in space while amplitude oscillates in time.
• Spatial frequency and temporal frequency follow the generating traveling waves.
• Energy density alternates between kinetic-like and potential-like forms across a period.

$$\text{nodes at } x = n\pi/k$$

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🎯 Special Cases and Limits

• Boundary nodes: $k = n\pi/L$ enforces nodes at $x=0,L$ for integer $n$.
• If $\omega=0$, the pattern reduces to a static sinusoid.
• Unequal counterpropagating amplitudes produce partial standing waves: nodes weaken but the interference structure persists.

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🔗 Related Functions

Single sine or cosine modes represent pure standing components; complex exponentials capture the underlying traveling waves; multi-tone superpositions build more complex standing patterns.

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Usage in Oakfield

Oakfield provides standing waves as a built-in stimulus variant:

• stimulus_standing operator generates standing-wave patterns (sinusoidal family) with configurable wavenumber, frequency, and phase, and can write into real or complex fields.
• Standing-wave stimuli are commonly used for quick diagnostics of oscillatory dynamics and interference patterns under operator graphs.

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Historical Foundations

📜 Vibrating Strings and Modes

Standing wave modes arise naturally in classical studies of vibrating strings and resonant cavities, where boundary conditions quantize admissible wavenumbers.

🌍 Modern Perspective

Standing waves are a standard basis for modal decomposition and spectral methods on bounded domains.

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📚 References

• Strauss, Partial Differential Equations
• Stein & Shakarchi, Fourier Analysis