Wave Physics: Acoustics, Elastic Waves, and Dispersion

Manual: General · Subject: Applied Physics

Develop advanced wave theory for propagation, superposition, dispersion, and scattering in real media.

General Wave Equations

Canonical form

Many systems reduce to $\partial_t^2 \psi = v^2 \nabla^2 \psi$ or a generalized dispersive form. In realistic media, attenuation and frequency dependence make the phase velocity $v_p=\omega/k$ and group velocity $v_g=d\omega/dk$ distinct.

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Dispersion

Dispersion means that different Fourier components travel at different speeds, causing pulse broadening and waveform distortion.

Wave quantities

Quantity	Expression	Meaning
Phase velocity	$v_p=\omega/k$	Speed of constant phase
Group velocity	$v_g=d\omega/dk$	Energy/envelope speed
Wavenumber	$k=2\pi/\lambda$	Spatial frequency
Angular frequency	$\omega=2\pi f$	Temporal frequency

Elastic wave families

In solids, longitudinal and transverse modes arise from compressional and shear restoring forces. In anisotropic media, the wave-speed tensor leads to birefringence, mode conversion, and direction-dependent attenuation.

What quantity is defined as $d\omega/dk$ ?

Phase velocity Group velocity Sound speed only Impedance

For a non-dispersive medium, what relationship holds between $\omega$ and $k$ ?

Analyzing a pulse in a dispersive medium

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Step 1: Decompose the signal into Fourier components.
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Step 2: Apply the dispersion relation $\omega(k)$ to each component.
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Step 3: Track phase accumulation $e^{i(kx-\omega t)}$ .
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Step 4: Reconstruct the signal and evaluate pulse spreading.

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Practical consequence

Dispersion engineering is crucial in telecommunications, phononic crystals, ultrafast optics, and seismic wave analysis.

Pulse broadening in a dispersive medium is most directly caused by differences in what across frequency components?

Mass Propagation speed Charge Entropy

Neuroanatomy Explorer

Memory Deck

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Concept Constellation

Wave Physics: Acoustics, Elastic Waves, and Dispersion

General Wave Equations

Canonical form

Elastic wave families

Analyzing a pulse in a dispersive medium