Crystal Lattice Waves

Lesson 1: Phononic Crystals and Band Theory

Graduate Level • Advanced Mechanics

The Silence Gap

"Can we build a material that is completely silent to specific frequencies of sound, regardless of how loud the source is?"

Conventional Insulation

Absorptive materials dissipate energy into heat. Thickness determines effectiveness.

Phononic Crystals

Periodic structure prevents wave propagation entirely through destructive interference.

The Quantum Analogy

Electronic Band Theory

Potential: \(V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})\)

Schrödinger Eq. solution via Bloch's Theorem.

Acoustic Band Theory

Density: \(\rho(\mathbf{r}) = \rho(\mathbf{r} + \mathbf{R})\)

Modulus: \(C(\mathbf{r}) = C(\mathbf{r} + \mathbf{R})\)

Key Parallel: Bloch's Theorem

\[ \psi(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_\mathbf{k}(\mathbf{r}) \] Where \(u_\mathbf{k}(\mathbf{r})\) inherits the periodicity of the lattice.

Governing Wave Equations

For an inhomogeneous, non-dispersive fluid medium:

\[ \nabla \cdot \left[ \frac{1}{\rho(\mathbf{r})} \nabla p(\mathbf{r}, t) \right] - \frac{1}{B(\mathbf{r})} \frac{\partial^2 p(\mathbf{r}, t)}{\partial t^2} = 0 \]

Mass Density \(\rho\)

Varies periodically in space.

Bulk Modulus \(B\)

Varies periodically in space.

The Brillouin Zone

Reciprocal Space

The First Brillouin Zone (FBZ) is the uniquely defined primitive cell in reciprocal space.

Reduces redundant wave vector \(\mathbf{k}\) information.
Zone boundaries defined by Bragg condition.
Band structure is plotted along high-symmetry paths (\(\Gamma \rightarrow X \rightarrow M \rightarrow \Gamma\)).

Standard 2D Square Lattice FBZ

Visualizing Band Gaps

ΓXMΓ

Forbidden Frequencies

Frequencies inside the gap cannot propagate as real-valued wave vectors.

Evanescent Decay

Amplitude decays exponentially within the crystal structure.

Numerical Tools

PWE Method

Plane Wave Expansion

Convert spatial differentials into algebraic eigenvalue problems in reciprocal space.

FEM Analysis

Finite Element Method

Model complex geometries (scatterers) with high precision using Floquet-Bloch boundary conditions.

TMM

Transfer Matrix Method

Ideal for 1D multilayered structures; relates pressure/velocity at layer boundaries.

Discussion Prompt

"Given a bandgap from 2kHz to 5kHz, how would you engineer a material to shift this gap to protect a submarine from 1kHz sonar?"

Lattice Scaling

Material Density

Inclusion Geometry

Acoustic Metamaterials and Non-Linear Mechanics

Crystal Lattice Waves

Refraction Reversed

Invisible Sound Paths

Eternal Solitary Waves

Unstoppable Edge Waves

Crystal Lattice Slides

Phononic Bandgap Teacher Guide

Brillouin Zone Lab Sheet

Backward Bending Slides

Negative Index Teacher Guide

Snell Simulation Task Worksheet

Invisible Sound Slides

Transformation Mapping Teacher Guide

Cloaking Shell Designer Worksheet

Eternal Solitary Waves Slides

Non-Linear Mechanics Teacher Guide

KdV Equation Problem Set Worksheet

Unstoppable Sound Slides

Topological Phase Teacher Guide

Edge State Analysis Worksheet

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