# ⚛  L1 Principle — Acoustic Wave Equation — time-domain pressure field reconstruction

**ID:** `L1-257` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *3D linear acoustic wave propagation*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D pressure field spacetime
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554035

---

## 🧠 1. Introduction

**Acoustic Wave Equation — time-domain pressure field reconstruction** is a **linear inverse problem** whose unknown lives in **3D pressure field spacetime** space, within the **3D linear acoustic wave propagation** sub-domain of **Acoustics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **pressure receivers** sensing mechanism.

The forward operator applies, in order: L · initial condition operator; L · wave propagate operator; S · record · receivers operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · initial condition → L · wave propagate → S · record · receivers → Temporal integration → **y** (detector).

```
y = ∫_t dt `S.record.receivers` `L.wave_propagate` `L.initial_condition` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.initial_condition` | L · initial condition operator |
| `L.wave_propagate` | L · wave propagate operator |
| `S.record.receivers` | S · record · receivers operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | 3D linear acoustic wave propagation |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | 3D_pressure_field_spacetime |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `acoustic_wave` · **Forward operator:** `acoustic_wave_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 30 |
| Fs khz | kHz | 100 |
| C error | — | 0 |
| N samples | — | 512 |
| Rho error | — | 0 |
| N receivers | — | 64 |
| Receiver pos error | — | 0 |
| Boundary reflectivity | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 512 |
| W | px | 64 – 512 |
| Z | — | 32 – 256 |
| Snr db | dB | 0 – 40 |
| Fs khz | kHz | 10 – 1000 |
| C error | — | 0.0 – 0.1 |
| N samples | — | 128 – 4096 |
| Rho error | — | 0.0 – 0.1 |
| N receivers | — | 16 – 512 |
| Receiver pos error | — | 0.0 – 0.02 |
| Boundary reflectivity | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x61eda6d05bf44aae410231c7f4aab76fade90cf4c14f16b7fca3f1f32a80ce19`
- **Chain tx hash:** `0xef08073047b4e18f6d23fd7bd5e7b1edd900d26277384f73f0daab71ac1ef75b`
- **Chain block:** `41554035`

---

## File Mapping

This bundle consists of: `L1-257.md`, `L1-257.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-257.md` | Source of truth — edit this | Human or LLM |
| `L1-257.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-257`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._