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civilboth

Linear Wave Theory

Solves the linear wave dispersion relation using Newton-Raphson iteration to compute wavelength, phase celerity, and group velocity for monochromatic waves. Also derives wave steepness and relative depth (d/L) for water depth classification. Follows the USACE Coastal Engineering Manual Part II Chapter 1 formulation applicable across deep, intermediate, and shallow water regimes.

both: The model must reason numerically unaided.

Standards

USACE Coastal Engineering Manual (CEM)

How this task is generated

One template produces many comparable benchmark tasks while keeping the scoring contract fixed.

01
Template
The reusable contract shown on this page.
02
Scenario
An archetype and site context are sampled.
03
Difficulty tier
Inputs may be hidden at harder tiers.
04
Task prompt
The model responds with the declared outputs.

Parameters

Inputs the model receives, and the outputs it is scored on.

Inputs

3 inputs

Always given

Included directly in every task prompt.

Water depth
water_depth_m
Water depth d
0.5 – 100 m

Derived from scenario

Sampled from the scenario and inferable from its description.

Wave height
wave_height_m
Wave height H
Derived from the archetype scenario.
0.1 – 10 m

Hidden at higher difficulty

Visible in easier tasks and withheld in one or more harder tiers.

Wave period
wave_period_s
Wave period T
Hidden at hard difficulty.
3 – 18 s

Scored outputs

5 outputs

Wavelength

wavelength_m

Wavelength L (m)

Scores if within ±5% of the reference value.

Wave celerity m per s

wave_celerity_m_per_s

Wave phase celerity C = L / T (m/s)

Scores if within ±5% of the reference value.

Group velocity m per s

group_velocity_m_per_s

Group velocity C_g = n * C (m/s)

Scores if within ±5% of the reference value.

Wave steepness

wave_steepness

Wave steepness S = H / L (dimensionless)

Scores if within ±2% of the reference value.

Relative depth

relative_depth

Relative depth d/L (dimensionless); >0.5 deep, <0.05 shallow, else intermediate

Scores if within ±2% of the reference value.

Difficulty

Each template is sampled at three tiers. Harder tiers may hide inputs, forcing the model to infer them from the scenario description.

easy

All inputs given

Moderate period and depth — all parameters given, straightforward dispersion solve

medium

All inputs given

Extreme depth ratio conditions — all parameters given, deep or very shallow regime

hard

Some inputs hidden

Wave period hidden — agent must estimate from wave height via empirical wind-wave relationships (T approx 3.5-5.5 * sqrt(H))

Hidden inputs

Wave period swave_period_s

Prompt replacement text

The site is {{ archetype.description }} ({{ archetype.site_context }}). Estimate the wave period from the given wave height using standard empirical relationships for the wave regime.

Task bundle

The exact instruction and parameter contract used to generate this task, pinned to the published library source.

/workspace

instruction.md

Teal lines show Jinja input conditions, not task visibility policy. A line renders only when that input or tool is visible.

1You are a senior coastal engineer specializing in wave mechanics and nearshore hydrodynamics.2 3## Problem4 5Calculate wave properties using linear (Airy) wave theory by solving the dispersion relation for the given wave period and water depth.6 7## Given8 9| Parameter | Value | Unit |10|-----------|-------|------|11| Wave period (T) | {{ wave_period_s }} | s |12| Water depth (d) | {{ water_depth_m }} | m |13{% if wave_height_m is defined %}14| Wave height (H) | {{ wave_height_m }} | m |15{% endif %}16{% if archetype_description is defined %}17 18### Site Conditions19 20{{ archetype_description }}21{% endif %}22 23{% if tool_available %}24## Available Tool25 26A linear wave theory calculation tool is available at `/workspace/{{ meta.name }}_calc.py`. Run it with:27 28```bash29python3 /workspace/{{ meta.name }}_calc.py --help30```31 32You may use this tool to verify your calculations or compute values directly.33{% endif %}34 35## Required36 37Calculate the following:38 391. Wavelength L (m)402. Wave phase celerity C (m/s)413. Group velocity C_g (m/s)424. Wave steepness S = H / L (dimensionless)435. Relative depth d/L (dimensionless)44 45## Applicable Standards46 47- USACE Coastal Engineering Manual (CEM) Part II Chapter 148 49## Constraints50 51- No internet access is available. Work from engineering knowledge and the provided tool.52- Use the following procedure:53  1. **Deep-water wavelength** (initial estimate):54     - L₀ = g × T² / (2π)55     - where g = 9.81 m/s²56  2. **Solve the dispersion relation** iteratively (Newton-Raphson):57     - L = L₀ × tanh(2πd / L)58     - Define f(L) = L − L₀ × tanh(2πd / L)59     - Derivative: f'(L) = 1 + L₀ × (2πd / L²) / cosh²(2πd / L)60     - Update: L_{n+1} = L_n − f(L_n) / f'(L_n)61     - Start from L₀ and iterate until convergence (|L_{n+1} − L_n| < 10⁻⁶)62  3. **Wave celerity**:63     - C = L / T64  4. **Group velocity**:65     - k = 2π / L66     - n = 0.5 × (1 + 2kd / sinh(2kd))67     - C_g = n × C68  5. **Wave steepness**:69     - S = H / L70  6. **Relative depth**:71     - d/L (classify: > 0.5 deep, < 0.05 shallow, else intermediate)72 73## Output Format74 75Show your step-by-step working in Markdown. At the end of your solution, include a JSON block with your final answers in exactly this format:76 77```json78{79  "wavelength_m": <numeric_value>,80  "wave_celerity_m_per_s": <numeric_value>,81  "group_velocity_m_per_s": <numeric_value>,82  "wave_steepness": <numeric_value>,83  "relative_depth": <numeric_value>84}85```86 87Write your complete solution to `/workspace/output.md`.88

Scenario archetypes

Each generated task is drawn from one of these realistic scenario bands.

Site contexts ground each scenario in a real locale the model can use to infer hidden values.

Deep ocean swell

deep_ocean_swell

Deep ocean swell with long period waves in open water

tasman-sea-offshoregreat-australian-bightcoral-sea-deep

Parameter ranges

wave_period_s: 10 – 18
water_depth_m: 50 – 100
wave_height_m: 0.5 – 4

Continental shelf

continental_shelf

Continental shelf with moderate depth and intermediate-period waves

gold-coast-shelfsydney-offshore-buoyperth-canyon-approach

Parameter ranges

wave_period_s: 6 – 12
water_depth_m: 10 – 40
wave_height_m: 0.5 – 5

Nearshore approach

nearshore_approach

Nearshore zone with waves approaching the coastline

bondi-beach-nearshoremanly-surf-zonenoosa-heads-approach

Parameter ranges

wave_period_s: 5 – 12
water_depth_m: 2 – 10
wave_height_m: 0.3 – 3

Shallow estuary

shallow_estuary

Shallow estuary or harbour entrance with short-period waves

moreton-bay-entrancedarwin-harbour-channelport-phillip-heads

Parameter ranges

wave_period_s: 3 – 7
water_depth_m: 0.5 – 4
wave_height_m: 0.1 – 1.5

Reef platform

reef_platform

Shallow reef platform with long-period ocean swell

ningaloo-reef-flatgreat-barrier-reef-lagoonheron-island-platform

Parameter ranges

wave_period_s: 8 – 16
water_depth_m: 1 – 5
wave_height_m: 0.3 – 2.5

Example task

tasman-sea-offshore-deep-ocean-swell-preview — hard difficulty, some inputs hidden.

Deep ocean swell with long period waves in open water. tasman-sea-offshore. Required outputs: wavelength_m, wave_celerity_m_per_s, group_velocity_m_per_s, wave_steepness, relative_depth

The model sees

Scenario context and visible inputs.

water_depth_m: 50 to 100 m
wave_height_m: 0.5 to 4 m

Executable tool: linear-wave-theory_calc.py

The model must infer

Inputs withheld at this difficulty.

Wave period s
wave_period_s

Stand-in text in the prompt

The site is {{ archetype.description }} ({{ archetype.site_context }}). Estimate the wave period from the given wave height using standard empirical relationships for the wave regime.

The model must produce

The scored JSON answer schema.

{
  "wavelength_m": <number>,
  "wave_celerity_m_per_s": <number>,
  "group_velocity_m_per_s": <number>,
  "wave_steepness": <number>,
  "relative_depth": <number>
}

wavelength_m · scored within ±5%
wave_celerity_m_per_s · scored within ±5%
group_velocity_m_per_s · scored within ±5%
wave_steepness · scored within ±2%
relative_depth · scored within ±2%

Linear Wave Theory

How this task is generated

Template

Scenario

Difficulty tier

Task prompt

Parameters

Inputs

Scored outputs

Wavelength

Wave celerity m per s

Group velocity m per s

Wave steepness

Relative depth

Difficulty

easy

medium

hard

Task bundle

Scenario archetypes

Deep ocean swell

Continental shelf

Nearshore approach

Shallow estuary

Reef platform

Example task

The model sees

The model must infer

The model must produce