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monopole_sphere

Compute exact radiated power and far-field pressure for a pulsating sphere from radius, frequency, and surface velocity. Use as a closed-form benchmark for acoustic BEM radiation solves.

Instructions

Exact pulsating (monopole) sphere radiated power + far-field pressure (NO solver) — the closed-form twin the Bempp exterior-acoustics BEM radiation solve (acoustic_radiation_submit) is gated against. A sphere of radius a_m vibrating with uniform surface normal velocity u_amp at freq_hz radiates W = (ρc/2)|U|²(4πa²)(ka)²/(1+(ka)²) and, at range r_m, |p(r)| = ρc|U|·ka/√(1+(ka)²)·(a/r) — both EXACT. The radiation efficiency σ=(ka)²/(1+(ka)²) → 0 (poor sub-wavelength radiator) as ka→0 and → 1 as ka→∞. rho/c default to air at 20 °C. A BEM Neumann (velocity) solve must reproduce W and |p(r)|.

Returns {a_m, freq_hz, k_per_m, ka, u_amp, rho, c, radiation_efficiency, radiated_power_w, surface_pressure_abs, r_m, farfield_pressure_abs, farfield_pressure_x_r, fidelity, band_pct, valid_range_ok, warnings, escalate_to}.

Input Schema

TableJSON Schema
NameRequiredDescriptionDefault
cNo
a_mYes
r_mNo
rhoNo
u_ampNo
freq_hzYes

Schema Changelog

Changes observed during successful MCP inspections.

  1. First observed

TDQS

A4.6/5.0
Behavior5/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

With no annotations, the description carries the full behavioral burden and does so well: it states the tool is exact, solver-free, gives the closed-form formulas, notes that rho/c default to air at 20 °C, and describes the expected BEM reproduction relationship. For a pure analytical calculation, this is unusually transparent.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The purpose is front-loaded, and the formulas, efficiency asymptotics, defaults, and return fields all earn their place. Since there is no output schema, listing the returned fields is necessary rather than redundant.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

The description is highly informative for a six-parameter analytical tool with no annotations or output schema. It still leaves some details implicit, such as units, the behavior when `r_m` is omitted, and the meaning of return fields like `fidelity`, `band_pct`, and `escalate_to`.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters4/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 0%, so the description must define the six parameters. It gives physical meaning for `a_m`, `u_amp`, `freq_hz`, and `r_m`, and notes that `rho`/`c` default to air at 20 °C. It does not fully explain the optional `r_m` null behavior or state units explicitly, leaving a gap despite good overall coverage.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific computation: exact monopole-sphere radiated power and far-field pressure without a solver. It explicitly positions itself as the closed-form twin that `acoustic_radiation_submit` is gated against, so an agent can distinguish it from the BEM sibling.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines4/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

It names the BEM alternative and says this tool is the closed-form reference against which that solve is gated, which gives clear usage context. It does not explicitly state when not to use this tool or give a direct 'use this instead of X when Y' rule, so it falls short of a 5.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.

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