Physics calculator

Wave Celerity Calculator

Compute pipe-wave celerity c = √[(K/ρ)/(1+(K D)/(E e))]. Runs locally. Not Joukowsky Δp, not Darcy f, and not minor-loss K.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · c = √[(K/ρ)/(1+(K D)/(E e))] (Korteweg–Hall) with bulk and Young's solves; + O3 mpmath tabulated c.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: c = √[(K/ρ)/(1+(K D)/(E e))] (Korteweg–Hall) with bulk and Young's solves; + O3 mpmath tabulated c.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (celerity, solve, alias, awkward, invalid-domain) · Matrix
Deferred: Not Joukowsky Δp, not Darcy, not minor-loss K, not Poisson restraint ψ, and not Q=Av.
Numerical scope: O2: Korteweg–Hall c=√[(K/ρ)/(1+(K D)/(E e))] vs a separate-module identity (≤2 ULP). Not Joukowsky Δp, not Darcy, not minor-loss K, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated wave-celerity vectors. Not catalog K, not pipe-entrance geometry, not sudden expansion (v1−v2)², not Darcy, and not Q=Av.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Korteweg–Hall pipe-wave celerity from fluid bulk modulus, density, diameter, wall thickness, and wall Young’s modulus. Invert K or E.
Scope: Korteweg–Hall wave celerity: c = √[(K/ρ)/(1+(K D)/(E e))]; default ρ=1000 kg/m³; thin wall; ψ=1
Verification: Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · c = √[(K/ρ)/(1+(K D)/(E e))] (Korteweg–Hall) with bulk and Young's solves; + O3 mpmath tabulated c.· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision: 2026-09-26.o2-o3 · 1/1 property · digest f30a3f84b3d2
Legacy regression: 17/17 tests · Production surface contract 5/5
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 4/4 golden · 10/10 CVP boundary · 11/11 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources: ISO 80000-4 — Mechanics — Pressure, density, speed, modulus
Wylie, E.B. and Streeter, V.L. Fluid Transients — Wave speed in a liquid-filled elastic pipe
Sources
Evidence: 2 legacy golden · 10 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 11/11 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Korteweg–Hall celerityc = √[(K/ρ) / (1 + (K D)/(E e))]

Bulk invertK = c² / (1/ρ − c² D/(E e))

Young invertE = K D / (e (K/(ρ c²) − 1))

iThin-wall liquid line, ψ=1. K is fluid bulk modulus, not minor-loss K. E is wall Young’s modulus, not specific energy. Not Joukowsky Δp.

How to use

Enter K

Fluid bulk modulus in Pa. Invert for K on the K tab.

Enter D, e, and E

Inner diameter, wall thickness, and wall Young’s modulus. Invert E on the E tab.

Read c

c = √[(K/ρ)/(1+(K D)/(E e))]. Default ρ=1000 kg/m³.

Example calculations

Common configurations with formula and result.

Steel water main

K=2.2e9 Pa, D=1 m, e=0.01 m, E=2e11 Pa

c = √[(2.2e9/1000) / (1 + 2.2e9×1/(2e11×0.01))]

c ≈ 1023.53 m/s

Invert K

c≈1023.53 m/s with the same D, e, E

K = c² / (1/ρ − c² D/(E e))

K = 2.2e9 Pa

Wave Celerity calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 17/17 tests · Production surface contract 5/5
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: Korteweg–Hall pipe wave speed: c = √[(K/ρ)/(1+(K D)/(E e))] with default ρ=1000 kg/m³ and restraint ψ=1. This page computes c from fluid bulk modulus and wall elasticity, not Joukowsky Δp=ρcΔv.
What it calculates: Korteweg–Hall pipe-wave celerity from fluid bulk modulus, density, diameter, wall thickness, and wall Young’s modulus. Invert K or E.
Inputs: mode?
K|c
D
e
E|c
rho?
Outputs: c
rho
K
D
e
E
mode
model
Formula: c = √[(K/ρ)/(1+(K D)/(E e))]
Assumptions: Korteweg–Hall wave celerity: c = √[(K/ρ)/(1+(K D)/(E e))]; default ρ=1000 kg/m³; thin wall; ψ=1
K is fluid bulk modulus, E is wall Young’s modulus; not Joukowsky Δp=ρcΔv
Not Darcy f, not minor-loss K, not Q=Av, not Y=(F/A)/(ΔL/L0)
Units: SI; c in m/s; K and E in Pa; D and e in m; ρ in kg/m³
Boundary conditions: missing K, D, e, E, or c when required → MISSING_REQUIRED_INPUT
K, D, e, E, c, or ρ ≤ 0 → VALUE_MUST_BE_POSITIVE
c ≥ √(K/ρ) on E invert, or c ≥ √(E e/(ρ D)) on K invert → VALUE_OUT_OF_RANGE
dP/dV, f/Re, K_loss, n, A/Q, v, L, ψ → MIXED_INPUT_ENCODING
joukowsky, darcy, or minor as mode → INVALID_MODE
Example: K=2.2e9 D=1 e=0.01 E=2e11 → c≈1023.53
Validation cases: 3 published on this page · 17/17 tests · Production surface contract 5/5 · View evidence
K=2.2e9 D=1 e=0.01 E=2e11 → c≈1023.53
mode=bulk c≈1023.53 D=1 e=0.01 E=2e11 → K=2.2e9
K=2.2e9 D=1 e=0.01 E=2e11 dP=1e6 → MIXED_INPUT_ENCODING
Sources: ISO 80000-4 — Mechanics — Pressure, density, speed, modulus
Supports: K and E are moduli; ρ is a density; c is a speed; D and e are lengths
Wylie, E.B. and Streeter, V.L. Fluid Transients — Wave speed in a liquid-filled elastic pipe
Supports: c = √[(K/ρ)/(1+(K/E)(D/e))] for a thin-wall line with restraint factor 1
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute Korteweg–Hall wave celerity.

Supported and not supported

Supported — c from K, ρ, D, e, and E · invert K · invert E · API physics.fluid.wave_celerity

Not supported — Joukowsky Δp=ρcΔv, Darcy hf, minor-loss K, Young’s Y from F and ΔL, restraint ψ≠1

Agent / API notes

Capability id: physics.fluid.wave_celerity · tool id: wave-celerity · pin 1.0.0.

{ "K": 2.2e9, "D": 1, "e": 0.01, "E": 2e11 }

Modes: celerity (default), bulk, youngs. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/wave-celerity. There is no /calc/fluid.

Other calculators in this family: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator . Explore all Fluid Mechanics.

Frequently asked questions

Key distinctions behind the calculation.

Is this Joukowsky water hammer?

No. Surge pressure Δp=ρcΔv is /calc/physics/water-hammer. Sending Δp or Δv is MIXED_INPUT_ENCODING.

Is K a minor-loss coefficient?

No. K here is fluid bulk modulus in Pa. Fitting loss is /calc/physics/minor-loss. Sending K_loss is MIXED_INPUT_ENCODING.

Is this Young’s modulus from stress and strain?

No. Y=(F/A)/(ΔL/L0) is /calc/physics/youngs. This page takes wall E as an input, or inverts E from c.

Do you include Poisson restraint ψ?

No. This seed freezes ψ=1 (thin wall, axially constrained). Sending ν or ψ is MIXED_INPUT_ENCODING.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine waveCelerity wrapper.