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

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 17/17 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 17/17 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
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 golden · 10 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 163 CURRENT) @ 2026-09-17T13:12:27.589Z
Semantic contract: PASS

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 3/3
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 3/3 · 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: Water hammer, Darcy–Weisbach, Minor loss, Young's modulus .

}

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.