A laboratory flume holds water of depth

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Long waves of finite height: bores, jumps and solitary waves

Problem

h = 0.40\,\mathrm{m}

. A paddle stroke launches a single hump of amplitude

A = 0.10\,\mathrm{m}

that runs along the flume without changing shape. ((a)) Estimate its speed

V

and width

L

using the solitary-wave relations

V = \sqrt{gh}(1+A/2h)

and

L = \sqrt{4h^3/3A}

. ((b)) Taking the wave's inverse width,

k \sim 1/L

, as its effective wavenumber, estimate its Ursell number

\mathrm{Ur} = (A/h)/(kh)^2

, show that this value does not depend on

A

h

, and say what that tells us about the balance between steepening and dispersion.

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