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ENVIRONMENTAL FLUID MECHANICS

Waves

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

The Great Wave off Kanagawa, woodblock print by the Japanese artist Hokusai

Ripple waves in a pond

What we notice in this video clip:

- Structured waves emerge from messy turbulence.- Turbulence does not travel, but waves do.- Corollary: Turbulence is unable to transport energy,

but waves are.- Longer waves travel faster than shorter waves.

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2D surface waves

Governing equations:

gz

p

z

ww

x

wu

t

w

x

p

z

uw

x

uu

t

uz

w

x

u

0

0

1

1

0

Assume:

Frictionless dynamics 0,0,0/ vy constant

for

),,(

),,(

),,(

tzxp

tzxw

tzxu

Additional simplification:

Small-amplitude waves to permit linearization of the problem.

Mathematically: Small-amplitude wave weak velocity , the wave speed.

→ Linearized equations:

gz

p

t

w

x

p

t

uz

w

x

u

0

0

1

1

0

gz

p

z

ww

x

wu

t

w

x

p

z

uw

x

uu

t

uz

w

x

u

0

0

1

1

0

,L

U W cT

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Boundary conditions:

Free surface at top:

Impermeable bottom:

No need for initial conditions and boundary in x.We will impose periodicity in both x and t.

0at0 zw

),(at

),(at

txaHzx

au

t

aw

txaHzpp atm

Hzt

aw

Hzgapp atm

at

at0Linearize

Look for periodic waves by seeking a solution of the type

0

( , , ) ( ) cos( )

( , , ) ( ) sin( )

( , , ) ( ) ( ) cos( )

( , ) cos( )atm

u x z t U z kx t

w x z t W z kx t

p x z t p g H z P z kx t

a x t A kx t

T

k

2

2

with k = wavenumber, = wavelength

= angular frequency, T = period

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Dispersion relation

Relation between wave speed and wavelength

Particle trajectories

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World War IIaction over theMediterranean Sea

In June 1943, torpedoes from the British submarine United, submerged at (A), struck a 5,000-ton Italian supply ship at (B). Waves from the explosion are noted at the edges of the photograph. Torpedoes that missed the target continued on past the ship along path (B)-(D).

The disabled ship struggled on only to sink a while later, at point (C).

The submerged United was then attacked by an Italian airplane. A bomb was dropped at (E), but the British submarine had moved away and returned safely to port.

A Royal Air Force Spitfire on reconnaissance caught the outcome on film. This photo was classified for decades.

Wave refraction

Shallow-water waves propagate more slowly in shallower water (closer to shore) than in deeper water (further offshore), causing the wave fronts to turn.

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In areas where waves from storms in the open ocean approach shallower waters the waves can be refracted and diffracted.

Aerial photograph of an area near Kiberg on the coast of Finnmark (northern Norway), taken on 12 June 1976 by Fjellanger Widerøe A.S.

Wave diffraction

Explain what you see

We notice that outward radiating waves originate from the two smaller boats. The wave patterns are concentric, centered on the boats.

The explanation is that these two small boats are being rocked by incoming waves, and their rocking generates new sets of waves.

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Seiches

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A tsunami is essentially a half-wave, with a crest and no cancelling trough to follow.

Upon approaching a shore, a tsunami behaves as a shallow-water wave with no dispersion (= unchanging shape) and travelling at speed , with h = tsunami surge height. For h = 2 m, the speed is 4.4 m/s = 10 mph.

gh

Tsunamis

Another example of a half wave is the tidal bore in some estuaries.

Tidal Bores

Moncton River, New Brunswick, Canada

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Internal Waves

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Manifestation of internal waves in a cloudy atmosphere

Mountain waves

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Sand storms

A sand storm approaching Riyadh, Saudi Arabia

First-mode internal seiche in a rectangular basin

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Internal seiching in Lake Cayuga (Ithaca, New York)