WavesSurface Waves

Introduction

• Common Phenomena

– Seismic, electromagnetic– Earth, ocean and atmosphere

• In Ocean: travel through density discontinuities

• Surface (air – sea interface)

• Internal (thermocline)

Generated by Disturbing Forces and Propagated by Restoring Forces

• Disturbing forces– Wind (most common)– Sea floor motion due to seismic activity (tsunamis)– Ice falls, submarine landslides (gravity waves)– Tides, river outflows (gravity waves)

• Restoring forces

– Surface tension (capillary waves)– Gravity (wind waves, tsunamis)– Coriolis (tidal waves)

Wave Motion

• Extremely complex, especially in ocean– Diverse forcing mechanisms

– Superimposed frequencies / wavelengths

• Study simplified system (ideal waves) and apply lessons to real waves in nature

Wave Definition• Waves transfer kinetic energy through a

medium (air, water) without displacement

• Kinetic Energy = ½ * M * V2

– M = mass (kg)

– V = velocity (m / sec)

– E = mass * acceleration = kg m2 sec-2 (joules)

Ideal Waves

Wave descriptors:

wave height (H) & wave amplitude (H / 2)wave length (L) & wave steepness (H / L)wave period (T) and wave frequency (1 / T)

Ideal Wave Example

NOTE: frequency in cycles per second (hertz)

• Given a wave length (L) of 10 meters

• And a wave period (T) of 10 seconds

• Wave frequency equals ?

1 / T = 0.1 seconds-1 (hertz)

Ideal Progressive Wave Example

• Progressive – energy moving across or though a medium– water oscillates in a rotary fashion

• Wave Number = k = cycles / distanceWave Number = k = 2Π / L

Kinds of Waves Based on Motion

• Standing – surface moves up and

down between stationary “nodes” with maximum oscillation at “antinodes”

• Examples:

– Two progressive waves moving in opposite directions that cancel individual motions

– Oscillation of sea surface around a stable node– http://www.physicsclassroom.com/mmedia/waves/harm4.cfm

Progressive Waves (With Rotary Motion)

• Wave length described by wave number:

(K) = 2π / L (radians m-1)

(example: if L = 10 m, K = 0.68 radians m-1)• Wavenumber measures rotation (in radians) per meter

• NOTE: There are 2π radians / 360 degrees and a radian is the arc length of the radius of a circle

• Wave frequency described by radian frequency

(ω; omega) = 2π / T (radians sec-1)

(example: if T = 10 sec, ω = 0.68 radians sec-1)

Celerity (C)

• Celerity is the speed of energy motion

• Used to describe wave speed

– Simple description of celerity

• C = L / T

–Example: 10 m /10 sec = 1m/sec

• C = ω / k

–0.68 rads sec-1 / 0.68 rads m-1 = 1 m/sec

Kind of Waves Based on Wave Period

Capillary Waves• Smallest waves (L < 2cm)

• Disturbing force generally wind

• Restoring force surface tension

• Rapidly dissipate if they do not grow

• Don’t display typical sinusoidal shape

Wind Waves

• Disturbing force: wind

• Restoring force: gravity

• Medium size waves

• Chop and ocean swell

• Wavelength:

2 cm to 100’s of m

• Height up to 40 m

Seiches (Standing Waves)• Disturbing force:

wind, tsunami, tides• Restoring force:

gravity• Wavelength:

10’s km to 100’s of m

• Example:Lake Erie after strong wind

Tsunamis

• Disturbing force: vertical motion of the seafloor earthquakes, landslides

• Restoring force: gravity

• Wavelength: 100’s km

• Very small amplitude

• “Shallow water” waves, that pile up at shorelines

Tidal Waves

• Disturbing force: gravity and inertial forces

• Restoring force: coriolis

• Wavelength:up to 20,000 km

(half Earth circumference)

Wave Size and Shape

Sinusodial Wave Size and Shape

Sinusodial Wave Size and Shape

Trochoidal Wave Size and ShapeThe peak becomes narrower and steeper as the amplitude increases for a given wavelength.

Experimental evidence from wave tanks shows that a ratio of 1:7 for peak height to wavelength is the maximum and that an angle of 120° is the minimum angle for a peak.

Above this ratio the peaks became unstable.

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Water Motion in Waves

• Energy moves forward in progressive waves, with little permanent water displacement

• Circular (or orbital) water motion

• At surface diameter of orbit equal to h

• Orbits decrease in diameter with depth

• Orbital motion ceases completely at L/2

• Rotation or cycle time constant with depth

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Water Motion in Progressive Waves

Slight surface forward motion called Stokes Drift.

However, there is a equal and opposite motion at depth (not shown here).

NO NET WATER MOVEMENT

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Deep, Shallow and Transitional Waves• Waves categorized on

basis of ratio of wavelength depth to water depth:

• Deep water wave (depth > L / 2 “wave base”)

• Shallow water wave (depth < L / 20)

• Transitional wave (L / 2 > depth > L / 20

(shallower than L/2 but deeper than L/20)

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Waves and Seafloor Depth

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Wave Generation by Wind• Exact mechanism not well understood• Theories cannot fully account for rapid

build-up occasionally seen in nature

• The process:– Wind generates capillary waves– Continued wind gravity waves– Continued wind larger waves– Continued wind white capping– Eventually energy input balanced

by friction in white capping

• When this balance achieved sea called fully developed

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Wind Wave Formation

Wave development by the windSheltering model

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“Fully Developed Sea” and the Wave Spectrum

• A theoretical construct

• Rarely occurs due to changing sea and atmosphere conditions– Wind direction and speed

– Waves move away from generation area

• Factors used to predict wave heights:– wind speed

– wind duration

– wind fetch

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Wave Spectrum

NOTE: Higher frequency equals shorter period

commonly used term“Significant wave height”

Highest 1/3 of waves

Largest Recorded Waves:

Pacific Oceansteady wind > 60 mph

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Wind Wave Generation

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Wave Dispersion

storm generating area away from storm

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Wave Energy

• Energy = 1/8 * ρ * g * h2

• This yields joules m-2 (the energy “density” contained in the waves per m2 of ocean surface)

• “Energy density” (joules / m2)

is a function of wave height

• In the open ocean wave height is not directly related to L or T due to the spread of wave energy over the ocean as waves travel from region of generation and spread out perpendicular to the direction of wave motion

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Wave Energy

• Wave energy in surface ocean ~ 1018 joules• Wave energy dissipated on coasts ~ 2 *1012 W

• Comparison to civilization energy demand– USA energy consumption:

3.5 * 1012 watts– Total global energy consumption:

1.6 * 1013 watts

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Wave Interference & Group Velocity

• individual waves interfere

• can increase or decrease wave heights

• can produce random variations and occasionally rogue waves

Evidence of a 100-meter wide “wall of water” moving at 40 miles per hour

(Donelan & Magnusson, 2017)

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Wind Wave Sets

• Wave trains spread energy over the sea as they move from area of generation

• They travel according to the dispersion relation (wavelength and speed)

• Bigger waves are faster than smaller ones

• With enough travelled distance the larger waves leave smaller ones behind

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Wave Trains

• generate wave trains where wave energy travels half as fast as individual waves

Global Wave Heights

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Wave Domains in Hawaii

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Wave Domains in Hawaii

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Fate of Waves

• As waves enter shallow water, wave speed function of water depth, and waves slow

• Wave length decreases and wave height increases, but period remains constant

• When height reaches 1/7 of length, wave becomes unstable and breaks

• Waves generally break when H/L exceeds 1/7

• Energy released as friction when wave breaks

• Turbulence (mixing) and net transport

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Wave Speed

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Wave Speed

As waves enter shallow water(defined as a ratio of D:L)

speed decreaseswavelength decreases

height increases

But period remains constant

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Wave Refraction

• Like other waves, ocean waves change direction when their speed changes

• Waves wrap around headlands and islands

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Wave Refraction

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Wave Difraction

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Island – Wave Interactions

• Ocean waves are influenced by shallow bathymetry and coastlines, including islands

• Waves wrap around and bounce off islands

waves and currents

• Waves are also affected by currents• Key to understanding interaction is to remember that T remains

constant• Currents moving in direction of progressive waves expand

wavelength and lower wave height• Currents moving in opposite direction of progressive waves shorten

wave length and increase wave height• Where ocean moves L=(C+Cocean)/T• If Cocean is negative (opposite direction of wave motion), L decreases

and h increases• If current is fast enough waves eventually break even in deep water

– Particularly important in Agulhas Current

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Division on the basis of depth based upon calculation of wave speed

• This is obviously a practical subject

• When will storm waves reach a coast?

• When will a tsunami arrive?

• Wave speed generally denoted by C (celerity) which means speed in performance of work

• Simple wave speed C=L/T

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General Equation of Wave Speed

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Know this equation

Hyperbolic Tangent

When x < 0.3 the Tanh(x) = xWhen x > 3.0 the Tanh(x) = 1

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Shallow water wave speed

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Depth in meters and C in m sec-1

Derive this equation

Deep water wave speed

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Again L in meters and C in m sec-1Derive this equation

Relationship between T and L for deep water waves

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C= 1.25 *L1/2

and C=L/TOr 1.25 *L1/2 = L/TOr 1.56*L = L2/T2

OrL=1.56 *T2

Therefore longer waves always have a longer T and longer waves move fasterTherefore waves “disperse” over the sea with longer waves moving ahead of shorter ones

Note: this is for deep water wavesAlso Note: Wave height not a function of L or T

Dispersion relation

• Equation for deep water wave speed is often called the “Dispersion Relation”

• Generally written as

• But same as this expression

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Called a dispersion relation because these waves “disperse” over the ocean surface

depending upon L and/or T

Dispersive and non-dispersive

• Deep and shallow water waves sometime also called dispersive and non-dispersive respectively

• Dispersive (i.e., deep water)

– C dependent on L or T

• Non-dispersive (i.e., shallow water)

– C dependent upon depth & independent of L or T

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