Motion in the Sea -- Waves

• wind not only produces currents, it creates waves: as wind blows across the smooth water surface, the friction or drag between the air and the water tends to stretch the surface, resulting in wrinkles
• surface tension acts on these wrinkles to restore the smooth surface -- these are waves
• as waves form, the surface becomes rougher and it is easier for the wind to grip the roughened water surface and intensify the waves
• highest part of the wave is called the crest; lowest part that is depressed beneath the surface is called the trough; wave height is the overall vertical change in height between the crest and the trough (= 2 x amplitude)
• distance between two successive crests is the length of the wave or wavelength (L)

• the time required for two successive crests or two successive troughs to pass a point in space is called the period (T); the number of peaks (or troughs) that pass a fixed point per second is the frequency
• steepness is the wave height divided by the wave length (H/L) and is NOT the same thing as the slope between a wave crest and its adjacent trough

 

 

I. Characteristics of Wave Motion

1. a wave transfers a disturbance from one part of a material to another (i.e., the disturbance caused by a stone dropping into a pond is transmitted across the pond by ripples)
2. the disturbance is propagated through the material without any substantial overall motion of the material itself
• when under a wave crest, the water moves up and forward; under the troughs, it moves down and back; thus, on the whole, water particles don't go anywhere at all as the wave passes, but move in circles -- however, the energy of this movement is transmitted to succeeding water particles
• it is this orbital motion of the water particles that causes an object to bob up and down, forward and backward as waves pass under it; however, this motion does not extend far downward (the particles experience a displacing force and a restoring force)
3. the disturbance is propagated without any significant distortion of the wave form
4. the disturbance appears to be propagated with constant speed

 

 

II. Wave Forms -- Idealized

• assume the wave form is sinusoidal
• thus the wave's displacement is a cyclical variation in the water level caused by the wave's passage
• but wave form is really trochoidal (line form traced by a point on a rolling wheel)
• water particles in a wave over deep water move in an almost closed circular path: in the crests the particles are moving in the same direction as wave propagation, whereas in the trough, they are moving in the opposite direction
• however, the orbits are only approximately circular -- there is a small net component of forward motion, particularly in large amplitude waves

 

 

this small net forward movement is called wave drift or mass transport and occurs when the depth of the water is ¼ 1/2 the wavelength
	in shallow water, where depth is less than one-half the wavelength, orbits are progressively flattened with depth and there is little wave drift             water just above the seafloor cannot move in a circular path and can only move back and forth

• shallow water waves are found in waters shallower than 1/20th of the original wavelength
• transitional waves travel through water deeper than 1/20th the original wavelength, but shallower than 1/2
• wind waves/deep water waves travel through water deeper than 1/2 the original wavelength

 

 

III. Wave Speed

• in deep water (>1/2 wavelength), wave speed can be represented by the equation:
c = 

• where the acceleration due to gravity g is 9.8 m/sec²; L=wavelength

• in shallow water (<1/20 wavelength), water depth is the only variable that affects wave speed:
c = 

• where the acceleration due to gravity g is 9.8 m/sec² and d=depth

• in transitional depths (<1/2 wavelength, >1/20 wavelength), the equation is much more complex:
c = 

• where tanh is a mathematical function known as the hyperbolic tangent and L=wavelength; here you would need to know the wavelength and depth and have access to hyperbolic tangent tables

 

 

• because wavelength is very difficult to measure in the field, we estimate the wavelength by measuring the period (generally the wavelength increases as the period increases and so does the speed):

where T=period and g=9.8 m/sec² (or 32 ft/sec²) -- the equation can be simplified to L = 1.56 T² (for meters) or 5.12 T² (for feet)

• deep water waves that have the greatest wavelengths and longest periods, travel the fastest and are the first to arrive in regions distant from the storm which generated them
• if you toss a stone into a pond, a band of ripples is produced -- this band gets wider with increasing distance from the original disturbance and the ripples of greater wavelength progressively outdistance those with shorter wavelengths (they outrun the shorter ones)
• this is dispersion (separation of waves by their differing rates of travel) and it produces swells

 

• progressive groups of swell with the same origin and wavelength are called wave trains
• the leading wave in a train is drained of energy (because it must set up the circular motion), but after the wave train passes, it leaves behind enough energy to generate a new wave
• as a wave advances through the group, each wave moves through the group to die out at the front -- the distance traveled by each individual wave as it travels from rear to front of the group is twice that traveled by the group as a whole
• hence the wave speed is 2 x the group speed or the group speed is 1/2 the wave speed

 

• it is inevitable that swells from different disturbances will run together -- when this happens, they will interfere with each other
• when crests of two wave trains coincide, the amplitudes are added (constructive interference)
• when the wave trains are out of phase--the crests coincide with the troughs -- the amplitudes cancel out (destructive interference); get mixed interference when crests and troughs do not coincide
• thus, the wave trains interact, lose their individual identities and form a new group of waves separated by regions free from waves

• in shallow water, wavelength becomes less important, so wave speed becomes closer to group speed and eventually all waves travel at the same depth-determined speed -- here there is no interference and each wave in essence represents its own group

 

 

IV. Wave Types

• all waves are progressive waves -- that is energy moves through or across the surface of the material
• body waves are those that travel through the material (P & S seismic waves are examples; surface waves are another but they occur at the interface between two bodies of fluid {and thus include internal waves which occur between two layers of ocean water of differing density})
• there are two restoring forces that maintain wave progress: a) gravitational force exerted by the earth; b) surface tension of water molecules (keeps water sticky)
• thus surface waves are divided into capillary waves which have wavelengths less than 1.7 cm and are principally maintained by surface tension and gravity waves which have wavelengths greater than 1.7 cm and are principally maintained by gravity forces
• capillary waves are the first to form when the wind blows -- they are important for transferring energy from air to water to drive currents, but have little energy
• the capillary wave becomes a wind/gravity wave when its wavelength exceeds 1.7 cm
• if the wind continues to blow in deep water, the wave becomes larger because its crest is thrust higher and higher into faster wind and it extracts more energy from the moving air -- these irregularly peaked waves in the area of wind wave formation are called sea
• wave energy is proportional to wave height squared
• 3 factors affect the growth of wind waves or of sea:
1. wind strength --wind must be moving faster than the wave crests for energy to be transferred
2. wind duration --high winds that do not blow for long will not generate large waves
3. fetch --the uninterrupted distance over which the wind blows without significant change in direction
• a strong wind must blow continuously in one direction for 3 days to general the largest waves -- a fully developed sea is the maximal wave size theoretically possible
• but this doesn't happen frequently and even in gale winds, wave growth ceases because energy is lost (attenuated) in 3 ways:
1. some of the wind energy is transferred to the ocean surface to produce a surface current
2. some wind energy is dissipated by friction
3. energy is lost via white capping -- the breaking of the tip of the wave crest because it is being driven forward by the wind faster than the wave itself is traveling (this relates to the 1:7 ratio of wave height to wavelength -- if the exceeds this ratio, the wave will break, dissipating the excess energy from the wind)
• most wind waves eventually reach land and dissipate all their order and energy
• this process changes the form of the wave from a deep water wave to a transitional wave to a shallow water wave
• when the water depth is <1/2 the wavelength, the wave "feels" the bottom; this interrupts the circular motion of the water molecules and flattens it near the bottom
• the wave's energy must now be packed into less water depth, so the wave crests become peaked

• interaction with the bottom slows the wave, but waves behind this first wave continue at their original speed -- thus period remains unchanged while wavelength decreases; the wave then becomes too high for its decreasing wavelength
• depth continues to decrease and the crest of the wave breaks -- the turbulent mass of water rushing shoreward after the break is known as surf and the surf zone is that region between the shore and the breaking waves
• there are 4 types of breaking waves:
1. spilling breakers -- the crest moves forward faster than the wave as a whole; foam eventually covers the leading face of the wave; characteristically found on a gently sloping shoreline
2. plunging breakers -- the crest curls over and plunges downward with considerable force
3. collapsing breakers -- similar to plunging breakers, but instead of the crest curling over, it collapses; occur on beaches with moderately steep slopes
4. surging breakers -- crests remain relatively unbroken as the waves slide up the steep beach

V. Wave Refraction

• when a wave line approaches a shore at an angle, the line does not break simultaneously
• the part of the wave line in shallow water slows down but the attached segment in deeper water does not, so the wave bends or refracts
• produces odd surf patterns

VI. Seiches (sayshes)

• caused by rocking of water confined in a small space -- this rocking occurs at a specific resonant frequency
• because the wave oscillates vertically with no progressive movement, it is called a standing wave
• standing waves are the sum of two progressive waves of equal dimension but traveling in opposite direction (ex. seiches); occur in lakes, bays, estuaries and harbors which are open to the sea at one end
• at either end of the basin the water level is alternately high and low -- where the water level is constant (the node), the horizontal flow of water from one end of the basin to the other is greatest
• where the fluctuation of water is greatest (antinode), there is minimal horizontal movement of water

 

• but if a basin is open at one end, it is possible for a node to occur at the entrance of the basin and an antinode at the landward end; water levels will then rhythmically oscillate between node and antinode and will result in damage on the landward side

VII. Tsunami

• are long-wavelength, shallow water progressive waves caused by rapid displacement of ocean water
• displacement is caused by the sudden vertical movement of the earth along a fault line (which is really is a seismic sea wave); can also be caused by landslides, icebergs falling from glaciers, and volcanic eruptions
• are shallow water waves because their wavelengths are so long -- deep water waves are found in water deeper than 1/2 their wavelength, but if a tsunami has a wavelength of 200 km, it would need 100 km of depth and the deepest oceans rarely exceed 11 km
• thus, their speeds are described by c = 
• if generated in a typical Pacific abyssal depth of 4,600 m, the speed would be 9.8 m/sec X 4,600 m or 212 meters per second; at this speed a wave generated near the Aleutian Islands would take 5 hours to travel to Hawaii
• when it reaches shore, it will behave as follows: its period will remain constant, while its wavelength (which is already long) will decrease
• velocity will decrease
• furthermore, the energy of the wave remains the same and this increases the wave height
• in some cases the wave height can reach 30 m and it proceeds towards shore as an onrushing flood of water
• in addition, because the disturbance will generally set up wave trains, these trains will travel in groups with intervals of generally 15 minutes; thus after one tsunami hits the coast, all is not over