Water Waves

1

2

Herein we begin with a general review of waves. Recognize that in terms of their oceanic spatial and temporal scales, all waves are represented on a portion of this graph. It is not an ordinary linear x-y plot. It is three-dimensional in length [L], time [P], and has relative energy expressed topographically in the vertical direction. The horizontal plane is logarithmic [base 10] in both directions and has the virtue of covering an enormous range of distance, in centimeters, and range of time, in seconds. Note that the longest distance one can measure, point-to-point, is bounded by the size of the Earth. The time scale runs from less than a minute to thousands of years. Astronomical tides with lengths spanning ocean basin widths are represented by the high energy ridges at 12 and 24 hours. Adjacent to the tidal terms is the ocean current realm and the annual variations associated with seasonal climatological differences.

3

Oceanographers define, analyze, and describe ocean waves, processes, and their effects by using specific spatial and temporal characteristics of an ideal mathematical entity; a simple harmonic wave form. These are the characteristics:

• amplitude : the vertical distance between the highest [or lowest] part of a wave and an horizontal reference level [we use the dashed, horizontal line above ]

• height : the vertical distance between the apex of the crest and the lowest point of the trough [H]

• wavelength : the horizontal distance between two equivalent dynamic points on consecutive waves [L]

• steepness : H / L : the ratio of wave height to wavelength

• period : the time it takes for a wavelength-worth to pass a fixed point [P]

• speed [celerity] : L / P : the ratio of wavelength to period

A wave definition general enough for us is: a disturbance that propagates energy as it travels along an interface or within a medium.

4

There are a number of ways to classify waves, here are a few : and an example for each

• sources of energy that cause disturbances : atmosphere

• forces that dispel and eventually eliminate disturbances : gravity

• the manner in which energy is transmitted : progressive

Progressive waves are the most common and they can be categorized further. The ocean developes wind waves that are superimposed on an existing wave state when the wind begins to blow over the ocean. The effect of wind can be estimated in terms of its speed, duration and fetch. Fetch needs definition : it is the sea surface area directly under the influence of the wind.

The named categories of wind waves are capillary, chop and swell. Capillary waves have periods of less than a second. Chop have periods of one to ten seconds and are within the fetch. Swell have periods greater than ten seconds and develop outside the fetch.

Suppose the wind begins to gust and blow over an ocean initially at rest, a state practically impossible to find. At first, there appear what we normal folk call cat’s paws, that is, capillary waves, that can be tiny and short-lived. Unless the wind persists, capillary waves die off as the ocean surface is restored flat by the water’s stickiness, surface tension. The cat’s paws, small bumps on the ocean surface, cause the wind to move unevenly over the surface and push against the surface unevenly. So there are fetch areas that have a little higher or a little lower generating force associated with them and this tends to amplify a local wave’s height and length, that is, there is uneven but increasing wave growth on the ocean’s surface. As growth progresses, the re-storing force that tends to smooth out the waves gradually shifts in influence from surface tension to gravity and the wave nomenclature by which they are known shifts from capillary waves to gravity waves, that is, the chop and swell. And then the wind wave’s energy continues to grow by further transfer of momentum from wind to water. At any given moment, there are many different waves with different wavelengths, periods, amplitudes and energies. We call this result a wave spectrum.

5

There is more in the way of thinking about waves :

• ocean waves have energy of two types : kinetic due to movement and potential due to the wave’s surface position in the earth’s gravitational field relative to a flat sea surface [ the dashed line introduced in slide 3 ]

• while ocean waves travel the fetch, over relatively deep water, their passage does not leave the water permanently displaced; what shape are the water orbits?

• during a wave’s passage, the water involved in its movement describes an orbital path that is not completely closed in shallow depths, kind of like in loops, each of which moves a small amount in the direction of the wave disturbance’s propagation

• the above mass transport is inconsequential in very deep water but becomes significant in extent as waves progress into shallower water; what shape are the orbits then?

So what is “deep” water as far as waves are concerned ? That depends on the ratio of the water depth to the wave’s wavelength :

• deep water waves : waves for which the ratio is greater than half; short wavelengths

• shallow water waves : ratio is less than 5 one-hundreths; long wavelengths

Remember : not all shallow water waves occur near the shore, there are very long wavelength entities on the ocean : tsunami for instance, and tides as well.

6

When there Is a match between the an imposed force and the natural modes of oscillation of an object the result can produce a response that far exceeds ordinary expectations. As an example, a television commercial thirty-or-so years ago asked “is it actually Ella or Ella on memorex tape ?”. A crystal wine glass was subjected to Ella Fitzgerald holding a single note; she had perfect pitch and an incredible diaphram, the note being a natural mode of the glass. In the response to her sustained note and power, the wine glass walls began to vibrate and, since the situation was a

resonance one, the vibrations continued to grow in amplitude until the glass shattered. A number of singers have done this including Mythbusters’ Jamie Vendera.

Bridge builders met resonance head-on in Dec 1940 when the first Tacoma Narrows suspension bridge tore itself apart in response to a sustained wind system. The amplitude of the torsion that twisted the road bed increased to a point that the cables failed and the road bed fell into the Narrows. The second bridge was built after wind tunnel tests suggested wider roads and an air space that ran the length of the bridge between the lanes. These structural changes altered the bridge’s natural modes of oscillation and reduced the potential for resonance.

Resonance …….

Galloping Gertie does the twist

7

We can examine a geophysical reson-ance phenomena that has been repeated time and time again in Lake Michigan at Chicago. Continental air masses that pass from Canada over Illinois can take the form of squall lines, rapidly moving atmospheric pressure jumps that can sweep out over the lake from west to east. When the speed of such a system over the lake is above a depth contour that yields an equivalent shallow water wave speed [square root of the product of the acceleration of gravity and the depth], resonance is achieved. A water wave is generated that continues to grow in height as long as the pressure jump is traveling over the depth contour. Pressure jump and wave are “locked” together while momentum is transferred to the lake; a resonance situation. When the squall and wave reach the opposite eastern shore at Michigan City, the squall continues east and the wave is reflected back toward Chicago. The water level record taken at a break-water there shows a four-foot excursion crest-to-trough that repeatedly traversed the lake, west-east and east-west. The two-foot amplitude following an initial draw-down has taken the unwary walking the lake bottom for offshore swims several times in the last few decades despite warnings.

8

in the Arabian Sea …. 90,000 tons of diplomacy

9

Right whale and calf … for us to understand and conserve