Transformation of Waves

We have tried to see Ocean Waves in the NATURE page through the eyes of a scientist and a poet. In this piece let us try to see how the born waves are transformed as it propagates – this time through the eyes of an applied scientist or engineer. I will mainly focus on ocean waves, although the discussed transformation processes apply to all different types of waves.   
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1. Waves Intro
A natural water wave is the undulated response of water to transport the energy it gains – through the attractions of Sun and Moon as in a tide, sudden massive ocean floor rupture as in a tsunami, hurricane storms piling huge water mass as in a storm surge and, of course, through the blowing winds causing ubiquitous ocean wind-waves.
The born waves are visible in the duality of crests and troughs, but are composed of multiple waves of different periods and heights built within them (as are vivid in harmonic decomposition of tidal waves, as well demonstrated in my 1991 COPEDEC-PIANC paper: Tidal observations and spectral analyses of water level data in the mouth of the Ganges-Brahmaputra-Meghna river system). Natural waves are spectromatic not only in terms of periods, heights and phases but also in directions. Some basics of wind-wave hindcasting and forecasting methods are reviewed in one of my publications {Wave Hindcasting. Encyclopedia of Coastal Science, Springer, 2005. Updated 2017}.
As soon as a wave is born, it is subjected to the transformation processes as the energy propagates. The processes can best be described by three important parameters: wave height H, the height from trough to crest; local wave length L or wave period T, measured simply from crest to crest; and the local still water depth d. A very useful parameter proposed by Fritz Joseph Ursell (1923 – 2012) does use these three parameters to classify waves - to indicate whether they can be approximated by the 1st order processes of Linear Wave Theory proposed by George Biddell Airy (1801 – 1892). This parameter named in his honor as Ursell Number describes a threshold combining the two ratios H/d and d/L, and is given by U = HL^2/d^3. It turns out that for all practical purposes, waves can be assumed to be symmetric or linear, when U is equal to or less than 5.0.
Further, it is important to know that a wave cannot sustain itself when it reaches the steepness at the threshold value of 1/7 or higher – steepness being the ratio of H over L. The unsustainability results in wave breaking and losses of energy – which is mostly seen as whitecaps in noisy wind waves. A criterion proposed by Miche {Le pou voir reflechissant des ouvrages maritimes exposes a l’action de la houle. 1951} captures the wave breaking thresholds not only due to the limiting steepness, but also due to the limiting water depth.
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2. Transformation Zone Delineation
Next, it is useful to delineate the wave transformation region for convenience. A deep-water zone covers the areas where depths are higher than half of the wave length, L/2. The contiguous shallower region is known as the refraction zone and it starts from the deep-water limiting depth to the region where wave breaks, roughly at d = 1.28H in flat bottom. In reality, wave spectrum and beach slope also play a role in controlling depth-limited wave breaking. In the refraction zone waves start to feel the bottom and several changes happen in this zone – in particular the symmetry of the sinusoidal deep-water wave is lost by distortion – turning it into asymmetric at U = 5.0 or higher. In this piece we will be mostly talking about these two zones, but only the symmetric aspect of it. The nonlinear asymmetric wave phenomenon is highly elaborate by its own right, and hope that I will visit it at some other time.
After breaking, the transformation processes result in dissipation of the entire wave energy in the surf zone, and to up in the beach slope in the swash zone. The dissipated wave energy rushes up the shore as a bore of very high velocity. As the energy is dissipated, water rushes down the slope by gravitational pull and by the suction of trough. Vigorous turbulence and sand movement characterize this zone shaping and reshaping the shoreline morphology. Again, it is helpful to focus on these processes separately and I hope to come back to it at a later time.                      
With these understandings let us turn our attention to see how waves are transformed. The dynamic equilibrium of things requires that the characteristics of transformed waves are determined by the balance of incoming energy against those due to dissipation, transmission and reflection.
To keep this discussion to a manageable level, I will try to focus on wind waves. Waves developed by wind storms are highly irregular, noisy and short-crested, and the energy dissipation starts immediately by the interactions between the born waves and the lashing wind prompting steepness related breaking.  
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3. Frequency Dispersion and Spatial Evolution
Among others, three other notable processes also start immediately in deep-water wave motions. The first is known as the frequency dispersion – frequency is the reciprocal of period, which means that a wave with a period of 10 second has a frequency of 0.1 hertz. This process involves separation of the long period components as they start to travel faster leaving behind the short-period ones. This is the reason why we see the fore-runners – long-period long-crested swells on coasts that have traveled far away from their sources of origin at a storm field.
The second is the spectral evolution of waves through the wave to wave interactions. In a spectrographic display, peak wave energy proportional to H^2, is seen as concentrated at a certain frequency. The evolution process accounts for spreading out this energy in both ways toward the higher and lower frequencies, and also in directions. Those transferred to the higher frequency domain mostly break as the limiting steepness is reached, and can be seen as whitecaps. 
The third is the grouping of waves seen in the togetherness of high ones separate from the low ones. As pointed out in the Ocean Waves blog in the NATURE page, this process appears to result from the superimposition of component waves of different heights, periods and phases. The grouping, in essence imposes an additional long period wave envelope.
What are the simplest things that happen in the refraction zone? Let us list some of the simple processes in this zone. With the decreasing water depth toward the shore, the wave speed or the celerity of the wave energy slows down, the wave length decreases, but the wave orbital velocity increases. 
To demonstrate the processes, I have constructed a simple image shown in this piece. In this image, a 1-meter, 8-second wave approaching the shore-parallel contour at a 30-degree angle is followed on its journey to transformation. The shown analysis is limited within a region where the wave processes can be approximated by the Linear Wave Theory at U = 5.0. As the waves begin to lose its properties of symmetry shoreward, the processes start to become complicated, and I hope to come back to that at some other time. But I like to touch upon one simple aspect of non-linearity known as the Stokes Drift.
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4. Stokes Drift
Stokes Drift named after the British mathematician George Gabriel Stokes (1819 – 1903) represents a residual current, also known as the mass transport velocity heading in the direction of wave propagation. The residual current is about 1-order less than the peak orbital velocity – which means that for a peak orbital velocity of 1 meter per second, the Stokes Drift would be about 0.1 meter per second. This non-linear effect was shown by Stokes by solving the unsteady Bernoulli (1700 – 1782) equation to the 2nd order. Stokes Drift explains the reasons why objects floating in the ocean finally end up on the shore, even in the absence of an appreciable ocean current.       
Apart from the nonlinear transformation, at least four other processes happen contributing to the evolution of waves in the refraction zone. These are reshuffling of wave energy, wave bending or refraction that accounts for wave crests becoming parallel to the shore, shoaling of waves, and frictional dissipation of wave energy at the bottom. It turns out that the frictional dissipation is very negligible in the linear wave region.
While spreading out of the peak energy occurs in deep-water, in the shallow refraction zone the wave to wave interactions results in a one way process – the transfer of the peak energy to the high frequencies. The process results in a spectral evolution that approaches the specter of a solitary monochromatic wave. This is one of the reasons why the ratio of group celerity to individual wave celerity increases from n = 0.5 in deep-water to n = 1.0 in very shallow water.        
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5. Refraction and Shoaling
Let us now turn our attention to the processes of refraction and shoaling. The attached image shows how the wave approaching at an oblique angle of 30 degrees is reduced to less than 28 degrees. This refraction process occurs due to the slowing down of wave celerity as it crosses the boundary from relatively deep water to a shallower area. It has been formulated by Dutch Astronomer Willebrord Snellius (1580 – 1626) for optics, and the refraction process is known as the Snell’s Law. As obvious in optics, refraction causes a brilliant light to become softer – so does a water wave – becoming less energetic by losing some of its energy in the process.
While the refraction process causes the waves to lose energy, another process known as shoaling causes the waves to amplify. This process can simply be explained by the fact that as the wave length decreases, and the celerity slows down, the height has to increase to balance the energy. The contrasting influences of the refraction and shoaling processes results in a transformation that indicates the initial loss of energy density or lessening of wave height if the wave approach angle is high, and heightening of waves further shoreward. While this happens, the peak bottom orbital velocity gains strength as both the depth and wave length continue to decrease.
​What is the implication of reducing wave celerity and increasing orbital velocity? With waves becoming nonlinear, depth-limited wave breaking is initiated when a wave form can no longer be sustained because the increasing orbital velocity overtakes the decreasing celerity at some point. In a short article I have demonstrated how that happens in the case of an uprushing tsunami {Discussion of Maximum Fluid Forces in the Tsunami Runup Zone. ASCE Journal of Waterway, Port, Coastal and Ocean Engineering, 2008}.
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Here is an anecdote to ponder:
The disciple asked the master, “Sir, how am I today from yesterday?”
The master smiled, “My friend that is a difficult question. But note this, as long as life’s energy flows, the response in humans occurs through the rise and fall of emotions and thought processes, like the crest and trough of a wave. And while all that happens, a human is transformed from one state to another. So the past you may not be the same you today, although your physical body is same like the water on which wave propagates. This is just how the dynamics of life, society and Nature work.”
“Could I have been worse or better than yesterday?”
“It all depends. Like the gravity tries to restore the wave undulations, so does the foundation in all of us. If you are built upon strong humane values, chances are that you are heading toward improving yourself. But a single day is too short a time to show any change in life.”  

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- by Dr. Dilip K. Barua, 4 August 2016      

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