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It follows from the review on classical wave models that the asymmetry of crest and trough is the direct cause for wave drift. Based on this, a new model of Lagrangian form is constructed. Relative to the Gerstner model, its improvement is reflected in the horizontal motion which includes an explicit drift term. On the one hand, the depth-decay factor for the new drift accords well with that of the particle’s horizontal velocity. It is more rational than that of Stokes drift. On the other hand, the new formula needs no Taylor expansion as for Stokes drift and is applicable for the waves with big slopes. In addition, the new formula can also yield a more rational magnitude for the surface drift than that of Stokes.

The drift caused by water wave was firstly studied by George Gabriel Stokes in 1847. His approximate formula based on small-amplitude wave is known as “Stokes drift” nowadays. Is the wave drift caused by the asymmetry of crest and trough? If the answer is true, then not only the nonlinear Stokes wave with finite amplitude but also the Gerstner wave with large amplitude exists wave drift. Thus the doubt “Do we observe Gerstner waves in wave tank experiments?” in [

In order to understand the wave mechanism, there is a necessity for us to review the wave studies. Historically speaking, the study of water wave can be dated back to the year 1687 when Newton did an experiment with U-tube and got the result “the frequency of deep-water waves must be proportional to the inverse of the square root of the wave length”. As reviewed in [

As for the study which takes Stokes drift as a special topic [

As the problem concerned, the default model should be the inviscid and incompressible Navier-Stokes equations. But the solving of these equations involves in determining the upper surface boundary condition which is just the wave to look for [

The classical linear wave theory illustrated in nowadays textbooks [

On the assumption that the amplitude A is infinitely small relative to the wave-length

here

together with a dispersion relation

According to the web of Wikipedia [

Within the framework of linear theory, the motion distance is very short and the particle’s Lagrangian location

Based on this together with Taylor expansion technique, the Stokes drift is then estimated by:

with

From the above analysis we see the formula for Stokes drift only holds for

In case

with

For this case, the horizontal and vertical velocities are also in the forms of Equation (2.4). But the substitution of

Relative to the linear wave, the Stokes wave looses the range of wave steepness to

On the assumption that the particle’s trajectory is a circle, Gerstner (1802) found a rotational trochoid wave:

with a dispersion relation

For this case, the water pressure is in a particular form [

which has noting to do with the variables a and t. Here the last term reflect the effect from the fact that the equilibrium is higher than the motionless water level due to the asymmetry of crest and trough. This shows the water pressure is merely in the depth-dependent form

We note that the Gerstner model (2.8) is actually an alternative form of the approximate linear model (2.5) with a translation on the phase angle by

In addition, it follows from Equation (2.8) that the particle’s horizontal velocity at the wave crest equals to

In addition, it is easy to check that

is also an exact solution to the Lagrangian equations in case a steady flow U exists. However, it follows from [

From the previous analysis we know Airy, Stokes and Gerstner adopted a same approach, that is, to take the conjectured wave forms as the preconditions. What is more, the water pressures are given as corollaries in the last. Here we take an inverse approach to do so. Let the wave model be the object, the conjecture is done on the pressure.

Take one water particle as the research object, we describe it by Lagrangian coordinates

For a hydrostatic case with constant density, the water pressure increases linearly along with the water-layer thickness s, that is,

this is the so-called “quasi-hydrostatic approximation” adopted in physical oceanography [

There is another case, might as well, call it by “gravitational approximation” which takes the gravity as the main restoring force. For this case, there should be

In fact, the quasi-hydrostatic and gravitational approximations are two extreme cases: the vertical pressure gradient force is too strong for the first case and too weak for the second case. Notice that the pressure formulas (2.3) and (2.10) for the linear, Stokes and Gerstner waves are deduced from the Navier- Stokes equations and their forms are very objective, we follow them and estimate the pressure by

Here the preconditioned sine or cosine function is substituted by an undetermined free surface

To insert the pressure expression (3.3) into Equations (3.1) it yields

Notice that the wave is a synthesis of transversal and longitudinal waves, with the aid of these two equations we model them separately. To denote

then

with

These mean the horizontal motion is due to the pressure-gradient force caused by the slant water body and the vertical motion is due to the variation of the surface elevation itself (can be understood as the variation in the previous period, it squeezes the water body and leads to new vertical motion).

Before deriving the model of traveling-wave form we take no account of

here

It accords well with our common sense.

The horizontal motion of the surface particle is determined by the first equation in (3.6). It is associated with partial derivative of the undetermined surface wave which is insoluble in essence. In the following we estimate its solution by approximating the wave slope

Let

here the position of wave trough is set on

Notice that the vertical motion

To inset this into the first equation of (3.6) it yields an estimation below:

where

In addition, there is an interesting phenomenon that

The remainder work is to find the relations between

It follows from Equations (3.12) and (3.13) together with the relationship

Their variations are depicted in

Now that the particle’s horizontal and vertical motions are constructed, it is time for us to recall back the transformation (3.5). Since the equilibrium

To substitute

where

The corresponding dispersion relation still remains

The above two equations compose a new water wave model. It differs from the linear model, nonlinear Stokes model and Gerstner model. From

It follows from Equation (3.16) that, on each period of time T all the particles propagate forward with the same length

for

Relative to the Stokes drift

the modifications of new formula are reflected in the depth-decay and slope- dependent factors. Since the horizontal velocity of the water particle has a depth- decay factor

In the following we compare the newly derived formula with that of Stokes drift by numerical approach. Here only the surface drift is considered. It follows from

By reviewing the classical linear wave, Stokes wave and Gerstner wave we have found that the asymmetry of crest and trough is the direct cause for wave drift. Based on this, a new model of Lagrangian form is constructed. Relative to the Gerstner model, its improvement is reflected in the horizontal component which includes an explicit drift term. The newly derived drift formula depends not only on the wave amplitude A, but also on the average wave slope

On the one hand, the depth-decay factor

To estimate the drift of big waves at sea is valuable for ocean engineering. A good formula should be able to yield a reliable magnitude for it. The numerical simulations show that the newly derived formula yields a more rational surface drift (

We thank the supports from the National Natural Science Fund of China (No.41376030).

Wang, J.-L. and Li, H.-F. (2017) Ocean Wave Model and Wave Drift Caused by the Asymmetry of Crest and Trough. Open Journal of Marine Science, 7, 343-356. https://doi.org/10.4236/ojms.2017.73025