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Integral Equations

Nekrasov's Equation

An important integral equation in hydrodynamics is that derived by Nekrasov [12] to describe waves moving on the surface of a fluid. Assuming the fluid to be inviscid and the flow to be irrotational, the form of this equation that is relevant when the depth of the fluid is infinite is

Here the dependent variable is the angle between the tangent to the wave surface and the horizontal, while the independent variable is the velocity potential of the flow along the wave surface. We assume that we are dealing with a periodic wave train and translating with the wave so that it has a fixed shape in our frame of reference, and then has as a period. With corresponding to the crest of a wave, so , it can be shown that is necessarily an odd function of if there is just one crest per period, and the waves are therefore symmetric about their crests with and corresponding to troughs; thus . Granted this, it is usual to deal with the above equation in the slightly modified form

The parameter is given in terms of physical variables by

where is the acceleration due to gravity, is the wavelength, is the speed with which the wave is progressing (since we are moving with the wave, this is actually the speed of the fluid at infinite depth), and is the fluid speed at the crest.

Whatever the value of , there is always the solution , and this is the only solution for small values . As increases, there is a bifurcation at with a genuine waveform having just one crest per period appearing. Because of the nature of , increasing from to corresponds to moving from right to left along the wave surface, with the usual orientation of axes, so this solution has for and for . For larger , there are other possible solutions corresponding to there being several crests between and , but we can select the one with just one crest there by choosing an initial approximation with the correct variation. We use

There is a logarithmic singularity of the integrand in this form of Nekrasov's equation. This can be transformed into a removable singularity by performing an integration by parts but, as with Theodorsen's equation, Mathematica seems to prefer the singular form provided it is given a little help.

With this example, we will compromise on accuracy to avoid excessive computation times; we shall see that the problem becomes more difficult as increases, and any method of solution then requires lengthy procedures to obtain accurate results. With this in mind, we use NDSolve to compute the integral in the denominator of the integrand as a function of just once during each iteration, rather than using NIntegrate every time we evaluate this integrand. The basic philosophy behind the iteration is much as before; we use

We have Interpolation here rather than InterpolatingPolynomial or Fit for reasons to be given later. We also have an integer governing the number of interpolation points we use, and new[s] to allow us to use a scaling to redistribute these points, but we begin with a moderate value of the former and no scaling for the latter.

Now for a moderate value of .

We obtain the relevant approximation and save it for later use.

We can now plot the result.

We see that there has been only moderate distortion of our original approximation as part of a sine wave, but there is a steepening near the origin. Anticipating that there will be more such distortion for larger , we increase the value of for the next case.

The distortion has indeed increased somewhat and we note that the solution looks roughly like a cubic near , but is virtually linear towards the ends of the interval. Neither a polynomial nor a finite trigonometric sum is suitable for approximating such a function, which is why we have used the cubic splines produced by Interpolation. Since it is now clear that the neighborhood of the origin is becoming more important, we will introduce a scaling that pushes more interpolation points into this region as we increase further, and also increase .

We will perform the calculations for just one more value of , modifying both and new[s] again.

We can now bring together the four solutions computed thus far.

The trend seems to be clear; as increases further, we might expect the solution to approach one where this graph has a discontinuity at . This would mean that the surface of the wave is no longer smooth at the crest, but has a corner there. Now Stokes showed that one could have such corners only if the angle (as seen by the fluid) is , which here corresponds to jumping from to as increases through 0, and conjectured that the limit of this family of waveforms as is, indeed, such a wave with a corner. This is supported by the fact that the flow in such a corner has zero fluid speed actually at the corner, for letting in the definition of above we obtain . This Stokes's conjecture (now proved) is also illustrated by the previous graph.

We can use the solutions obtained to plot the actual shape of the waves. For convenience, we normalize the wavelengths to 2 and choose the horizontal axis to pass through the troughs, so the profiles covering one period start at the point and end at the point .

Again we see evidence that the waveform is approaching a state with a corner as . The point marked 0.282 on the vertical axis is where the corner forming the crest lies in this limiting wave of greatest height. (To illustrate the difficulty of calculations in this area, however they are done and whatever theoretical approach is used, while there is agreement on this value to 3 decimal places in the literature, there still seems to be some doubt over what the fourth digit should be.)

The previous two graphs also suggest a further conjecture, due to Krasovskii, to the effect that, for all waves of this family, we have for all values of . In fact, this conjecture is now known to be false. For large values of , the graph of acquires small ripples near the maximum and minimum in our figure, and these are responsible for values of that are slightly greater than at points close to the crest. To demonstrate these in a numerical approach based on Nekrasov's equation, calculations performed with showed a maximum value for of . Since the effect being investigated looks very like the familiar Gibbs phenomenon associated with approximations, one must be careful to produce a real effect and not one artificially introduced by the method used. With such large values of involved, it is evident that asymptotic methods are more appropriate for such investigations. A recent article concerning this problem is Byatt-Smith [13], and earlier references can be traced from this source.