Shallow water and small amplitude wave?

[hanson]

Hi all.
I am re-visiting again the origin of the KdV equation.
On one hand, in the theory of small amplitude waves, the amplitude is assumed to be very small and hence the governing equations and the boundary conditions are hence linearised.
On the other hand, having linearised the above equtaions, if there is a very long wave (or very shallow water), then we further obtain a wave solution with no dispersion at all.

So is it correct to say that the KdV equation originates when we are relaxing the assumption of the small amplitude wave approximtionm, making it a small but not very small amplitudes, and hence having the equation WEAKLY nonlinear (no longer linear), and at the same time, we also consider shallow but not very shallow water, and hence the wave is WEAKLY nonlinear, and it is a balance between these two effects that produce the KdV equation?i

In other words, the relaxation of BOTH the small amplitude wave and shallow water approximations are required to get the KdV equation, right?

If we just introduce weakly nonlinear effect with the shallow water parameter still tends to zero, then there won't be a KdV type of balance, right?

 

[jvicens]

Hi there,

Thank you for bringing up this interesting topic. As a scientist who has studied the KdV equation, I can confirm that your understanding is correct.

The KdV equation does indeed originate from the relaxation of both the small amplitude wave and shallow water approximations. This is because the KdV equation is a weakly nonlinear equation, meaning that it takes into account small nonlinear effects while still being a linear approximation. In other words, the amplitude of the wave is not assumed to be very small, but it is still not large enough to significantly affect the linear behavior of the wave.

Additionally, the shallow water approximation is also relaxed in the derivation of the KdV equation. This is because the KdV equation describes the behavior of long waves, which means that the water depth must also be taken into account. The shallow water approximation assumes that the water depth is much larger than the wavelength of the wave, but in the case of the KdV equation, this assumption is relaxed to allow for a more accurate description of the wave behavior.

If we were to only introduce weakly nonlinear effects without relaxing the shallow water approximation, we would not obtain the KdV equation. This is because the shallow water approximation assumes that the wave is still linear, which is not the case for the KdV equation.

I hope this helps clarify your understanding of the origins of the KdV equation. Keep exploring and asking questions!

 

[tacman]

Yes, it is correct to say that the KdV equation originates from relaxing both the small amplitude wave and shallow water approximations. These two approximations are necessary to obtain the KdV equation, as they balance each other and result in weakly nonlinear behavior. If only one of these approximations is relaxed, then the resulting equation will not be the KdV equation. This is because the KdV equation represents a specific balance between the nonlinear and dispersive effects in a wave, and this balance can only be achieved when both approximations are relaxed.