Limiting criteria of permanent progressive waves

Model Reduction of a Higher-Order KdV Equation for Shallow Water Waves

“…This is larger than what is observed in practice, where solitary water waves cannot grow (in our normalized units) beyond a height h D 1:38 [15], and also larger than the theoretical upper limit 1:95 [31][32][33]. This may be due to the particular values of˛andˇused in our study.…”

“…The lower curve shows the restraining influence of the higherorder terms in˛andˇ. Note: The wave amplitudes achievable in water experiments are restricted to the narrow zone h Ä 1:38 (or according to theory, h Ä 1:95) and corresponding wave velocities c Ä 1:3 (or c Ä 1:4), where the two curves have hardly separated yet [15,[31][32][33] …”

“…The maximum amplitude (height above the quiescent water surface) that is actually measured in practice is somewhere around 0:55h 0 [15], while the theoretical upper limit is 0:78h 0 [31][32][33]. For higher amplitudes the velocity of the water particles on the wave crest becomes larger than the propagation speed, hence the wave breaks.…”

The Shape of Soliton-Like Solutions of a Higher-Order Kdv Equation Describing Water Waves

“…The choice of α implies that a/h 0 = 2 3 α = 0.40, which is not much smaller than 1, so that there is good reason to include the higher order terms in α. It should be noted that this choice is still within the realistic range for solitary water waves over a horizontal bottom; the maximum value of a/h 0 that can be achieved in practice lies somewhere around 0.55 [30], while the theoretical upper limit is 0.78 [31][32][33]. (In our equations this means that u = η/a should not exceed 0.55h 0 /a = 1.375 or, theoretically, 0.78h 0 /a = 1.95.…”