Existence Theorems of Permanent Gravity Waves on Water of Uniform Depth

As for gravity waves of permanent type on water surface, existence theorems have been presented by many authors since Levi-Civita (1925). But yet it seems to have no simple proof which covers all cases. On the other hand, the calculation scheme for practical computation of these waves has been unified in a simple formula (ClH)j ^ it has already been shown that the result of the approximate calculation is in a good agreement with experimental fact (H2]). This unified calculation formula is based on a formulation to determine a holomorphic function in a unit circle with some nonlinear boundary condition on the circumference, and in the case of infinite water depth it just agrees with Levi-Civita's formulation itself. Hence we want to extend the proof of the latter to our general case. But as for the case of solitary wave, i.e., the case of infinite wave length, and of the highest wave, i.e., the case having angular crests, the asking holomorphic function has the corresponding singular point on the unit circle, so it is difficult formally to extend, and our effort is yet unsuccessful. Therefore in the following we deal with the case of the periodic wave having round crests and our method is based on extending directly Herbert Beckert's proof (C^H) concerning with the case of infinite depth to our case. The author wishes to acknowledge her indebtedness to Hikoji Yamada for suggesting this problem and also to Reiko Sakamoto, under whose