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The expansion method introduced by Friedrichs (1948) for the systematic development of shallow-water theory for water waves of large wavelength was used by Keller (1948) to obtain the first approximation for the finite-amplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well as for periodic waves of permanent type, corresponding to the cnoidal waves of Korteweg & de Vries (1895).The present investigation extends Friedrich's method so as to include terms up to the fourth order from shallow-water theory for a flat horizontal bottom, and thereby obtains the complete second approximations to both cnoidal and solitary waves. These second approximations show that, unlike the first approximation, the vertical motions cannot be considered as negligible, and that the pressure variation is no longer hydrostatic.

Tms book is au excellent t ransla tion of the original Germ an text published as the " TheOl'ie schallnaher Strol1lungen" by Springer-Yerl ag in H)57. Several minor errors h av e been correc ted , bu t 11 0 new references have b een added. The b ook sh ould still be of grent usc, 1l 0L only to fluid dy nami cists, bu t also to applied mathemat.ieians who a re interes ted in nonl inear (qu asilinenr) p a.rt.ia l difl'erent.i al equat ions wi t h bOllndary-value prohl ems co ntaining interacting ellip tic and hyp erb oli c domains.

The second approximation to cnoidal waves is compared with the third approximation for Stokes waves of permanent form in water of finite depth. The comparison clearly indicates that cnoidal wave theory should not be applied to finite amplitude waves if their wavelengths are shorter than 5 times the depth. It is shown how the limiting heights of cnoidal waves are also related to the vanishing of the pressure gradient near the wave crest. The third approximation to Stokes waves in finite water depths is verified by the use of the classical small‐perturbation expansion method which is best suited for small wave amplitudes. For finite amplitude waves the series expansion in terms of the infinitesimal‐wave parameter is found to be most suitable for wavelengths shorter than 8 times the depth.