### Research Article

### S. C. De^{a1}

^{a1 }Department of Mathematics University of Manchester

Abstract

The well-known Stokes theory (**9, 10**) of waves of permanent form in water of finite depth has been extended to the fifth order of approximation. The solutions have been first obtained in the form of equations for the space coordinates *x* and *y* as functions of the velocity potential *Φ* and stream function *ψ*. Expressions for the complex potential *W* in terms of the complex variable *z* ( = *x* + *iy*), the form of the wave profile, and the square of the wave velocity have been obtained to the fifth order.

Expressions for the three physical quantities *Q, R* and *S*, where *Q* is the volume flow rate per unit span, *R* is the energy per unit mass (i.e. *g* times the total head, measuring heights from the bottom and pressures from atmospheric) and *S* is the momentum flow rate per unit spaa, corrected for pressure forces and divided by density, have been obtained to the fifth order. The values for the dimensionless quantities *r* = *R/R*_{c} and *s = S/S*_{c}, where *R*_{c} and *S*_{c} refer to the values of *R* and *S* for a critical stream of volume flow *Q*, are tabulated for certain values of the ratios mean depth to wavelength and amplitude to wavelength. The values of *r* and *s* thus obtained have been used to calculate the ratios of mean depth to wavelength and of wave height to wavelength according to the cnoidal wave theory as recently presented by Benjamin and Light-hill(**1**), and the results are found to be in satisfactory agreement with that from Stokes's theory for waves longer than six times the depth.

The (*r, s*) diagram introduced in the recent work of Benjamin and Lighthill(**1**) has been further considered, and the unshaded part of the diagram referred to in that paper has been mapped with a network of curves for constant values of the ratios of mean depth to wavelength and of wave height to wavelength (Fig. 2). The third barrier to the existence of steady flows, corresponding to ‘waves of greatest height’ referred to in that paper, has also been indicated in Fig. 2.

(Received December 31 1954)