Water Waves: Theory and Experiments

Abstract

The analytical results of the nonlinear theory of wave packets are tested against experiments performed in a water tank and compared with the analytical results of the linear theory of low-amplitude waves and the theory of weakly nonlinear gravitational waves on the free fluid surface infinite in extent. The results of experiments and observations well-known in the literature are used for testing.

CLASSICAL EQUATIONS IN THE CURVILINEAR COORDINATES \(\sigma ,\theta \)

Points $P$ and Q are mentioned in the integral for the velocity potential; point $P$ has the curvilinear coordinates $\sigma ,\theta$ and point Q has the coordinates ${{\sigma }_{1}},\;{{\theta }_{1}}$ at ${{\sigma }_{1}} = 0$.

In the curvilinear coordinates the velocity potential can be written as follow:

$$\Phi (\sigma ,\theta ,t) = \frac{f}{{{\text{|}}f{\text{|}}}}\frac{1}{{2\pi }}\int\limits_{ - \pi /2}^{\pi /2} \,\nu ({{\theta }_{1}},t)A(\sigma ,{{\sigma }_{1}},\theta ,{{\theta }_{1}},t)\mathop {\left. {\frac{{d{{\theta }_{1}}}}{R}} \right|}\nolimits_{{{\sigma }_{1}} = 0} ,$$

(A1)

where

$$A = (\sigma - {{\sigma }_{1}} + W - {{W}_{1}})(f - {{W}_{1}}) + \frac{{\sigma - f + W}}{{{{\sigma }_{1}} - f + {{W}_{1}}}}\frac{{\partial {{W}_{1}}}}{{\partial {{\theta }_{1}}}}(\tan\theta {{\cos}^{2}}{{\theta }_{1}} - \sin{{\theta }_{1}}\cos{{\theta }_{1}}),$$

$$R = {{(\sigma - {{\sigma }_{1}} + W - {{W}_{1}})}^{2}}{{\cos}^{2}}{{\theta }_{1}} + {{[(\sigma - f + W)\tan\theta \cos{{\theta }_{1}} - ({{\sigma }_{1}} - f + {{W}_{1}})\sin{{\theta }_{1}}]}^{2}},$$

$$W = W(\theta ,t),\quad {{W}_{1}} = W({{\theta }_{1}},t).$$

Nonlinear equations (A2) and (A3) are, respectively, the kinematic condition on the free fluid surface and the pressure continuity condition on this surface

$$\frac{{\partial W}}{{\partial t}} = {{D}_{2}}\frac{{\partial \Phi }}{{\partial {{\sigma }_{ - }}}} - {{D}_{1}}\frac{{\partial \Phi }}{{\partial {{\theta }_{ - }}}},$$

(A2)

$$\frac{{\partial \Phi }}{{\partial {{t}_{ - }}}} - \;\frac{1}{2}{{D}_{2}}\mathop {\left( {\frac{{\partial \Phi }}{{\partial {{\sigma }_{ - }}}}} \right)}\nolimits^2 + \frac{1}{2}\mathop {\left( {D\frac{{\partial \Phi }}{{\partial {{\theta }_{ - }}}}} \right)}\nolimits^2 + W(\theta ,t) = 0,$$

(A3)

$$D = \frac{{\cos\theta }}{{W - f}},\quad {{D}_{1}} = \left( {\sin\theta + D\frac{{\partial W}}{{\partial \theta }}} \right)D,$$

$${{D}_{2}} = 1 + ({{D}_{1}} + D\sin\theta )\frac{{\partial W}}{{\partial \theta }}.$$

The conditions at infinity can be taken in the form:

$${\text{|}}W(\theta ,t){\text{|}} < C(t){{\cos}^{2}}\theta ,\quad \mathop {\lim}\limits_{\cos\theta \to 0} \frac{{\partial W}}{{\partial \theta }} = 0,\quad {\text{|}}\nu (\theta ,t){\text{|}} < C(t).$$

(A4)

Conditions (A4) guarantee that the perturbation source in fluid imparts finite energy to the fluid.