Linear Wave-Body Interaction

Linear wave-body interactions

• Consider a plane progressive regular wave interacting with a floating body in two dimensions.

• The main concepts survive almost with no change in the more practical three-dimensional problem

[math]\displaystyle{ \zeta(t): \quad \mbox{ambient wave elevation. Regular or random with definitions to be given below}. \, }[/math] [math]\displaystyle{ \xi_1(t): \quad \mbox{Body surge displacement} \, }[/math] [math]\displaystyle{ \xi_3(t): \quad \mbox{Body heave displacement} \, }[/math] [math]\displaystyle{ \xi_4(t): \quad \mbox{Body roll displacement} \, }[/math]

Linear theory

• Assume: [math]\displaystyle{ \left| \frac{\partial\zeta}{\partial x} \right| = O(\varepsilon) \ll 1 \, }[/math]

Small wave steepness. Very good assumption for gravity waves in most cases, except when waves are near breaking conditions.

• Assume [math]\displaystyle{ \left| \frac{\xi_1}{A} \right| = O(\varepsilon) \ll 1 \, }[/math]