Physics

The key is recognizing that viscosity dissipates mechanical energy through viscous stress doing negative work on fluid elements. You'll start with the Navier-Stokes equation, multiply by velocity to get a power balance, and integrate over a volume to find the rate at which kinetic energy converts to heat via the viscous stress tensor $\boldsymbol{\tau}$. For surface waves, the small-viscosity assumption means dissipation is confined to a thin boundary layer near the interface where velocity gradients are steep—this lets you approximate the energy loss without solving the full viscous problem. The condition "small viscosity" quantifies when the viscous penetration depth is much smaller than the wavelength.

The key is recognizing that viscosity dissipates mechanical energy through viscous stress doing negative work on fluid elements. You'll start with the Navier-Stokes equation, multiply by velocity to get a power balance, and integrate over a volume to find the rate at which kinetic energy converts to heat via the viscous stress tensor $\boldsymbol{\tau}$. For surface waves, the small-viscosity assumption means dissipation is confined to a thin boundary layer near the interface where velocity gradients are steep—this lets you approximate the energy loss without solving the full viscous problem. The condition "small viscosity" quantifies when the viscous penetration depth is much smaller than the wavelength.