Mean motions and impulse of a guided internal gravity wave packet

<jats:p>Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumber<jats:italic>k</jats:italic>) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequency<jats:italic>N</jats:italic>, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)</jats:p><jats:p>To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by<jats:disp-formula id="S0022112073000480_frm1">\[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \]</jats:disp-formula>Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/<jats:italic>k</jats:italic>, and<jats:italic>c</jats:italic><jats:sub><jats:italic>g</jats:italic></jats:sub>=<jats:italic>c</jats:italic>(<jats:italic>N</jats:italic><jats:sup>2</jats:sup>–ω<jats:sup>2</jats:sup>)/<jats:italic>N</jats:italic><jats:sup>2</jats:sup>, the group velocity.</jats:p><jats:p>If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic waves.)</jats:p>