 Solution

Use the attenuation coefficient read from the PDF,where here $k$ is the beam wave-number magnitude and the angle convention has $\sin \theta >0$ for the chosen forward attenuation coordinate. The omitted denominator in the TeX is essential both physically and dimensionally. Work in the prescribed along-beam kinematic model, with small parcel excursion compared with ${\lambda }^{-1}$.

The first-order along-beam fluid displacement, initially zero, is $\delta \xi =A\sin (\omega t)/\omega$. Therefore the leading displacement correction to the along-beam velocity isThis oscillatory drift of an attenuated internal-wave beam is oscillatory, soIndeed the scalar parcel equation $\dot{\Xi }=U_0{e}^{-\lambda \Xi }\cos \omega t$ can be integrated exactly:For excursions small enough that the right side stays positive, the parcel returns to its initial along-beam coordinate after each period. This demonstrates the zero net drift in this supplied model. It is not a claim that all components of a viscous beam, or a separately generated Eulerian mean flow, vanish; the question specifies the along-beam component only.