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A Taylor expansion of the tangential no-slip boundary condition gives $u_{2x}(0)=-\sin \xi {\partial }_y{u}_{1x}(0)$. Since $f^{\prime \prime }(0)=-s$,For matrix-relative Brinkman velocity, the mean second-order tangential flow isThe mean tangential traction has no first-order geometric remainder: at the surface ${\partial }_y{\sigma }_{xy,1}=s(s+1)\sin \xi$ and ${\sigma }_{xx,1}=s(s+1)\cos \xi$, so the mean displaced-boundary and tilted-normal corrections cancel. Thus the force-free condition gives ${\overline{u}}_{2x}^{\prime }(0)=0$, hence $C=0$. In the $A=0$ limit boundedness likewise excludes mean shear. The swimming speed of a Brinkman sheet is thereforeIt is larger than the simple-fluid speed by $\sqrt{1+(\alpha /k{)}^2}$ for fixed waveform and frequency. The matrix offers a reaction structure for the transverse stroke and more strongly confines the induced motion; it acts as a footing against which the wave pushes. This is a prescribed-stroke comparison, not a guarantee at fixed motor power. The small-amplitude expansion is at fixed $A$; a large-$A$ use also needs $ϵs\ll 1$ to control variation across the displaced boundary.