 Solution

The out-of-plane displacements in the wag transform as $y$, hence as $B_2$. A one-quantum combination of two distinct normal modes transforms as the product of their irreducible representations. The observed feature satisfiesA combination of the wag with an $A_1$ stretch has symmetry $B_2\otimes A_1=B_2$ and can absorb through ${\mu }_y$. In contrast, $B_2\otimes B_1=A_2$ has no electric-dipole component in $C_{2v}$ and is forbidden in the stated symmetry model. The missing feature near $4010=1167+2843\mathrm{c}{\mathrm{m}}^{-1}$ therefore identifies the $2843$ stretch as $B_1$.

The completed assignments areThe allowed combination obtains intensity through anharmonic/dipole-coupling terms; a purely separable harmonic model with a strictly linear dipole has no combination bands. The argument uses both the observed combination and the selection rule: absence of a feature by itself would not generally prove a forbidden symmetry.^[ref]