Airy stress function (${\tau }_{xx}=A_{yy},{\tau }_{yy}=A_{xx},{\tau }_{xy}=-A_{xy}$)

The displayed scalar stress potential automatically satisfies the two in-plane static momentum equations without body force, because the mixed third derivatives cancel. For homogeneous isotropic linear elasticity, strain compatibility further makes $A$ biharmonic. Since stresses depend only on second derivatives, $A$ is unique only up to $ax+by+c$ on a connected domain. Conversely a function with all second derivatives zero is affine, proving that this is exactly the gauge freedom. It does not license adding an arbitrary biharmonic function, which generally changes stress.

For plane strain in $y>0$ under surface normal traction $-P(x)$ and zero surface shear, use the forward Fourier transform phase $e^{ikx}$. At $k\ne 0$, biharmonicity gives $({\partial }_y^2-k^2{)}^2\hat{A}=0$, whose decaying part is $(C+Dy)e^{-|k|y}$. Boundary traction fixes $C=\hat{P}/k^2$, $D=|k|C$, proving the displayed formula. Constitutive inversion and a decaying displacement mode give $2\mu \hat{v}=[y+2(1-\nu )/|k|]\hat{P}{e}^{-|k|y}$. Multiplication by $-ik$ produces an unambiguous displacement-gradient convolution kernel.

The zero mode is separate: a nonzero resultant load $\hat{P}(0)$ conflicts with an ordinary fully decaying displacement normalization in two dimensions and leads to a logarithmic reference issue. Balanced loading or a distributional/reference-displacement treatment is required before extending the transformed formulas globally. Dividing by $k^2$ and assigning a finite value at $k=0$ is invalid.