The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of the Green's function.
The far-field approximations to the derivatives of the Green's function have been used without derivation and verification in the previous work. In this work, we provide the detailed derivations of the far-field approximations to the derivatives of the Green's function. The binomial expansions for the derivatives of the Green's function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives. The approximations to the derivatives of the Green's function are systemically verified by using the benchmark two dimensional convecting vortex and the co-rotating vortex pair. In addition, we provide the derivations of the approximations to the multiple integrals of the Green's function by using the far-field approximations to the derivatives.