Computation of the high-order derivatives of Green’s function for the FW-H equation is required to eliminate the spurious sound associated with the quadrupole sources. The approximations to the derivatives of Green’s function in the frequency domain have been used without derivation in previous work. This work provides the detailed derivations of the approximations to the derivatives of Green’s function. The essential expressions and formulations associated with the binomial expression of the derivatives of Green’s function and the far-field condition to obtain the approximations are provided in detail. The benchmark flows of the two-dimensional convecting vortex and co-rotating vortex pair are used to verify the approximations. The results show that the approximations can be accurate at different Mach numbers and observer directions as long as the distance is large enough. The derivations of the approximations to integrals of Green’s function by using approximations to the derivatives are also reported in detail.

### 4.1 Appendix A: Approximations to the integrals of the Green’s function

The approximations to the integrals of Green’s function have been used by Zhou et al. [17] to fix the divergence problem of the frequency-domain surface correction integral [20]. However, the detailed derivations of the approximations to the integrals of Green’s function have not been provided. In this Appendix, we show that the approximations to the integrals of Green’s function can be derived based on the approximations to the derivatives of Green’s function.

The approximations to the integrals of Green’s function with respect to the variable *y*_{1} at the far field used in the work of Zhou et al. [17] are given as follows

$$ \begin{aligned} \frac{{{\partial }^{2}}G{^{q,n}}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}\approx \left(\frac{\partial \varphi_{} (\mathbf{x};\mathbf{y})}{\partial {{y}_{q}}} \right)^{\text{-}{n}}\frac{{{\partial }^{2}}G^{{}}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}, \end{aligned} $$

(40)

where

$$ {}\begin{aligned} \frac{{{\partial^{2}}{G^{q,n}}(\mathbf{x};\mathbf{y})}}{{\partial {y_{i}}\partial {y_{j}}}}& = {I^{q,n}}\left({\frac{{{\partial^{2}}G(\mathbf{x};\mathbf{y})}}{{\partial {y_{i}}\partial {y_{j}}}}} \right)\\ &= \int_{\infty}^{{y_{q}}} {\left({\int_{\infty}^{{\xi_{n}}} {\left({\int_{\infty}^{{\xi_{n - 1}}} {\left({ \cdots \int_{\infty}^{{\xi_{3}}} {\left({\int_{\infty}^{{\xi_{2}}} {Kd{\xi_{1}}}} \right)d{\xi_{2}}} \cdots} \right)} d{\xi_{n - 2}}} \right)d{\xi_{n - 1}}}} \right)} d{\xi_{n}},\\ K(\mathbf{x};\mathbf{y}) &= \frac{{{\partial^{2}}G(\mathbf{x};\mathbf{y})}}{{\partial {y_{i}}\partial {y_{j}}}} \end{aligned} $$

(41)

is the multiple integral of \(\frac {{{\partial }^{2}}{{G}}\left (\mathbf {x};\mathbf {y}\right)}{\partial {{y}_{i}}\partial {{y}_{j}}}\) with respect to the variable *y*_{q}. For the *i*th integral (*i*<*n*) from the inner of Eq. (41), the independent variable is *ξ*_{i} and the upper limit of the integral is *ξ*_{i+1}. When *i*=*n*, the independent variable is *ξ*_{n}, and the upper limit of the integral is *y*_{q}.

Equation (40) is obtained by proving the following equation

$$ \begin{aligned} \frac{{{\partial^{2}}G_{}^{q,l}(\mathbf{x};\mathbf{y})}}{{\partial {y_{i}}\partial {y_{j}}}}{\mathrm{ = }}\frac{{{\partial^{2}}\left({{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{q}}}}} \right)}^{- l}}G_{}^{}(\mathbf{x};\mathbf{y})} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}. \end{aligned} $$

(42)

We give the details of the derivation of Eq. (42) for the two-dimensional flows by using the mathematical induction method and the approximations of Eq. (4). The derivation of Eq. (42) for the three-dimensional flows can be obtained in a similar manner.

To use the mathematical induction method, we first prove that Eq. (42) is valid when *l*=1 and *q*=1. For *l*=1 and *q*=1, Eq. (41) reduces to

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}=\int_{\infty }^{{{y}_{1}}}{\left({{\left. \frac{{{\partial }^{2}}G_{^{2D}}^{{}}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}} \right|}_{{{y}_{1}}={{\xi }_{1}}}} \right)d{{\xi }_{1}}}. \end{aligned} $$

(43)

After transforming the partial derivative with respect to **y** to **x** and using integration by parts, the right-hand side of Eq. (43) becomes

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}=&\frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\exp }^{\varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}d{\xi_{1}}}\\ =&\frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\frac{\partial }{{\partial {\xi_{1}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\left({\frac{{\partial \varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}{{\partial {\xi_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}} \right]d{\xi_{1}}} \\ { }&-\frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\left[ {{{\left({\frac{{\partial \varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}{{\partial {\xi_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}} \right]\frac{\partial }{{\partial {\xi_{1}}}}\left({\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}} \right)d{\xi_{1}}}. \\ \end{aligned} $$

(44)

According to Eq. (16), the integrand in the last term on the right-hand side of Eq. (44) is of order *O*(*R*^{−3/2}). Meanwhile, the left-hand side of Eq. (44) is of order *O*(*R*^{−1/2}). Thus, the last term on the right-hand side of Eq. (44) can be neglected compared with the left-hand side of Eq. (44), so that we have

$$ {}\begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}\approx \frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\frac{\partial }{{\partial {\xi_{1}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\left({\frac{{\partial \varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}{{\partial {\xi_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};{\xi_{1}},{y_{2}})}}} \right]d{\xi_{1}}}. \end{aligned} $$

(45)

Considering the limit of Green’s function at infinity and transforming the partial derivative with respect to **x** back to **y**, Eq. (45) becomes

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}&\approx \frac{{{\partial^{2}}}}{{\partial {y_{i}}\partial {y_{j}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right] \\ & =\frac{{{\partial }^{2}}\left({{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-1}}G_{^{2D}}^{{}}(\mathbf{x};\mathbf{y}) \right)}{\partial {{y}_{i}}\partial {{y}_{j}}}, \\ \end{aligned} $$

(46)

proving that Eq. (42) is valid when *l*=1 and *q*=1.

According to the procedures of the mathematical induction method, we assume that Eq. (42) is valid when *l*=*h* and *q*=1.

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,h}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}\approx \frac{{{\partial }^{2}}\left[ {{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-h}}G_{^{2D}}^{{}}(\mathbf{x};\mathbf{y}) \right]}{\partial {{y}_{i}}\partial {{y}_{j}}}. \end{aligned} $$

(47)

According to Eqs. (41) and (47), we have

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,h+1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}\approx \int_{\infty }^{{{y}_{h}}}{\left({{\left. \frac{{{\partial }^{2}}\left[ {{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}}G_{^{2D}}^{{}}(\mathbf{x};\mathbf{y}) \right]}{\partial {{y}_{i}}\partial {{y}_{j}}} \right|}_{{{y}_{1}}={{\xi }_{1}}}} \right)d{{\xi }_{1}}}. \end{aligned} $$

(48)

Using integration by parts, we reformulate Eq. (48) as follows

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,h+1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}&\approx \frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\left({{{\left. {\frac{i}{{4\beta }}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- h}}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right|}_{{y_{1}} = {\xi_{1}}}}} \right)d{\xi_{1}}} \\ &=\frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\left\{ {{{\left. {\frac{\partial }{{\partial {y_{1}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- h}}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right]} \right|}_{{y_{1}} = {\xi_{1}}}}} \right\}d{\xi_{1}}} \\ &\quad-\frac{{{\partial^{2}}}}{{\partial {x_{i}}\partial {x_{j}}}}\int_{\infty}^{{y_{1}}} {\left\{ {{{\left. {\left[ {{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- 1}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right]\frac{\partial }{{\partial {y_{1}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- h}}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}} \right]} \right|}_{{y_{1}} = {\xi_{1}}}}} \right\}d{\xi_{1}}}. \\ \end{aligned} $$

(49)

From Eq. (16), we know that the last term on the right-hand side of Eq. (49) can be neglected compared with the left-hand side of Eq. (49). Similar to the derivation of Eq. (46), Eq. (49) can be approximated by

$$ \begin{aligned} \frac{{{\partial }^{2}}G_{^{2D}}^{1,h+1}(\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}\partial {{y}_{j}}}\approx \frac{{{\partial^{2}}}}{{\partial {y_{i}}\partial {y_{j}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi kR}}} \right)}^{1/2}}{{\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)}^{- (h + 1)}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right]. \end{aligned} $$

(50)

Equation (50) shows that Eq. (42) is valid when *l*=*h*+1 and *q*=1.

By using the Leibniz product rule, the right-hand side of Eq. (42) is equal to

$$ {}\begin{aligned} & \frac{{{\partial }^{2}}}{\partial {{y}_{i}}\partial {{y}_{j}}}\left[ \frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left({{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}} \right){{\exp }^{\varphi (\mathbf{x};\mathbf{y})}} \right] \\ &= \frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left[ {{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}\left(\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{i}}}\right)\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{j}}}\right)+o(1) \right) \right] \\ &\quad+ \frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left[ \frac{\partial \left({{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}} \right)}{\partial {{y}_{i}}}\frac{\partial \left({{\exp }^{\varphi (\mathbf{x};\mathbf{y})}} \right)}{\partial {{y}_{j}}} \right] \\ & \quad+\frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left[ \frac{\partial \left({{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}} \right)}{\partial {{y}_{j}}}\frac{\partial \left({{\exp }^{\varphi (\mathbf{x};\mathbf{y})}} \right)}{\partial {{y}_{i}}} \right] \\ & \quad+\frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left[ \frac{{{\partial }^{2}}\left({{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}} \right)}{\partial {{y}_{i}}\partial {{y}_{j}}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}} \right]. \\ \end{aligned} $$

(51)

In the far field, \(\frac {\partial \varphi (\mathbf {x};\mathbf {y})}{\partial {{y}_{i}}}\) is of order *O*(1), and the derivative of \({{R}^{-1/2}}{{\left (\frac {\partial \varphi (\mathbf {x};\mathbf {y})}{\partial {{y}_{1}}}\right)}^{-l}}\) can be expressed by \(o\left ({{R^{- 1/2}}{{\left ({\frac {{\partial \varphi (\mathbf {x};\mathbf {y})}}{{\partial {y_{1}}}}} \right)}^{- l}}} \right)\). Thus, the last three terms of Eq. (51) can be neglected compared with the first term on the right-hand side. We ignore the derivatives corresponding to \({{\left (\frac {\partial \varphi (\mathbf {x};\mathbf {y})}{\partial {{y}_{1}}}\right)}^{-l}}\) and obtain

$$ \begin{aligned} &\frac{{{\partial }^{2}}}{\partial {{y}_{i}}\partial {{y}_{j}}}\left[ \frac{i}{4\beta }{{\left(\frac{2{{\beta }^{2}}}{\pi k}\right)}^{1/2}}\left({{R}^{-1/2}}{{\left(\frac{\partial \varphi (\mathbf{x};\mathbf{y})}{\partial {{y}_{1}}}\right)}^{-l}} \right){{\exp }^{\varphi (\mathbf{x};\mathbf{y})}} \right]\\&\approx {\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)^{- l}}\frac{{{\partial^{2}}}}{{\partial {y_{i}}\partial {y_{j}}}}\left[ {\frac{i}{{4\beta }}{{\left({\frac{{2{\beta^{2}}}}{{\pi k}}} \right)}^{1/2}}{R^{- 1/2}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}} \right]\\ &={\left({\frac{{\partial \varphi (\mathbf{x};\mathbf{y})}}{{\partial {y_{1}}}}} \right)^{- l}}\frac{{{\partial^{2}}{G_{{\mathrm{2}}D}}}}{{\partial {y_{i}}\partial {y_{j}}}}. \end{aligned} $$

(52)

Finally, substituting Eq. (52) in Eq. (50) yields Eq. (40). Thus, approximations to the integral of Green’s function are derived.

We use a homogeneous vortical flow with a uniform Lighthill stress tensor to verify the far-field approximations to the integrals of Green’s function. We define the uniform Lighthill stress tensor as *T*_{ij}=*A* cos(2*π**f**t*), where *A* is 1/s ^{2}. The uniform vortical flow moves within a domain *V* of [0,10]×[0,10]. The frequency *f* is taken as 1Hz. The speed of sound is taken as *c*_{0}=340m/s, and the density is taken as *ρ*=1kg/m^{3}. The far-field observer is located at (340000m,340000m), which is sufficiently large compared to the wavelength of sound. The freestream Mach number is 0.3 along the *o*−*y*_{1} axis. The contribution from the Lighthill stress tensor to the far-field sound can be computed by using the quadrupole source term in the FW-H integral as follows

$$ \begin{aligned} {I_{Q{\mathrm,FW-H}}} = \int\limits_{V} {{T_{ij}}\frac{{{\partial^{2}}{G_{2D}}\left({\mathbf{x};\mathbf{y}} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}} {\mathrm{d}}\mathbf{y}. \end{aligned} $$

(53)

By using the far-field approximations to the integrals of Green’s function (Eq. (40)), the far-field sound can be approximated by

$$ \begin{aligned} {I_{Q{\mathrm,appr}}} &= {T_{ij}}\int\limits_{V} {\frac{{{\partial^{2}}{G_{2D}}\left({\mathbf{x};\mathbf{y}} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}} {\mathrm{d}}\mathbf{y}\\ &=\frac{1}{2} {T_{ij}}\int\limits_{V} {\frac{\partial }{{\partial {y_{1}}}}\left({{{\left({\frac{{\partial {\varphi_{2D}}}}{{\partial {y_{1}}}}} \right)}^{- 1}}\frac{{{\partial^{2}}{G_{2D}}\left({\mathbf{x};\mathbf{y}} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}} \right)} {{ + }}\frac{\partial }{{\partial {y_{2}}}}\left({{{\left({\frac{{\partial {\varphi_{2D}}}}{{\partial {y_{2}}}}} \right)}^{- 1}}\frac{{{\partial^{2}}{G_{2D}}\left({\mathbf{x};\mathbf{y}} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}} \right){\mathrm{d}}\mathbf{y}\\ &=\frac{1}{2} {T_{ij}}\int\limits_{S} {{{\left({\frac{{\partial {\varphi_{2D}}}}{{\partial {y_{q}}}}} \right)}^{- 1}}\frac{{{\partial^{2}}{G_{2D}}\left({\mathbf{x};\mathbf{y}} \right)}}{{\partial {y_{i}}\partial {y_{j}}}}} {n_{q}}{\mathrm{d}}\mathbf{y}. \end{aligned} $$

(54)

Figure 11 shows the results calculated by using Eqs. (53) and (54), respectively. The results computed by using the surface integral (Eq. (54)) match well with the volume integral (Eq. (53)), indicating that the approximation to the integrals of Green’s function is valid.

### 4.2 Appendix B: Derivation of Eqs. (15) and (16)

The detailed derivation of Eqs. (15) and (16) in Section 2.2 are given in this Appendix. Here, we start from the derivation of Eq. (15).

According to Eqs. (7) and (10), we rewrite *R* as

$$ \begin{aligned} R &= {\left({{g_{l}}r_{l}^{2}} \right)^{1/2}},\\ {g_{l}} &= 1 + \left({{\beta^{2}} - 1} \right)\left({1 - {\delta_{1l}}} \right),\\ {r_{l}} &= {y_{l}} - {x_{l}}, \end{aligned} $$

(55)

where the subscript *l*=1,2 in two dimension and *l*=1,2,3 in three dimension. The Einstein summation convention is used in Eq. (55). \(\beta =\sqrt {1-{{M}^{2}}}\) is the Prantle-Glauert factor. *M* is the Mach number of the freestream flow. *δ*_{ij} is the Kronecker delta function.

By defining \({\lambda _{1}} = g_{1}^{1/2}{r_{1}}\), the 1st- to 3rd-order derivatives of the distance *R* can be reformulated as follows

$$ \begin{aligned} \frac{{\partial R}}{{\partial y_{1}^{}}} = \frac{{\partial R}}{{\partial \lambda_{1}^{}}}\frac{{\partial {\lambda_{1}}}}{{\partial y_{1}^{}}} = g_{l}^{1/2}\frac{{\partial R}}{{\partial \lambda_{1}^{}}}, \end{aligned} $$

(56)

$$ \begin{aligned} \frac{{{\partial^{2}}R}}{{\partial y_{1}^{2}}} = \frac{\partial }{{\partial \lambda_{1}^{}}}\left({\frac{{\partial R}}{{\partial y_{1}^{}}}} \right)\frac{{\partial {\lambda_{1}}}}{{\partial y_{1}^{}}} = \frac{\partial }{{\partial \lambda_{1}^{}}}\left({g_{l}^{1/2}\frac{{\partial R}}{{\partial \lambda_{1}^{}}}} \right)\frac{{\partial {\lambda_{1}}}}{{\partial y_{1}^{}}} = g_{l}^{}\frac{{{\partial^{2}}R}}{{\partial \lambda_{1}^{2}}}, \end{aligned} $$

(57)

$$ \begin{aligned} \frac{{{\partial^{3}}R}}{{\partial y_{1}^{3}}} = \frac{\partial }{{\partial \lambda_{1}^{}}}\left({\frac{{{\partial^{2}}R}}{{\partial y_{1}^{2}}}} \right)\frac{{\partial {\lambda_{1}}}}{{\partial y_{1}^{}}} = \frac{\partial }{{\partial \lambda_{1}^{}}}\left({g_{l}^{}\frac{{{\partial^{2}}R}}{{\partial \lambda_{1}^{2}}}} \right)\frac{{\partial {\lambda_{1}}}}{{\partial y_{1}^{}}} = g_{l}^{3/2}\frac{{{\partial^{3}}R}}{{\partial \lambda_{1}^{3}}}. \end{aligned} $$

(58)

For the high-order derivatives of the distance *R*, we have

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}R}}{{\partial y_{1}^{{k_{1}}}}} = g_{l}^{{k_{1}}/2}\frac{{{\partial^{{k_{1}}}}R}}{{\partial \lambda_{1}^{{k_{1}}}}}. \end{aligned} $$

(59)

We can estimate that *g*_{l}∼*O*(1), since 0≤*β*<1, *m**a**x*(*δ*_{1l})=1, and *m**i**n*(*δ*_{1l})=0. For the computation of far-field sound, we approximately estimate the order of magnitude by using the assumption *λ*_{1}∼*O*(*R*). We notice that this assumption is not true for the observer near the vertical direction. However, it gives a reasonable approximation to the order of magnitude for the observer at the most part of the region. The numerical result in Section 3 also shows that the above relations give good approximations to the far-field sound pressure. The detailed investigation to the effects of this assumption on the results is expected to be conducted in the future.

We denote *R* as \(D{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with *D*=1, *m*_{1}=1 and *n*_{1}=0. Further, the derivative of \({R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with respect to *λ*_{1} is \({m_{1}}{R^{{m_{1}} - 2}}\lambda _{1}^{{n_{1}} + 1} + {n_{1}}{R^{{m_{1}}}}\lambda _{1}^{{n_{1}} - 1}\). \({R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) and \({m_{1}}{R^{{m_{1}} - 2}}\lambda _{1}^{{n_{1}} + 1} + {n_{1}}{R^{{m_{1}}}}\lambda _{1}^{{n_{1}} - 1}\) are of the order \(\phantom {\dot {i}\!}O({R^{{m_{1}} + {n_{1}}}})\) and \(\phantom {\dot {i}\!}O({R^{{m_{1}} + {n_{1}}-1}})\), respectively. In the most part of the region where *λ*_{1}∼*O*(*R*), the derivative of \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with respect to *λ*_{1} is one order smaller than \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\). Therefore, the *k*_{1}-th order derivative of *R* with respect to *λ*_{1} is of the order \(\phantom {\dot {i}\!}O(R^{1-k_{1}})\). According to Eq. (59), \(\phantom {\dot {i}\!}\frac {{{\partial ^{{k_{1}}}}R}}{{\partial y_{1}^{{k_{1}}}}}\) is thus of the order \(\phantom {\dot {i}\!}O(R^{1-k_{1}})\). Therefore, we can express the *k*_{1}th-order derivative of *R* with respect to *y*_{1} as

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}R}}{{\partial y_{1}^{{k_{1}}}}} \approx O\left({{{{R^{{-k_{1}}}}}}} \right)R. \end{aligned} $$

(60)

It is noted that the first order derivative of *R* with respect to *y*_{1} is \(\frac {{\partial R}}{{\partial y_{1}^{}}} = \frac {{{\lambda _{1}}}}{R}g_{l}^{{1}/2} \sim O(1)\) in the most part of the region where *λ*_{1}∼*O*(*R*). For Eq. (16), a similar formulation to the Eq. (63) can be obtained as follows

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}{R^{- 1/2}}}}{{\partial y_{1}^{{k_{1}}}}} = \frac{{{\partial^{{k_{1}}}}{R^{- 1/2}}}}{{\partial \lambda_{1}^{{k_{1}}}}}g_{l}^{{k_{1}}/2}. \end{aligned} $$

(61)

The derivation of Eq. (61) can be obtained by a similar way to that of Eq. (59) with the use of chain rule. It is noted that *R*^{−1/2} can be written as \(\phantom {\dot {i}\!}D{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with *D*=1, *m*_{1}=−1/2 and *n*_{1}=0. The derivative of \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with respect to *λ*_{1} is \(\phantom {\dot {i}\!}{m_{1}}{R^{{m_{1}} - 2}}\lambda _{1}^{{n_{1}} + 1} + {n_{1}}{R^{{m_{1}}}}\lambda _{1}^{{n_{1}} - 1}\). \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) and \(\phantom {\dot {i}\!}{m_{1}}{R^{{m_{1}} - 2}}\lambda _{1}^{{n_{1}} + 1} + {n_{1}}{R^{{m_{1}}}}\lambda _{1}^{{n_{1}} - 1}\) are of the order \(\phantom {\dot {i}\!}O({R^{{m_{1}} + {n_{1}}}})\) and \(\phantom {\dot {i}\!}O({R^{{m_{1}} + {n_{1}}-1}})\), respectively. In the most part of the region where *λ*_{1}∼*O*(*R*), the derivative of \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\) with respect to *λ*_{1} is one order smaller than \(\phantom {\dot {i}\!}{R^{m_{1}}}{\lambda _{1}}^{n_{1}}\). Therefore, the *k*_{1}th-order derivative of *R*^{−1/2} with respect to *λ*_{1} is of the order \(\phantom {\dot {i}\!}O(R^{1-k_{1}})\). According to Eq. (61), \(\phantom {\dot {i}\!}\frac {{{\partial ^{{k_{1}}}}R^{-1/2}}}{{\partial y_{1}^{{k_{1}}}}}\) is thus of the order \(\phantom {\dot {i}\!}O(R^{-1/2-k_{1}})\). Therefore, we can express the *k*_{1}-th order derivative of *R* with respect to *y*_{1} as

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}R^{-1/2}}}{{\partial y_{1}^{{k_{1}}}}} \approx O\left({{{{R^{{-k_{1}}}}}}} \right)R^{-1/2}. \end{aligned} $$

(62)

### 4.3 Appendix C: Derivation of Eq. (17)

The detailed derivation of Eq. (17) is reported in this Appendix. We expand the *k*_{1} th-order derivative of exp*φ* as follows

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}}}{{\partial y_{1}^{{k_{1}}}}}{\exp^{\varphi}} &= {\left({\frac{{\partial {\varphi}}}{{\partial {y_{1}}}}} \right)^{{k_{1}}}}{\exp^{\varphi}} + {E_{1}}{\left({\frac{{\partial {\varphi}}}{{\partial {y_{1}}}}} \right)^{{k_{1}} - 1}}\frac{{{\partial^{2}}{\varphi}}}{{\partial y_{1}^{2}}}{\exp^{\varphi}} \\ &\quad+{E_{2}}{\left({\frac{{\partial {\varphi}}}{{\partial {y_{1}}}}} \right)^{{k_{1}} - 2}}\frac{{{\partial^{3}}{\varphi}}}{{\partial y_{1}^{3}}}{\exp^{\varphi}} + {E_{3}}{\left({\frac{{\partial {\varphi }}}{{\partial {y_{1}}}}} \right)^{{k_{1}} - 3}}\frac{{{\partial^{4}}{\varphi}}}{{\partial y_{1}^{4}}}{\exp^{\varphi} }\\ &\quad+{E_{4}}{\left({\frac{{\partial {\varphi }}}{{\partial {y_{1}}}}} \right)^{{k_{1}} - 4}}{\left({\frac{{{\partial^{2}}{\varphi}}}{{\partial y_{1}^{2}}}} \right)^{2}}{\exp^{\varphi}} +..., \end{aligned} $$

(63)

where *E*_{1}, *E*_{2}, *E*_{3}, *E*_{4}... are the coefficients corresponding to the number of derivative’s order *k*_{1}.

We start from the derivation from the Eq. (17) in the two-dimensional space. According to Eq. (6), the exponent *φ* of the 2D Green’s function for the FW-H equation is as follows

$$ {\varphi}(\mathbf{x};\mathbf{y}) = {\mathrm{i}}\left[ {\frac{{Mk({x_{1}} - {y_{1}})}}{{{\beta^{2}}}} + \frac{\pi }{4} - \frac{k}{{{\beta^{2}}}}R} \right],\\ R = \sqrt {{{({x_{1}} - {y_{1}})}^{2}} + {\beta^{2}}{{({x_{1}} - {y_{1}})}^{2}}}, $$

(64)

where \(\mathrm {i} = \sqrt {-1}\) is the unit of the imaginary number. *k*=*ω*/*c*_{o} is the acoustic wavenumber where *ω* and *c*_{o} are the angular frequency and the speed of sound, respectively. **x** and **y** are the coordinate of observer and source locations, respectively. \(\beta =\sqrt {1-{{M}^{2}}}\) is the Prantle-Glauert factor. *M* is the Mach number of the freestream flow.

The derivative of the exponent *φ* with respect to *y*_{1} is

$$ \frac{{\partial \varphi }}{{\partial y_{1}^{}}}=\frac{{ - {\mathrm{i}}Mk}}{{{\beta^{2}}}} - \frac{{\mathrm{i}}k}{{{\beta^{2}}}}\frac{{\partial R}}{{\partial {y_{1}}}}. $$

(65)

The *k*_{1}th-order (*k*_{1}≥2) derivative of *φ* with respect to *y*_{1} is

$$ \frac{{{\partial^{{k_{1}}}}{\varphi}}}{{\partial y_{1}^{{k_{1}}}}} = \frac{{ - {\mathrm{i}}k}}{{{\beta^{2}}}}\frac{{{\partial^{{k_{1}}}}R}}{{\partial y_{1}^{{k_{1}}}}}. $$

(66)

When \(\frac {{{\partial ^{{k_{1}}}}{\varphi }}}{{\partial y_{1}^{{k_{1}}}}} = 0\) (*k*_{1}≥2), Eq. (63) reduces to

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}}}{{\partial y_{1}^{{k_{1}}}}} = {\left({\frac{{{\partial^{}}\varphi (\mathbf{x};\mathbf{y})}}{{\partial y_{1}^{}}}} \right)^{{k_{1}}}}{\exp^{\varphi (\mathbf{x};\mathbf{y})}}{{ }}. \end{aligned} $$

(67)

When \(\frac {{{\partial ^{{k_{1}}}}{\varphi }}}{{\partial y_{1}^{{k_{1}}}}} \ne 0\) (*k*_{1}≥2), the ratio between \(\frac {{\partial \varphi }}{{\partial y_{1}^{}}}\) and \(\frac {{{\partial ^{{k_{1}}}}{\varphi }}}{{\partial y_{1}^{{k_{1}}}}}\) (*k*_{1}≥2) is

$$ \frac{\frac{{\partial \varphi }}{{\partial y_{1}^{}}}}{\frac{{{\partial^{{k_{1}}}}{\varphi}}}{{\partial y_{1}^{{k_{1}}}}}} = \frac{M+\frac{{\partial R}}{{\partial {y_{1}}}}}{\frac{{{\partial^{{k_{1}}}}R}}{{\partial y_{1}^{{k_{1}}}}}}. $$

(68)

For subsonic flows with 0≤*M*<1, we have \({\frac {{\partial \varphi }}{{\partial y_{1}^{}}}}/{\frac {{{\partial ^{{k_{1}}}}{\varphi }}}{{\partial y_{1}^{{k_{1}}}}}} \sim O(R^{k_{1}-1})\) according to Eq. (60), which means that \({\frac {{\partial \varphi }}{{\partial y_{1}^{}}}}\) is much larger than \({\frac {{{\partial ^{{k_{1}}}}{\varphi }}}{{\partial y_{1}^{{k_{1}}}}}}\) (*k*_{1}≥2) at the far-field (*R*≫1). It is noticed that *E*_{1}, *E*_{2}, *E*_{3}, *E*_{4}... in Eq. (63) are dependent on the order of the derivative *k*_{1} and independent of the distance *R*. For the computation of sound at far field, we can always find large enough distance *R* which is much larger than the coefficients *E*_{1}, *E*_{2}, *E*_{3}, *E*_{4}.... Therefore, by ignoring the terms including high order derivatives of exp*φ* in Eq. (63), Eq. (63) can be approximately computed as follows,

$$ \begin{aligned} \frac{{{\partial^{{k_{1}}}}{{\exp }^{\varphi (\mathbf{x};\mathbf{y})}}}}{{\partial y_{1}^{{k_{1}}}}} \approx {\left({\frac{{{\partial^{}}\varphi (\mathbf{x};\mathbf{y})}}{{\partial y_{1}^{}}}} \right)^{{k_{1}}}}{\exp^{\varphi (\mathbf{x};\mathbf{y})}}{{ }}. \end{aligned} $$

(69)

Equation (17) in Section 2 can be obtained by combining Eqs. (67) and (69). The derivation of Eq. (17) for in three-dimensional space can be obtained in a similar way.