The wave equation
\[ \Delta p(\mathbf{x}, t) = \frac{1}{c^2}\frac{\partial^2p(\mathbf{x}, t)}{\partial t^2}, \quad \mathbf{x} \in \Omega \subset \mathbb{R}^3, \]describes wave phenomena such as the ones encountered in linear acoustical problems. In the above \(p\) is the sound pressure, \(c\) is the wave propagation speed and \(\mathbf{x}\) and \(t\) the position and time respectively. For time-harmonic problems it is standard to introduce a time dependency as follows
\[ p(\mathbf{x}, t) = \text{Re}\left\{p(\mathbf{x})\mathrm{e}^{\alpha \mathrm{i} \omega t}\right\}, \]where \(\alpha = \pm 1\). The fact that the time dependency can be chosen with \(\alpha\) being either \(1\) or \(-1\) is a constant cause of confusing when reading papers within the field of acoustics. Now inserting the time-harmonic solution into the wave equation the Helmholtz equation appears
\[ \Delta p(\mathbf{x}) + k^2p(\mathbf{x}) = 0, \quad \mathbf{x} \in \omega. \]Note that the above is independent of the choice of \(\alpha\) (thereby the confusing). However, the relation between the normal derivative of the sound pressure and the normal velocity does depend on the chosen time dependency,
\[ \frac{\partial p(\mathbf{x})}{\partial n(\mathbf{x})} = s(\alpha)kv_n(\mathbf{x}), \quad \mathbf{x} \in \Gamma. \]Where it was chosen to introduce
\[ s(\alpha) = -\alpha\mathrm{i} \rho_0 c, \]with \(\rho_0\) being the ambient density of the fluid.
Multiplying the Helmholtz equation with a test function (\(q\)) and integrating over the domain of interest we get that
\[ \int_\Omega q(\mathbf{x})\left[\Delta p(\mathbf{x}) + k^2 p(\mathbf{x})\right]\ \mathrm{d}\Omega_\mathbf{x} = 0. \]Integrating by parts twice it follows that
\[ \begin{aligned} \int_\Omega q(\mathbf{x})\left[\Delta p(\mathbf{x}) + k^2 p(\mathbf{x})\right]\ \mathrm{d}\Omega_\mathbf{x} = s(\alpha)k&\int_\Gamma q(\mathbf{x})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} - \int_\Gamma\frac{\partial q(\mathbf{x})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x}\\ +&\int_\Omega p(\mathbf{x})\left[\Delta q(\mathbf{x}) + k^2q(\mathbf{x})\right]\ \mathrm{d}\Omega_\mathbf{x} = 0. \end{aligned} \]We now introduce the Green's function
\[ G(\beta, \alpha, \mathbf{x}, \mathbf{y}) = \alpha\beta \frac{\mathrm{e}^{-\alpha\mathrm{i} k \|\mathbf{x} - \mathbf{y}\|_2 }}{4\pi\|\mathbf{x} - \mathbf{y}\|_2}, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^3, \]as the solution to the modified Helmholtz equation
\[ \Delta G(\alpha, \mathbf{x}, \mathbf{y}) + k^2G(\alpha, \mathbf{x}, \mathbf{y}) = \beta\delta(\mathbf{x}, \mathbf{y}). \]Simply inserting \(q(\mathbf{x}) = G(\beta, \alpha, \mathbf{x}, \mathbf{y})\) into (7)
\[ \beta c(\mathbf{y})p(\mathbf{y}) - \int_\Gamma\frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma G(\alpha,\beta,\mathbf{x},\mathbf{y})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} = 0. \]It turns out that while there is freedom in choosing either \(\alpha\) or \(\beta\) they both can not be varied at the same time. In fact, it must be true that \(\alpha\beta=1\), meaning that the chosen time dependency defines the resulting Green's function.
What can cause confusing is that the integral equation can be written as
\[ \beta \left(c(\mathbf{y})p(\mathbf{y}) - \int_\Gamma\frac{\partial G(\alpha,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma G(\alpha,\mathbf{x},\mathbf{y})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x}\right) = 0, \]where
\[ G(\beta, \alpha, \mathbf{x}, \mathbf{y}) = \beta G(\alpha, \mathbf{x}, \mathbf{y}). \]Now (11) implies that
\[ c(\mathbf{y})p(\mathbf{y}) - \int_\Gamma\frac{\partial G(\alpha, \mathbf{x}, \mathbf{y})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} - s(\alpha)k\int_\Gamma G(\alpha,\mathbf{x},\mathbf{y})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} = 0. \]This can cause confusion since possible sign are hidden within \(G(\alpha, \mathbf{x}, \mathbf{y})\). What makes things worse is that the Green's function in litterature is usually presented as
\[ G(\mathbf{x},\mathbf{y}) = \frac{\mathrm{e}^{-\alpha\mathrm{i} k \|\mathbf{x} - \mathbf{y}\|_2 }}{4\pi\|\mathbf{x} - \mathbf{y}\|_2}, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^3, \]meaning that various signs of the integral in (13) can be found. To make things worse there is also various definitions of the directions of the normal vectors, which result in even more confusion in relation to the signs of the integrals.