Acoustic wave equation seeks to describe wavefield in terms of a volumetric radiation source, $s$, or a surface source. For the latter, an associated boundary-value problem yields a description for the wavefield in terms of a Kirchhoff-Helmholtz or Rayleigh-Sommerfeld integral. A rigid-baffle assumption for the surface source (aperture) gives an integral formula in terms of a monopole source $-\partial p /\partial \boldsymbol{n}$, or equivalently $\rho_0 \partial ( \boldsymbol{u} \cdot {\boldsymbol{n}}) / \partial t$. Here, $p$ is the pressure, $\rho_0$ is the ambient density, $\boldsymbol{u}$ is the velocity vector and $\boldsymbol{n}$ is a unit vector normal to the surface. Alternatively, a soft-baffle assumption yields an integral formula in terms of a dipole source $p$. One example is interior-field dipole formula, which describes back-projected wavefield in terms of measurements over a surface. It will be shown theoretically that the amplitude calculated by the dipole formula is a function of obliquity factor or equivalently solid angle, a parameter which has not been included in the state-of-the-art approaches for full-field approximation. It will be shown numerically for a specific case how inclusion of the obliquity factors in a full-field approximation of the dipole integral formula yields a solution which matches the associated analytic formula.

翻译：声波方程旨在通过体积辐射源$s$或表面源来描述波场。对于后者，相关的边值问题通过基尔霍夫-亥姆霍兹或瑞利-索末菲积分给出波场的描述。对表面源（孔径）采用刚性障板假设时，可得到以单极源$-\partial p /\partial \boldsymbol{n}$（等价于$\rho_0 \partial ( \boldsymbol{u} \cdot {\boldsymbol{n}}) / \partial t$）表示的积分公式。其中$p$为声压，$\rho_0$为环境密度，$\boldsymbol{u}$为速度矢量，$\boldsymbol{n$为表面法向单位矢量。若采用软障板假设，则得到以偶极源$p$表示的积分公式。例如内场偶极公式即通过表面测量值描述反投影波场。理论分析表明，偶极公式计算得到的振幅是倾斜因子（等价于立体角）的函数，该参数在当前先进的全场近似方法中均未纳入。通过特定案例的数值模拟将证明：在偶极积分公式的全场近似中纳入倾斜因子后，所得解与对应的解析公式完全吻合。