## 2 Materials and Methods
### The Spec-Radiation Method  
The spec-radiation method is based on the angular spectrum method. The following derivation is mainly based on [PERSON] [41], who described it for light, and on [PERSON] and [PERSON] [38], who described it for acoustics.  
The Helmholtz equation describes the propagation of acoustic pressure waves in an incompressible fluid:  
\[\
abla^{2}p(x,y,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t}p(x,y,z,t)=0, \tag{1}\]  
where \(x,y,z\) are the space coordinates, \(t\) is the time, \(c\) is the speed of sound in the corresponding medium, which is assumed to be constant here, and \(p(x,y,z,t)\) describes the acoustic pressure field in space and time. If Equation (1) is transferred to the Fourier domain, the following results are obtained:  
\[\
abla^{2}p(\mathbf{x},\omega)+kp(\mathbf{x},\omega)=0, \tag{2}\]  
with the space coordinate vector \(\mathbf{x}=[x,y,z]\) and:  
\[k=\frac{2\pi}{\lambda}=\frac{\omega}{c} \tag{3}\]  
where \(\omega\) is the time dependent angular frequency. The acoustic sound pressure \(p(\mathbf{k},\omega)\) can be written as follows:  
\[p(\mathbf{k},\omega)=\iiint\int_{-\infty}^{\infty}p(\mathbf{x},t)\mathrm{e}^{- \mathrm{i}(\mathbf{k}\cdot\mathbf{x}+\omega t)}\mathrm{d}x\ \mathrm{d}y\ \mathrm{d}z\ \mathrm{d}t. \tag{4}\]Here, the wave vector \(\mathbf{k}=[k_{x},k_{y},k_{z}]\) and \(k_{x},k_{y},k_{z}\) are the spatial angular frequencies and i is the complex unit. Equation (4) satisfies Equation (2) when the exponent factor also satisfies Equation (2), and we get:  
\[k_{z}=2\pi\sqrt{\left(\frac{1}{\lambda}\right)^{2}-f_{x}^{2}-f_{y}^{2}}, \tag{5}\]  
with \(k_{x}=2\pi f_{x}\) and \(k_{y}=2\pi f_{y}\). The sound pressure distribution in the plane \(z\):  
\[p(k_{x},k_{y},z,\omega)=\mathrm{e}^{-\mathrm{i}k_{z}z}\iint_{-\infty}^{\infty}p (x,y,z,\omega)\mathrm{e}^{-\mathrm{i}(k_{x}x+k_{y}y)}\mathrm{d}x\ \mathrm{d}y. \tag{6}\]  
This now describes the propagation in the \(z\)-direction with the spatial frequencies \(f_{x}\) and \(f_{y}\). Two cases must be distinguished:  
\[\begin{split} 1.& f_{x}^{2}+f_{y}^{2}\leq\frac{1}{ \lambda^{2}}\\ 2.& f_{x}^{2}+f_{y}^{2}>\frac{1}{\lambda^{2}}\end{split} \tag{7}\]  
The first case describes homogeneous waves and the second case describes exponentially decaying evanescent waves. If the spectrum of a plane, e.g., the measuring plane with the coordinates \(x=x^{\prime}\), \(y=y^{\prime}\) and \(z=z^{\prime}\), is known at \(z^{\prime}=0\):  
\[\mathfrak{F}(p(x^{\prime},y^{\prime},0,\omega))=\iint_{-\infty}^{\infty}p(x^{ \prime},y^{\prime},0,\omega)\mathrm{e}^{-\mathrm{i}(k_{x}x^{\prime}+k_{y}y^{ \prime})}\mathrm{d}x\ \mathrm{d}y \tag{8}\]  
with \(\mathfrak{F}\) as the Fourier transform operator and Equation (8) substituted into Equation (5), resulting in:  
\[p(k_{x},k_{y},z,\omega)=\mathfrak{F}(p(x^{\prime},y^{\prime},0,\omega))\ \mathrm{e}^{-\mathrm{i}k_{z}z}. \tag{9}\]  
To display the result in the spatial domain and not in the Fourier domain, an \(\mathfrak{F}^{-1}\) is necessary:  
\[p(x,y,z,\omega)=\mathfrak{F}^{-1}\left(\mathfrak{F}(p(x^{\prime},y^{\prime},0, \omega))\ \mathrm{e}^{-\mathrm{i}k_{z}z}\right). \tag{10}\]  
Depicted here is that only factor \(\mathrm{e}^{-\mathrm{i}k_{z}z}\) is responsible for the propagation of the sound wave. Equation (10) becomes with propagation function \(H(k_{x},k_{y},z)\):  
\[p(x,y,z,\omega)=\mathfrak{F}^{-1}\left(\mathfrak{F}(p(x^{\prime},y^{\prime},0, \omega))\ H(k_{x},k_{y},z)\right). \tag{11}\]  
The propagation function can now take two forms, depending on whether the sound waves are to be calculated forward in time or backward. For both forward or backward propagation, the propagation function is now defined as:  
\[H(k_{x},k_{y},z)=e^{-\mathrm{i}k_{z}z}. \tag{12}\]