# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 3 Particle-Velocity-Based Mixed-Source Expansion
### Spherical Harmonic Representation of Particle Velocity  
Particle velocity is a vector and it can be expressed in spherical coordinates, as  
\[\mathbf{V}(\mathbf{x},k)=V_{r}(\mathbf{x},k)\hat{\mathbf{r}}+V_{\theta}(\mathbf{x},k)\hat{\mathbf{ \theta}}+V_{\phi}(\mathbf{x},k)\hat{\mathbf{\phi}}, \tag{4}\]  
where \(\hat{\mathbf{r}}\), \(\hat{\mathbf{\theta}}\), and \(\hat{\mathbf{\phi}}\) are the unit vectors in the \(r\), \(\theta\), and \(\phi\) directions, respectively. We have recently formulated continuous particle velocity in the spherical harmonic domain [44], which decomposes each component of the particle velocity vector in terms of spherical harmonic functions, as  
\[V_{\Psi}(\mathbf{x},k)=\sum_{p=0}^{Q_{\Psi}}\sum_{q=-p}^{p}X_{pq}^{(\Psi)}(k,|\bm {x}|)Y_{pq}(\hat{\mathbf{x}});\quad\Psi\in\{r,\theta,\phi\}, \tag{5}\]  
where \(Q_{\Psi}\) is the particle velocity truncation order (the spherical harmonic decomposition of particle velocity also contains infinite summation terms, and we truncate it to an explicit order for implementation) in the \(\Psi\) direction, and \(X_{pq}^{(\Psi)}(k,|\mathbf{x}|)\) denote the velocity coefficients in the \(\Psi\) direction. According to the results in [44], the velocity coefficients are given as  
\[X_{pq}^{(r)}(k,|\mathbf{x}|)=\frac{i}{k\rho_{0}c}\alpha_{pq}(k)j_{p}^{q}(k|\mathbf{x}|), \tag{6}\]  
\[X_{pq}^{(\theta)}(k,|\mathbf{x}|)=\frac{2\pi i}{k\rho_{0}c}H_{pq}\sum_{n=|q|}^{N} H_{mq}\alpha_{nq}(k)\frac{j_{n}(k|\mathbf{x}|)}{|\mathbf{x}|}G_{1}, \tag{7}\]  
\[X_{pq}^{(\phi)}(k,|\mathbf{x}|)=-\frac{2\pi q}{k\rho_{0}c}H_{pq}\sum_{n=|q|}^{N}H_ {mq}\alpha_{mq}(k)\frac{j_{n}(k|\mathbf{x}|)}{|\mathbf{x}|}G_{2}, \tag{8}\]where \(\rho_{0}\) is the medium density, \(j^{\prime}_{p}(k|\mathbf{x}|)\) denotes the derivative of \(j_{p}(k|\mathbf{x}|)\) with respect to \(r\), and  
\[H_{pq}=\frac{(-1)^{\frac{q+|q|}{2}}}{2^{|q|}(|q|)!}\sqrt{\frac{(2p+1)}{4\pi} \frac{(p+|q|)!}{(p-|q|)!}}, \tag{9}\]  
\[\begin{split} G_{1}&=(n+|q|+1)\mathcal{G}(a,b+\frac {1}{2}-\delta_{n+|q|+1};-\mu_{1}-\delta_{n+|q|+1},-\mu_{2};v_{1}+\delta_{n+|q|},v_{2};\xi_{1},\xi_{2})\\ &-(n+1)\mathcal{G}(a,b+\frac{1}{2};-\mu_{1},-\mu_{2};v_{1},v_{2}; \xi_{1},\xi_{2}),\end{split} \tag{10}\]  
\[\begin{split} G_{2}&=\mathcal{G}(a,b;-\mu_{1},-\mu_ {2};v_{1},v_{2};\xi_{1},\xi_{2})\\ &=[(-1)^{2b+1}+1]\sum_{j_{1}=0}^{\mu_{1}}\sum_{j_{2}=0}^{\mu_{2}} \frac{(-\mu_{1})_{j_{1}}(v_{1})_{j_{1}}}{(\xi_{1})_{j_{1}}!}\frac{(-\mu_{2})_ {j_{2}}(v_{2})_{j_{2}}}{(\xi_{2})_{j_{2}}!}\frac{B(j_{1}+j_{2}+a,b)}{2},\end{split} \tag{11}\]  
with  
\[\delta_{M}=\begin{cases}1,&\text{if $M$ is even}\\ 0,&\text{if $M$ is odd},\end{cases} \tag{12}\]  
\[(a)_{j}=\begin{cases}1,&\text{if $j=0$}\\ a(a+1)\dots(a+j-1),&\text{if $j=1,2,\dots$},\end{cases} \tag{13}\]  
and \(a=(2|q|+1)/2\), \(b=(3-\delta_{n+|q|}-\delta_{p+|q|})/2\), \(\mu_{1}=-(1-n+|q|-\delta_{n+|q|})/2\), \(\mu_{2}=-(1-p+|q|-\delta_{p+|q|})/2\), \(\
u_{1}=(2+n+|q|-\delta_{n+|q|})/2\), \(v_{2}=(2+p+|q|-\delta_{p+|q|})/2\), \(\xi_{1}=\xi_{2}=|q|+1\), and \(B(\cdot)\) denotes the beta function. The detailed derivations of (6)-(8) can also be found in [44].  
Note that for a sound field the truncation order of the velocity field (i.e., \(\mathcal{Q}_{\mathbf{\tau}}\)) is usually larger than or equal to the truncation order of the pressure field (i.e., \(N\)) due to the truncation theorem of the velocity expressions [44]. The velocity coefficients represent the continuous particle velocity distribution of a sound field over space and they are directly determined by the pressure coefficients of the sound field, as well as the radial functions (i.e., the spherical Bessel functions). All other terms in the expressions of the velocity coefficients are either constants or indices-only dependent variables. In other words, we can obtain the continuous particle velocity distribution of a sound field over space directly from the pressure coefficients describing the sound field.