## 2 Analytical Model
### Root Means Square (RMS) Pressure
#### 2.3.2 Particular Case of a Harmonic Dipolar Source  
Incident, Reflected and Total RMS Pressure Fields  
First, one can note that all \(p_{s}\) terms in Equation (54) can be expressed under a similar form (with \(r^{*}=r,r^{\prime},r^{\prime\prime}\) and \(\theta^{*}=\theta,\theta^{\prime},\theta^{\prime\prime}\))  
\[p_{s}(r^{*},\theta^{*},t)\approx A^{*}sin(a^{*}-\omega t) \tag{55}\]  
In the above equation, \(A^{*}\) and \(a^{*}\) relate to the source directivity and wavenumber, respectively. The former term is given by  
\[A^{*}=A_{max}\bigg{[}\frac{cos(\theta^{*})}{\sqrt{r^{*}}}\bigg{]} \tag{56}\]  
On the other hand, the second term is expressed as  
\[a^{*}=k_{0}r^{*}+\frac{\pi}{4} \tag{57}\]  
Through simple trigonometric algebra (cf. Appendix A), it is easy to show that  
\[\frac{1}{T}\int_{0}^{T}\lvert sin(a-\omega t)\rvert^{2}dt=\frac{1}{2} \tag{58}\]  
and  
\[\frac{1}{T}\int_{0}^{T}sin(a-\omega t)sin(b-\omega t)dt=\frac{1}{2}cos(a-b) \tag{59}\]  
From there, one gets  
\[\frac{1}{T}\int_{0}^{T}\lvert p_{s}(r^{*},\theta^{*},t)\rvert^{2}dt=\frac{1} {2}\lvert A^{*}\rvert^{2} \tag{60}\]  
and  
\[\frac{1}{T}\int_{0}^{T}p_{s}(r^{*},\theta^{*},t)p_{s}(r^{**},\theta^{**},t)dt =\frac{A^{*}A^{**}}{2}cos(a^{*}-a^{**}) \tag{61}\]  
All in all, this leads to,  
\[\frac{1}{T}\int_{0}^{T}\lvert p(\mathbf{x},t)\rvert^{2}dt=\frac{1}{2}\Big{(}A^{2} +{A^{\prime}}^{2}+{A^{\prime}}^{2}\Big{)}+AA^{\prime}cos\big{(}a-a^{\prime} \big{)}+AA^{\prime\prime}cos(a-a^{\prime\prime})+A^{\prime}A^{\prime\prime} cos\big{(}a^{\prime}-a^{\prime\prime}\big{)} \tag{62}\]  
i.e.,  
\[p_{RMS}(\mathbf{x})=\sqrt{\frac{1}{2}\Big{(}A^{2}+{A^{\prime}}^{2}+{A^{\prime }}^{2}\Big{)}+AA^{\prime}cos(a-a^{\prime})+AA^{\prime\prime}cos(a-a^{\prime \prime})+A^{\prime}A^{\prime\prime}cos(a^{\prime}-a^{\prime\prime})} \tag{63}\]  
or  
\[p_{RMS}(\mathbf{x})=A_{max}\sqrt{ \begin{array}{c}\frac{1}{2} \bigg{[}\frac{cos(\theta)^{2}}{r}+\frac{cos(\theta^{\prime})^{2}}{r^{\prime}} +\frac{cos(\theta^{\prime\prime})^{2}}{r^{\prime\prime}}\bigg{]}+\frac{cos( \theta)cos(\theta^{\prime})}{\sqrt{rr^{\prime}}}cos\left(k_{0}(r-r^{\prime}) \right)\\ +\frac{cos(\theta)cos(\theta^{\prime\prime})}{\sqrt{rr^{\prime\prime}}}cos(k_ {0}(r-r^{\prime\prime}))+\frac{cos(\theta^{\prime})cos(\theta^{\prime\prime}) }{\sqrt{r^{\prime}r^{\prime\prime}}}cos(k_{0}(r^{\prime}-r^{\prime\prime})) \end{array} } \tag{64}\]  
In the above identity, the three first terms under the square root translate the contribution of the incident and the two reflected fields, respectively. The three last terms rather translate the interferences between these various fields, with either reinforcement or cancellation effects. Here, it is worth noticing that the three terms associated with either the incident or the reflected fields do _not_ depend on the source frequency, but solely on its location, via \((r^{*},\theta^{*})\). Oppositely, the three terms associated with these fields' interferences are driven byboth the source location and frequency_ (via the wavenumber, \(k_{0}\)). Finally, the amplitude term, \(A_{max}\), solely depends on the frequency (via the pulsation, \(\omega\)). Incident, Reflected and Total RMS Pressure Fields at the Wall  
For an observer that would be located on the top wall, the resulting overall RMS pressure can be obtained by either applying to Equation (48) the time integration of Equation (50) or, more simply, by recasting Equation (64) using the particular identities \({r_{+}}^{\prime}=r_{+}\) and \({\theta_{+}}^{\prime}=\theta_{+}\) that are verified at the wall (see Section 2.2.4).