I'm trying to model the propagation of a dipole source and I need to know what is the directivity function (i.e. the polar pattern function) which describes it. I've read in an article that a dipole can be described with this formula:
where the last exponential represent the Green's function right? So what is here the directivity function? Thank you :)
The derivation of the directivity function of the dipole below is that presented in "Acoustics - An Introduction" by Heinrich Kuttruff. I am very confident though that the same derivation can be found in other introductory textbooks of Acoustics.
For completeness we first present, without any kind of derivation or proof, some information about point/monopole sources needed in the derivation of the dipole source directivity function. The information presented here can also be found in many introductory textbooks.
In spherical coordinates, the pressure function of a point source (monopole) is given by
$$ p \left( r , t \right) = \frac{j \omega \rho Q}{4 \pi r} e^{j \left(\omega t - k r \right)} \tag{1} \label{monopole} $$
with $j$ denoting the imaginary unit for which $j^{2} = -1$ holds true, $\omega$ the radial frequency for which $\omega = 2 \pi f$ holds true with $f$ being the temporal frequency, $\rho$ the medium density (usually the unperturbed for linear acoustics), $Q$ the "strength" of the source (actually is the volume of fluid displaced by the source and for a pulsating sphere it equals the product of surface area $4 \pi \alpha^{2}$, with $\alpha$ the radius of the sphere and surface velocity $U_{0}$, $Q = 4 \pi \alpha^{2} U_{0}$), $r$ the distance from the source and $k$ the wave number for which $k = \frac{\omega}{c} = \frac{2 \pi}{\lambda}$ equals true, with $c$ the wave propagation speed and $\lambda$ the wavelength.
The dipole is modelled as the sum of two monopoles with amplitudes $Q$ and $-Q$ (with opposite phase, denote by the opposite signs) a distance $d$ apart. The distance $d$ should be substantially greater than the radius $a$ so that at the vicinity of each monopole, so that the acoustic pressure field in the vicinity of either source will be dominated by the $\frac{1}{r}$ term.
Below is a graph depicting the setup of a dipole source as described by the sum of two monopole sources (source: Acoustics - An Introduction by Heinrich Kuttruff)
Using equation \eqref{monopole} to express the pressure resulting from each source (the second with a negative sign) and adding them we get
$$ p \left( r, t \right) = \frac{j \omega \rho Q}{4 \pi} \left( \frac{e^{-j k r_{1}}}{r_{1}} - \frac{e^{-j k r_{2}}}{r_{2}} \right) e^{j \omega t} \tag{2} \label{sum-of-monopoles} $$
Assuming that $r_{1, 2} \gg d$ and using the approximation $r_{1, 2} \approx r \mp \frac{d}{2} \cos \left( \theta \right)$ we get from equation \eqref{sum-of-monopoles}
$$ p \left( r, t \right) = \frac{j \omega \rho Q}{4 \pi r} \left( e^{j \left( k d/2 \right) \cos \left( \theta \right)} - e^{-j \left( k d/2 \right) \cos \left( \theta \right) } \right) e^{j \left( \omega t - k r \right)} \implies \\ p \left( r, t \right) = -\frac{\omega \rho Q}{2 \pi r} \sin \left( \frac{k d}{2} \cos \left( \theta \right) \right) e^{j \left( \omega t - k r \right)} \tag{3} \label{dipole-complete} $$
In most cases, $d$ is very small compared to the wavelength $\lambda$ and thus $kd \ll 1$. In this case, $\sin \left( \frac{k d}{2} \cos \left( \theta \right) \right)$ can be replaced with its argument and equation \eqref{dipole-complete} becomes
$$ p \left( r, t \right) = -\frac{\omega \rho Q}{2 \pi r} \frac{k d}{2} \cos \left( \theta \right) e^{j \left( \omega t - k r \right)} \tag{4} \label{dipole-simple} $$
The directivity function (in almost all cases) taken as the far-field radiation characteristic function of the source. We have anyway assumed a rather large distance $r$ compared to the inter-element distance $d$ in order to "get rid" of the the $\sin$ function this is like stating we are in the far-field anyway.
Fixing the distance $r$ to some value, picking appropriate values for the frequency $f$, medium density $\rho$, source strength $Q$ and wave propagation speed $c$ (or directly calculating the wavenumber $k$) you can express equation \eqref{dipole-simple} like
$$ p \left( r, t \right) = -\frac{\omega \rho Q}{2 \pi r} \frac{k d}{2} e^{j \left( \omega t - k r \right)} \cos \left( \theta \right) \tag{5} \label{dipole-fixed} $$
where for either fixed time $t$, or for average value in time, the first part of equation \eqref{dipole-fixed}, $-\frac{\omega \rho Q}{2 \pi r} \frac{k d}{2} e^{j \left( \omega t - k r \right)}$ is just a constant value. Thus the last part, $\cos \left(\theta\right)$, constitutes the variability with angle $\theta$, which is the directivity function of the dipole source.
As stated above, you can fix all values taking place in the calculation of the acoustic pressure field generated by a dipole source. As it is rather apparent from equation \eqref{dipole-fixed}, the only difference this will make is on the amplitude of the pressure at the specified location (radial distance) since the variability with respect to the angle $\theta$ is only presented in the cosine function which is independent of any other parameters.
Thus, changing any of the aforementioned parameters the directivity will remain unchanged. This is due to the approximation of far-field radiation shown in the $k d \ll 1$ assumption. For a "more accurate" relation (whether this will provide any improvements to your problem depends on the problem itself) you could use equation \eqref{dipole-complete}. In this case, you can see that the dipole will be indeed a dipole-like source for small values of the argument of the sine function $\frac{k d}{2} \cos \left( \theta \right)$. For larger values of the argument "lobbing" behaviour will emerge which is more descriptive of real-life arraying behaviour of sources (for more info on that you'd have to consult array processing textbooks which is out-of-scope of this answer).
Below you can see an image of such behaviour. Equation \eqref{dipole-complete} is used to generate the polar plot for two dipoles where only the inter-element distances are different so that the product $k d$ in one case satisfies the assumption that leads to the simplification of the sine function and in the other case it does not. The parameters used are as follows: Frequency $f = 1 ~ kHz$, speed of propagation $c = 343 ~ \frac{m}{s}$, density $\rho = 1.21 ~ \frac{kg}{m^{3}}$, source strength $Q = 1$, $r = 10 ~ m$, inter-element distances $d_{1} = 10^{-5} ~ m$ and $d_{2} = 0.55 ~ m$ and time $t = 5 ~ s$. Please note that most of the parameters do not affect the radial behaviour and are used just for completeness. We could had plotted just the sine function (as this is the actual directivity function). Finally, please note that some of the needed quantities, such as the wavenumber $k$ and radial frequency $\omega$, are calculated from the parameters shown above.
