(reference:content:pointSource)=
# Point source
Acoustic monopoles are the simplest type of sound sources. They can be seen as single points in space, radiating sound waves in all directions. The mathematical expression of a monopole in the frequency domain can be written as follow:

$$
p(r) = jk\rho c Q \frac{e^{-jkr}}{4\pi r},
$$

where $r$ is the distance from the monopole, $k=\frac{2\pi f}{c}$ the wavenumber, $\rho$ the density of propagation medium, $c$ the speed of sound in said medium, and $Q$ the volume velocity. Using a point source is practical when solving problems that are not specifically source related, such as the acoustic of a room.

## Implementation
Monopole studies use the following syntax:

```python
system.study_acousticPointSource("monopole",         # reference
                                 monopolePosition,   # list or numpy array
                                 acoustic_radiator,  # driver / enclosure objects
                                 meshPath=path,      # optional boundary mesh
                                 domain="interior")  # domain of simulation
```

where:

- `monopolePosition` can either be a numpy array of size `(nSource, 3)`, or a vector of vectors (e.g. `[[x1, y1, z1], [x2, y2, z2], ..., [xN, yN, zN]]`),
- `acoustic_radiator` is defined in the same way as for `.study_acousticBEM()`. References to BEM need to be set even-though no mesh surface is radiating. Hence, `ref2bem` can be given arbitrarily,
- `meshPath` is optional: point source studies can be set with or without physical boundaries,
- `domain` is set to `"exterior"` or `"interior"`, similarly to `.study_acousticBEM()`.


## Practical example
Let's take a room of dimensions $L_x=3.15$ m, $L_y=5$ m and $L_z=2.3$ m. We set a single point source in a corner, and evaluation point in the opposite position. In that case, we consider the walls to be fully reflective.

```{figure} ./monopole_images/source_mic_position.png
    :name: monopole-system
    :width: 15cm

Shoebox room, visualization done in Salome.
```

As usual, we start by importing libraries, setting the mesh path and initializing the system object:

```python
import numpy as np
import electroacPy as ep
from electroacPy import gtb

#%% mesh data
room_mesh = "../geo/mesh/room.med"

#%% system initialization
frequency = np.arange(10, 200.25, 0.25)
sim = ep.loudspeakerSystem(frequency)
```

Now, the position of sources and evaluation points are written as vectors. The source has a unit velocity. In that case, `ref2bem` is set to `10` --- even-though this value is not "used" directly as a reference to a mesh surface, the post-processing function will use it.

```python
#%% source and microphone position
xSce = [2.85, 4.7, 0.3]
xMic_1 = [0.3, 0.3, 2] 
sim.lem_velocity("source", ref2bem=10)
```

Then, we create the study, add an evaluation and run simulations.

```python
#%% study creation
sim.study_acousticPointSource("room", [xSce], "source",
                              meshPath=room_mesh, domain="interior")
sim.evaluation_fieldPoint("room", "corner_mic", [xMic_1])

#%% run
sim.run()
```

The acoustic pressure at the corner microphone --- displayed in {numref}`corner-mic` --- can be extracted as follow:

```python
#%% extract data and plot
mic = sim.get_pMic("room", "corner_mic")
gtb.plot.FRF(frequency, mic, logx=False, xlim=(20, 200))
```

```{figure} ./monopole_images/corner_mic_response.svg
    :name: corner-mic

Corner microphone pressure.
```

We can also compare the BEM simulation with the analytical solution of modes in a shoebox room:

$$

f(n_x, n_y, n_z) = \frac{c}{2} \sqrt{\left(\frac{n_x}{L_x}\right)^2 + \left(\frac{n_y}{L_y}\right)^2 + \left(\frac{n_z}{L_z}\right)^2}

$$


```{figure} ./monopole_images/corner_mic_response_with_modes.svg
    :name: corner-mic-modes

Comparison between PSM/BEM acoustic pressure and analytical solution of room modes.
```

Finally, using `sim.plot_pressureMesh("room")` we display the pressure over the mesh:

```{figure} ./monopole_images/modes.png
    :name: room-mode-bem
    :width: 15cm
Some room modes.
```