Does a tower bell ringing prevent thunderstorms? Introduction
This is the beginning of an apparently physics-unrelated question, which involves 1700-1800 Italian law, atmospheric processes, sound waves propagating through fluid and clouds, and lightning strikes burning down churches.
The core of the question is figuring out if and how a sound wave influences the formation / disruption of clouds and eventual thunderstorms.
Background and Context
I live next to the European Alps, in northern Italy. Here weather is often cloudy and thunderstorms are not rare. For religious purposes, every town in this area has a church with a tall bell tower alongside it. And of course, a huge bronze bell hangs in each tower to ring every hour.
I've talked to a local elderly (who happens to manage the bell tower) about thunderstorms, and I was told that when a thunderstorm approaches he rings the bell repeatedly and vigorously to try to "break up the storm", in order to supposedly avoid rain and thunder falling on the town. This is a practice still widely used to this day in this area. Me being a engineering student, I wanted to figure out if that really made sense and what effects ringing the bell really had.
Question formalization
A bronze bell ringing multiple times produces sound. The kinetic energy of the bronze clapper is cast on the bell, which converts it into sound energy.
Specifically, the bronze bell vibrates and transfers the sharp vibration to the layer of air molecules wrapped around the bell, which initiates a chain transmission. This produces sound waves, which are longitudinal (compression) waves transmitted through a fluid (air).

Longitudinal waves are waves in which the vibration of the medium is
parallel ("along") to the direction the wave travels and displacement
of the medium is in the same (or opposite) direction of the wave
propagation. Mechanical longitudinal waves are also called
compressional or compression waves, because they produce compression
and rarefaction when traveling through a medium, and pressure waves,
because they produce increases and decreases in pressure. [source]


Those waves are then spread out more or less spherically (following the inverse square law) from the bell top point.
The natural questions to ask at this point are:

*

*What happens when the sound wave passes through the cloud?

*Does the wave have enough energy to reasonably alter the formation of the cloud and therefore prevent / allow the thunderstorm?

*Is there such thing as "breaking up the storm" or is it just a popular belief?

I genuinely have no idea since I have no first hand experience with this.
Thunderstorm formation looks like a whole different process, and I can't seem to find a connection with sound waves.

Thunderstorms form when warm, moist air rises into cold air. The warm
air becomes cooler, which causes moisture, called water vapor, to form
small water droplets - a process called condensation. The cooled air
drops lower in the atmosphere, warms and rises again. [source]

Historically relevant experiments
While doing my research I found papers from the late 1700s where Italian scientist were trying to answer one of our questions: does ringing the tower bell disrupt an incoming thunderstorm?
Their answer is the opposite of what I expected: they thought that ringing the bell actually amplified the storm.
You can find the original version of these papers here
Their results look extremely nonrigorous, but it can be a starting point to reflect on.
Note: I can do a full translation of the paper if someone is interested, right now I'll do my best to translate and summarize the key concepts (sparing you a ton of mid-1700 religious and legal drama)
Paper published in 1770
Key: Total nonsense

The vibration induced by the bell sharp sound has 2 effects on air: it
rubs together different parts [of the cloud] and it thins out the mass
[of the cloud]. The rubbing, as it is known, awakens the "electric
fire" (I have no clue about what that is). The thinning out attracts
it [the electric fire], diminishing air resistence. Vibrations in
general unite instead of breaking up.

It is also said that the bell sound (while storming) attracts the lightning strikes on the tower bell (which is pure nonsense), instead of letting lightning target houses or fields, and it is thought to be a good thing.
Plus, the bell sound would warn locals that a storm is approaching.
Act of 1783
Key: Law prohibits ringing bells during storms

A series of sad experiences says that the tower bell ring, instead of
dissipating the clouds and the storms, activates and puts in danger
the towers and the churches, which have been subject to numerous
strikes, fires and deaths during storms. It is therefore prohibited to
ring bells during storms.

Paper published in 1787:
Key: Evaluation of air vibration effects on clouds
This paper exposes 2 reasons backing up the law against ringing bells during storms.

1: Atmospheric Sound Ondulations.
Vibrations originated from
percussion of objects propagate spherically, generating "compression
and approachment of parts". Not heavy clouds break under this
condition. But a cloud is a electrified conductor. It is known that
the stress exercised from the electric fire is inversely proportional
to its capacity, and its capacity is directly proportional to the
surface area. If the cloud volume decreases, its surface area must
decrease too. Therefore, the capacity will decrease and the intensity
of the electric fire will increase.  From this we can infer that
vibrations, which compress the volume of a cloud, necessarily increase
the intensity of the electric fire, which will discharge on the
closest bodies, thus on the bell tower.


2: Air rarity around bell tower. Vibrations continuously remove air and turn it into thinner and thinner air. There is therefore an
air current towards the sound source. The electric fire will follow
that direction.

It ends with:

It is demonstrated that ringing the bell promotes the striking of a lightning and targets the strike on the source of the vibrations.


 A: The question whether ringing a bell on a tower can influence lightning strikes involves two aspects: hypothesizing physical processes which might be relevant, and then figuring out (or measuring) whether they are. Here are two physical processes I can think of: the sound waves in the air, and the bell itself with the clapper hitting it.
Sound waves could conceivably influence electrical conductivity of damp air, but I doubt whether this effect is large enough to have a non-negligible influence on the charge and field build-up which leads to a lightning strike.
The final image in the question shows a lightning rod (also called lightning conductor) not a bell. Such rods were introduced in the 1750s  (says Wiki). They work via two effects: a gradual one and a sudden one. The electric field near a charged conductor has a much larger value near a sharp point. So by using a pointed conductor one encourages modest currents in the air near the tip of the lightning rod which, gradually, tend to reduce the charge build-up in the cloud and thus reduce the overall frequency and strength of lightning strikes. This is not enough to entirely suppress strikes but overall it does help. These currents also have the effect of making the lightning more likely to strike the rod than other nearby things. This offers protection to the building as long as the long thick copper wire from ground to rod can itself conduct most of the current without melting.
Coming now to the ringing bell, if the clapper is metal then as it strikes the bell there might conceivably be an increase in the electrical emission from the bell (in the presence of the kind of charge build-up which occurs in thunderstorms). Thus it begins to play the first part of the role of a lightning rod. However I guess this effect is pretty small, and of course without the rest of the rod the bell offers no protection against an actual strike. As another answer has already pointed out, if such effects were non-negligible then one would think it would have been noticed and investigated by now, and bells would be recommended as a solution for some buildings. But last time I looked pretty much all church towers have lightning rods.
Finally, concerning cloud formation: this also is a big issue in many parts of the world so one would think that if bells could help then it would have been noticed and documented more fully and convincingly by now. Disturbing air which is already supersaturated will in general encourage (not discourage) droplets to form. However there have been many efforts to do this by seeding the air and it appears that it is hard to do on a large enough scale to have any useful impact.
A: There is enormous energy in a storm, which will be randomly oriented with respect to a bell tower.
The energy in the bell's sound will be radially falling with $1/r^2$ and is not directional. The energy that a man can input to the bell ringing is limited.
The physics answer must be: there is no connection.
Then the question becomes metaphysics, the bell ringing raises the religious impulses of people to diffuse the storm. Are prayers effective? Uneducated religious people in general seem to believe so. Physics has not reached the level of detecting whether there is anything in metaphysics, (from  telepathy to, in this case, manipulating storms)
Taking the expectation of bell ringing to defuse storms as a metaphsyics  experiment, the answer seems to be "prayers are not effective"

Churches and castles were often extremely dangerous buildings during thunderstorms in the days before the lightning rod was invented. Being so tall, they were highly vulnerable to lightning strikes: hundreds of bell-ringers across Europe were killed over the centuries in the mistaken belief that ringing the bells would ward off lightning. But an even greater hazard was the habit of storing gunpowder in castles and church vaults.

A: In addition to the energy considerations of a human generated sound wave, there is this to consider too:
As you state, sound is a longitudinal wave of pressure variation through the air.  Let's suppose we have the capacity to generate a VERY strong sound wave.
At the atmospheric level of the thundercloud, where water vapor is forming droplets, assuming the temperature stays relatively constant, the phase diagram of water would tell us that an increase in pressure will likely only increase the size of the water droplet (i.e. more water will be forced into the liquid portion of the phase diagram).  Conversely, a rarefaction or decrease in pressure, will likely only decrease the size of water droplets.
Since compressions and rarefactions follow one another, the net effect of this on water droplet size would likely be zero.
(Adding in the fact that things won't stay isothermal further supports this.  A pressure increase will lead to a temperature increase, which will work opposite that of the pressure increase: droplet size would likely decrease.  So the net effect of a pressure increase on droplet size is likely net zero.  Same for a pressure decrease when taking into account temperature changes due to the pressure change.)
A: The energy scale of thunderstorm is much larger than that of a bell (most other devices built by humans), so it is unlikely that such devices can affect thunderstorms in a controllable way.
A single bolt of lightning is can carry 108 to 1010  joules of energy. This is similar to the amount of electrical energy that the average US household uses in month (3×109 J). A complete thunderstorm has been estimated to carry 1015 J, which is enough energy to power 300,000 US households for one month or as much energy as would be released by 16 copies of the nuclear bomb dropped on Hiroshima, Japan in 1945.
A bell rung by a human can't use more than the power output of the human body which is about 400 W maximum. Even if this power was converted to sound with high efficiency and the person could maintain this level of effort for 10 minutes, the amount of energy released would only be 2.4×105 J. This is 4 billion times smaller than the estimated energy of the storm.
It is likely that only a device operating on similar energy scale on the time scale of thunderstorm could reliably control a thunderstorm. Nuclear weapons are the only technology we have that operates at this scale, but their energy can be easily controlled or directed with current technology and the collateral damage would be huge. Perhaps some tricks could be used to reduce the energy scale somewhat, which would be dependent on a very good understanding of storm dynamics and the ability to simulate them and potential interventions realistically. In any case, this is beyond the current technology level of humanity and probably will be for some time.
In any case, your question is an interesting case of people trying to find regularity in noise, with some thinking that ringing the bell makes the storm less dangerous and others thinking it makes the storm worse. The truth is that the bell is unlikely to have any consistent effect on the storm and the medieval people trying the bell method were finding patterns in random noise.
A: Of course a ringing bell has no effect on the weather. Imagine, if it did, bells would be a standard addition to any equipment or structure that is especially vulnerable to lightning strikes.
A: The sonic energy from thunder dwarfs any bell and I have seen no evidence that the pressure waves from thunder affect a storm.
A: The problem is, we don't know. Well, the rain and thunder can't be prevented, like a butterfly can't change the course of a storm. But it could prevent a lightning bolt from blitzing the ringer, while it could change the paths of the lighting strokes so he'll be blitzed down and stop his ringing, which has happened frequently.
So it could influence the storm and paths of lightning (chaos theory), but the energy thrown down from the clouds and the rain falling will not be influenced. Maybe if the ringer gives off a strong directed yell it will bump off the blitz...
So, while the ringing might influence the chaotic processed that are super sensitive to initial conditions, like lighting flashes are, it will have no effect on the clouds that run, the wind that blows and the rain that falls.
A: Ringing the bell will not stop a discharge of electricity between clouds but the intensity of sound waves generated thereby may well be reduced i.e.that of the clap.
