First, some background on sound intensity and its relation to pressure etc. can be found at:
https://physics.stackexchange.com/a/266046/59023
https://physics.info/intensity/
Second, this question turned out to be much more interesting and complicated than I had initially anticipated. Some great discussions/background on bell acoustics can be found in the following references:
- Fletcher, N.H., et al., "Bell clapper impact dynamics and the voicing of a carillon," J. Acoust. Soc. Am. 111(3), pp. 1437--1444, doi:10.1121/1.1448517, 2002.
- Woodhouse, J., et al., "The Dynamics of a Ringing Church Bell," Adv. Acoust. Vib. 2012, pp. 19, doi:10.1155/2012/681787, 2012.
Can I parameterize the diameter of the bell, its height, and the kinetic force of the knock to get the volume in decibels?
Yes, but it's not trivial, it turns out (see discussion in references above). We can, however, do some simple order of magnitude estimates.
Let's start with an absolute limit to get an idea of upper/lower bounds on values. Suppose the sound pressure intensity generated by this hypothetical, Bell of Lost Souls is the maximum possible in Earth's atmosphere, namely ~194 dB. This would correspond to a maximum gauge pressure, $\Delta P_{max}$ ~ 100,237.5 Pa. We know that the gauge pressure can be defined as:
$$
\Delta P = 2 \pi \ \rho \ f \ C_{s} \ \Delta s \tag{0}
$$
where $\rho$ is the mass density of air, $f$ is the frequency, $C_{s}$ is the speed of sound (because we assume sound waves), and $\Delta s$ is the displacement the particles in the sound wave from equilibrium. We know that at standard temperature and pressure (STP) that $\rho$ ~ 1.225 kg m-3 and $C_{s}$ ~ 331.3 m s-1. If we assume the sound emitted occurs at f = 1000 Hz, then we find that $\Delta s_{max}$ ~ 0.039 m. This displacement may look small, but normal sound waves that don't cause your head to explode only displace air particles by, at most, a few 10s of micrometers.
Note that while the sound wave propagates at $C_{s}$, the particles oscillating in the longitudinal mode have velocities that vary above and below this value. The magnitude of the velocity oscillations is given by:
$$
\Delta V = 2 \pi \ f \ \Delta s \tag{1}
$$
which then results in $\Delta V_{max}$ ~ 247 m s-1. As an aside, at this point you should start to be concerned about the applicability of our linear assumptions to this point since the velocity fluctuations are approaching the magnitude of the mean, i.e., $\Delta V_{max}/C_{s}$ > 70%.
The next part requires that we make some assumptions about the bell size and composition. Suppose the local area of the bell of interest is ~100 m by 100 m (or an area $A$ ~ 104 m2) with a thickness, $d$ ~ 1 m. We also assume the bell is made of steel so that we know its Young's modulus is $E$ ~ 200 GPa. We can estimate a rough potential energy per unit area of the bell by treating it as a thin plate such that:
$$
\frac{ U_{s} }{ A } \approx \frac{ E \ \Delta s^{2} }{ 2 \ s_{o} } \tag{2}
$$
where we assume $s_{o}$ ~ 1 m then we find (assuming 100% radiation efficiency) to be $U_{s}/A$ ~ 154 MJ m-2. If we multiply by the area we assumed above, then $U_{s}$ ~ 1.54 TJ.
As discussed by Woodhouse et al. [2012], the conversion of kinetic energy of the clapper (i.e., the object striking the bell surface) to vibrational energy is smaller than 100% (not by a tremendous amount, but definitely smaller). So we can get a lower bound on the impact speed of the clapper if we assume a mass. Let's assume the clapper has a mass of ~1000 kg then its speed must satisfy $V_{c}$ $\gtrsim$ 56 km s-1 (or $\gtrsim$125,000 mph). This is over five times the escape velocity of Earth's gravity, so not very realistic. So we increase the clapper mass to ~1 Mkg and then $V_{c}$ $\gtrsim$ 1.8 km s-1 (or $\gtrsim$4,026 mph), which is still absurd.
Note that even though the energy conversion efficiency is ~90%, there are numerous other issues that will alter these estimates (e.g., bell shape, shape of clapper surface at impact, effective masses associated with different excited normal modes, etc.).
Assuming that this bell is gargantuanly massive, like maybe the size of a large building or skyscraper being hit by 5 kN of bus sized hammer.
So we can estimate the force on the bell surface (very handwavy) as just $U_{s}/\Delta s_{max}$ ~ 39.3 TN. This is very likely a bad estimate as there is some deformation and displacement of the bell during impact, so the displacement value of $\Delta s_{max}$ is likely too large here. That is, the impacting force is almost certainly much larger than 39 TN. For a reference, Fletcher et al. [2002] provides some typical values on a normal sized bell in the area of ~20--150 kN (i.e., ~200 million times smaller).
I was reading about this Bell of Lost Souls, a bell as big as a cathedral, whose toll forced the populace into bunkers to escape death. Could the implications of such a massive bell be that bad?
Well, I think the human ear drum ruptures somewhere just north of ~155--160 dB. Every ~3 dB of intensity increase is basically doubling the intensity amplitude so 194 dB has an intensity ~2512 times larger. Note that it is very possible to kill a living organism with sound if it has sufficient intensity at certain frequencies. Shock waves are a nonlinear type of sound wave and the abrupt change in pressure can exert lethal forces on a human body.
