Double Bell 🛎️ 🛎️ Analogy with Δt Strike

Moar later… this is part of brainstorming series…

To analyze the spectral response of two classical brass church bells, each with a fundamental frequency of 938.272 Hz, rung simultaneously or with a slight time delay, we’ll model their harmonics and the resulting mixing spectrum. In the playful analogy to particle physics, we interpret the fundamental frequency as corresponding to the proton’s rest energy (938.272 MeV), extending our previous discussions where frequencies in Hz are mapped to energies in MeV. We’ll explore the effects of both bells being struck simultaneously (in-phase) and with a slight time delay (out-of-phase), analyze the spectral broadening due to the time delay, and compare the resulting spectrum to proton-proton resonance energies, such as the Δ(1232 MeV) and N(1440) (1440 MeV) resonances. The analysis will include equations, a spectral plot, and an HTML visualization, maintaining the “420 friendly” response style while keeping it concise and comprehensive.

Harmonic Structure of a Single Bell

Each bell has a fundamental frequency of 938.272 Hz, corresponding to the proton’s rest energy (938.272 MeV) in our analogy. For simplicity, we assume the bells produce a harmonic series (though real bells are slightly inharmonic), with frequencies:

[ f_n = n \cdot 938.272 , \text{Hz}, \quad n = 1, 2, 3, \ldots ]

The first ten harmonics are:

• ( f_1 = 938.272 , \text{Hz} ) (fundamental)
• ( f_2 = 1876.544 , \text{Hz} )
• ( f_3 = 2814.816 , \text{Hz} )
• ( f_4 = 3753.088 , \text{Hz} )
• ( f_5 = 4691.360 , \text{Hz} )
• ( f_6 = 5629.632 , \text{Hz} )
• ( f_7 = 6567.904 , \text{Hz} )
• ( f_8 = 7506.176 , \text{Hz} )
• ( f_9 = 8444.448 , \text{Hz} )
• ( f_{10} = 9382.720 , \text{Hz} )

In the energy analogy:

• ( E_n = n \cdot 938.272 , \text{MeV} )
• ( E_1 = 938.272 , \text{MeV} ) (proton)
• ( E_2 = 1876.544 , \text{MeV} ) (two-proton threshold)
• ( E_3 = 2814.816 , \text{MeV} ), etc.

The amplitude of each harmonic decreases as ( A_n = \frac{1}{n} ), reflecting the bell’s natural decay:

• ( A_1 = 1.000 ), ( A_2 = 0.500 ), ( A_3 = 0.333 ), ( A_4 = 0.250 ), ( A_5 = 0.200 ), ( A_6 = 0.167 ), ( A_7 = 0.143 ), ( A_8 = 0.125 ), ( A_9 = 0.111 ), ( A_{10} = 0.100 ).

Each harmonic is modeled as a Lorentzian resonance:

[ L_n(f) = \frac{A_n}{(f - f_n)^2 + \left( \frac{f_n}{2 Q} \right)^2} ]

where ( Q = 1000 ), giving a narrow peak with full width at half maximum (FWHM):

[ \text{FWHM} = \frac{f_n}{Q} ]

For the fundamental (( n=1 )):

[ \text{FWHM} = \frac{938.272}{1000} \approx 0.938 , \text{Hz} ]

In the energy analogy, this translates to ( \text{FWHM} = 0.938 , \text{MeV} ).

Case 1: Bells Struck Simultaneously (In-Phase)

When both bells are struck at the same time, their vibrations are in-phase, and their signals add constructively. Since both bells have identical fundamental frequencies (938.272 Hz), their harmonics align perfectly. The combined amplitude for each harmonic is the sum of the individual amplitudes:

[ A_{n,\text{total}} = A_n + A_n = 2 \cdot \frac{1}{n} ]

The spectral response is:

[ S_{\text{total}}(f) = \sum_{n=1}^{10} \frac{2 \cdot \frac{1}{n}}{(f - f_n)^2 + \left( \frac{f_n}{2 Q} \right)^2} ]

The frequencies remain unchanged, but the amplitudes double:

• ( n=1 ): 938.272 Hz, ( A = 2.000 )
• ( n=2 ): 1876.544 Hz, ( A = 1.000 )
• ( n=3 ): 2814.816 Hz, ( A = 0.667 )
• ( n=4 ): 3753.088 Hz, ( A = 0.500 )
• etc.

In the energy analogy (MeV), the spectrum has peaks at:

• 938.272 MeV (proton, ( n=1 ))
• 1876.544 MeV (two-proton threshold, ( n=2 ))
• 2814.816 MeV (( n=3 )), etc.

Comparison to Proton Resonances

• Δ(1232) Resonance: ~1232 MeV
• Nearest harmonic: ( n=1 ) (938.272 MeV, difference = ( 1232 - 938.272 = 293.728 , \text{MeV} )) or ( n=2 ) (1876.544 MeV, difference = ( 1876.544 - 1232 = 644.544 , \text{MeV} )).
• The Δ lies between ( n=1 ) and ( n=2 ).
• N(1440) Resonance: ~1440 MeV
• Nearest harmonic: ( n=2 ) (1876.544 MeV, difference = ( 1876.544 - 1440 = 436.544 , \text{MeV} )) or ( n=1 ) (938.272 MeV, difference = ( 1440 - 938.272 = 501.728 , \text{MeV} )).
• The N(1440) is also between ( n=1 ) and ( n=2 ).

The in-phase case amplifies the harmonic peaks but doesn’t introduce new frequencies or broaden the spectrum, so the comparison to resonances remains limited by the harmonic spacing (938.272 MeV).

Case 2: Bells Struck with a Slight Time Delay

Now, consider the bells struck with a small time delay ( \tau ). This introduces a phase difference between their signals, leading to interference effects and potential spectral broadening.

Time-Domain Signal

The signal from a single bell can be approximated as a sum of damped sinusoids:

[ s(t) = \sum_{n=1}^{10} A_n e^{-\alpha_n t} \cos(2\pi f_n t) ]

where ( \alpha_n = \frac{\pi f_n}{Q} ) is the damping rate, and ( f_n = n \cdot 938.272 , \text{Hz} ).

For two bells, with the second delayed by ( \tau ):

[ s_{\text{total}}(t) = s_1(t) + s_2(t - \tau) ]

The second bell’s contribution is:

[ s_2(t) = \sum_{n=1}^{10} A_n e^{-\alpha_n (t - \tau)} \cos(2\pi f_n (t - \tau)) ]

In the frequency domain, a time delay ( \tau ) introduces a phase shift. The Fourier transform of a delayed signal ( s(t - \tau) ) is:

[ S(f) e^{-i 2\pi f \tau} ]

For each harmonic, the combined spectrum involves interference:

[ S_{\text{total}}(f) = S_1(f) + S_2(f) e^{-i 2\pi f \tau} ]

The amplitude spectrum near each harmonic frequency ( f_n ):

[ |S_{\text{total}}(f)| = \left| \frac{A_n}{(f - f_n)^2 + \left( \frac{f_n}{2 Q} \right)^2} \cdot (1 + e^{-i 2\pi f \tau}) \right| ]

The interference term ( 1 + e^{-i 2\pi f \tau} ) has magnitude:

[ |1 + e^{-i 2\pi f \tau}| = \sqrt{2 (1 + \cos(2\pi f \tau))} ]

For small ( \tau ), the phase ( 2\pi f \tau ) is small near the fundamental, but grows with frequency (( f_n = n \cdot 938.272 )). Let’s choose a realistic time delay, say ( \tau = 0.001 , \text{s} ) (1 ms, a plausible human-scale delay for striking bells):

• At ( f_1 = 938.272 , \text{Hz} ): [ 2\pi f_1 \tau = 2\pi \cdot 938.272 \cdot 0.001 \approx 5.897 , \text{radians} \approx 337.9^\circ ] [ \cos(5.897) \approx -0.955 ] [ |1 + e^{-i 5.897}| \approx \sqrt{2 (1 - 0.955)} \approx 0.424 ]

The amplitude is reduced due to destructive interference, and this effect varies with frequency.

Spectral Broadening

The time delay causes interference that modulates the amplitude spectrum, effectively broadening the peaks. The interference pattern introduces sidebands around each harmonic, with spacing related to ( \frac{1}{\tau} ):

[ \Delta f_{\text{sideband}} \approx \frac{1}{0.001} = 1000 , \text{Hz} ]

In the energy analogy:

[ \Delta E_{\text{sideband}} \approx 1000 , \text{MeV} ]

This broadening is significant compared to the intrinsic width (( \text{FWHM} \approx 0.938 , \text{MeV} )) and the resonance energies. The spectrum now includes interference peaks, complicating direct matches to Δ(1232 MeV) or N(1440) (1440 MeV).

Spectral Plot

For visualization, we plot the in-phase case (simultaneous ringing) for simplicity, as the delayed case’s interference pattern is complex and frequency-dependent. Amplitudes are doubled for in-phase addition:

{

  "type": "bar",

  "data": {

    "labels": [938.3, 1876.5, 2814.8, 3753.1, 4691.4, 5629.6, 6567.9, 7506.2, 8444.4, 9382.7],

    "datasets": [{

      "label": "Harmonic Energies (In-Phase)",

      "data": [2.000, 1.000, 0.667, 0.500, 0.400, 0.333, 0.286, 0.250, 0.222, 0.200],

      "backgroundColor": "blue"

    }]

  },

  "options": {

    "scales": {

      "x": { "title": { "display": true, "text": "Energy (MeV)" } },

      "y": { "title": { "display": true, "text": "Amplitude" } }

    }

  }

}

For the delayed case, the spectrum is modulated, but a detailed plot requires numerical computation beyond a simple bar chart.

HTML Visualization

Here’s the HTML for the in-phase spectrum:

  

  

  

  

Comparison to Proton Resonances

• In-Phase: The spectrum has peaks at 938.272 MeV (proton), 1876.544 MeV (two-proton threshold), etc. The Δ(1232 MeV) and N(1440) (1440 MeV) fall between harmonics, with differences of 293.728 MeV and 501.728 MeV from the proton, respectively, not aligning closely with the 938.272 MeV spacing.
• Delayed Case: The time delay (( \tau = 0.001 , \text{s} )) introduces sidebands (~1000 MeV apart), further complicating direct matches to resonances. The broadening smears the peaks, making precise alignment with Δ or N(1440) less likely.

Conclusion

Ringing two bells with a fundamental of 938.272 Hz (938.272 MeV) models two colliding protons. In-phase ringing doubles the amplitude of harmonics, aligning the fundamental with the proton but not closely with Δ or N(1440). A time delay of 1 ms introduces interference, broadening the spectrum with ~1000 MeV sidebands, which disrupts precise resonance matching. The in-phase spectrum is visualized above, capturing the essence of two protons’ combined energy states in this cosmic analogy, dude!