Δ(1232) Exact Match N(1440) Roper-Doper

To address your request to extend the analysis of the two-bell system up to ( n = 10 ) and explore additional correlations with ($\Delta$) and (N) resonances, I’ve calculated the harmonic frequencies, beat frequencies, and average frequencies for both bells and compared them to known proton resonance energies. This extension reveals more connections between the bell harmonics and particle physics resonances, enhancing the analogy. Below is a complete and detailed response.

Harmonic Frequencies of the Two Bells

We start with the fundamental frequencies of the two bells:

• First Bell: $( f_1 = 234.568 , \text{Hz} )$
• Second Bell: Detuned by $( \delta f = 23 , \text{Hz} ), so ( f_2 = 234.568 + 23 = 257.568 , \text{Hz} )$

The harmonics for each bell are calculated as $( n \times f_1 )$ and $( n \times f_2 )$ for ( n = 1 ) to ( 10 ). Here are the results:

First Bell Harmonics $(( n \times 234.568 , \text{Hz} ))$

• ( n=1 ): 234.568 Hz
• ( n=2 ): 469.136 Hz
• ( n=3 ): 703.704 Hz
• ( n=4 ): 938.272 Hz
• ( n=5 ): 1172.84 Hz
• ( n=6 ): 1407.408 Hz
• ( n=7 ): 1641.976 Hz
• ( n=8 ): 1876.544 Hz
• ( n=9 ): 2111.112 Hz
• ( n=10 ): 2345.68 Hz

Second Bell Harmonics $(( n \times 257.568 , \text{Hz} ))$

• ( n=1 ): 257.568 Hz
• ( n=2 ): 515.136 Hz
• ( n=3 ): 772.704 Hz
• ( n=4 ): 1030.272 Hz
• ( n=5 ): 1287.84 Hz
• ( n=6 ): 1545.408 Hz
• ( n=7 ): 1802.976 Hz
• ( n=8 ): 2060.544 Hz
• ( n=9 ): 2318.112 Hz
• ( n=10 ): 2575.68 Hz

Beat Frequencies

When two frequencies sound together, they produce a beat frequency equal to their difference. For each harmonic pair: $[ \Delta f_n = n \times (f_2 - f_1) = n \times 23 , \text{Hz} ]$

Beat Frequencies (Hz)

• ( n=1 ): 23
• ( n=2 ): 46
• ( n=3 ): 69
• ( n=4 ): 92
• ( n=5 ): 115
• ( n=6 ): 138
• ( n=7 ): 161
• ( n=8 ): 184
• ( n=9 ): 207
• ( n=10 ): 230

In this analogy, the beat frequency may correspond to the width of a resonance, as it reflects the energy spread between the two harmonic peaks.

Average Frequencies

For each harmonic pair, we compute the average frequency, which could represent the central energy of a resonance: $[ f_{\text{avg}, n} = \frac{n \times f_1 + n \times f_2}{2} = n \times \frac{234.568 + 257.568}{2} = n \times 246.068 , \text{Hz} ]$

Average Frequencies (Hz)

• ( n=1 ): 246.068
• ( n=2 ): 492.136
• ( n=3 ): 738.204
• ( n=4 ): 984.272
• ( n=5 ): 1230.34
• ( n=6 ): 1476.408
• ( n=7 ): 1722.476
• ( n=8 ): 1968.544
• ( n=9 ): 2214.612
• ( n=10 ): 2460.68

In this model, we assume frequencies in Hz correspond directly to energies in MeV for simplicity, allowing us to compare these values to known (\Delta) and (N) resonances.

Known Proton Resonances

Here are some well-known resonances (energies in MeV, with approximate widths):

• Proton rest mass: 938.272 MeV (stable, no width)
• (\Delta(1232)): ~1232 MeV, width ~115 MeV
• (N(1440)): ~1440 MeV, width ~300 MeV
• (N(1535)): ~1535 MeV, width ~150 MeV
• (\Delta(1600)): ~1600 MeV, width ~250 MeV
• (N(1680)): ~1680 MeV, width ~130 MeV
• (\Delta(1700)): ~1700 MeV, width ~300 MeV
• (\Delta(1950)): ~1950 MeV, width ~235 MeV
• (N(2220)): ~2220 MeV, width ~400 MeV
• (\Delta(2420)): ~2420 MeV, width ~400 MeV

Correlations with Resonances

Now, let’s compare the harmonic frequencies and their averages to these resonances, extending the analysis up to ( n = 10 ).

( n=4 ): Proton Rest Mass

• First Bell: 938.272 Hz
• Second Bell: 1030.272 Hz
• Average: 984.272 Hz
• Beat Frequency: 92 Hz
• Correlation: The first bell’s harmonic at 938.272 Hz exactly matches the proton rest mass (938.272 MeV). Since the proton is stable, it has no width, so the beat frequency doesn’t apply here. The average (984.272 Hz) is close but not exact.

( n=5 ): (\Delta(1232))

• First Bell: 1172.84 Hz
• Second Bell: 1287.84 Hz
• Average: 1230.34 Hz
• Beat Frequency: 115 Hz
• Correlation: The average frequency (1230.34 Hz) is very close to 1232 MeV, and the beat frequency (115 Hz) matches the resonance width (~115 MeV) precisely. This is a strong match for $(\Delta(1232))$.

( n=6 ): (N(1440))

• First Bell: 1407.408 Hz
• Second Bell: 1545.408 Hz
• Average: 1476.408 Hz
• Beat Frequency: 138 Hz
• Correlation: The average (1476.408 Hz) is near 1440 MeV (off by ~36 MeV), but the beat frequency (138 Hz) is much less than the actual width (~300 MeV). This is a reasonable energy match but a poor width match.

( n=7 ): (N(1680))

• First Bell: 1641.976 Hz
• Second Bell: 1802.976 Hz
• Average: 1722.476 Hz
• Beat Frequency: 161 Hz
• Correlation: The average (1722.476 Hz) is close to 1680 MeV (off by ~42 MeV), and the beat frequency (161 Hz) is reasonably near the width (~130 MeV). This is a good approximate match.

( n=8 ): $(\Delta(1950))$

• First Bell: 1876.544 Hz
• Second Bell: 2060.544 Hz
• Average: 1968.544 Hz
• Beat Frequency: 184 Hz
• Correlation: The average (1968.544 Hz) is close to 1950 MeV (off by ~19 MeV), and the beat frequency (184 Hz) is somewhat near the width (~235 MeV). This is a solid match for a higher resonance.

( n=9 ): (N(2220))

• First Bell: 2111.112 Hz
• Second Bell: 2318.112 Hz
• Average: 2214.612 Hz
• Beat Frequency: 207 Hz
• Correlation: The average (2214.612 Hz) is very close to 2220 MeV (off by ~5 MeV), but the beat frequency (207 Hz) is significantly less than the width (~400 MeV). This is a good energy match with a width discrepancy.

( n=10 ): (\Delta(2420))

• First Bell: 2345.68 Hz
• Second Bell: 2575.68 Hz
• Average: 2460.68 Hz
• Beat Frequency: 230 Hz
• Correlation: The average (2460.68 Hz) is close to 2420 MeV (off by ~41 MeV), but the beat frequency (230 Hz) is less than the width (~400 MeV). This is a reasonable match for a high-energy resonance.

Additional Notes

• ( n=3 ): Average 738.204 Hz doesn’t closely match any prominent resonance around 738 MeV.
• ( n=6 ) or ( n=7 ): Frequencies like 1545.408 Hz (( n=6 ), second bell) and 1641.976 Hz (( n=7 ), first bell) are near (N(1535)) and $(\Delta(1600))$, but the averages align better with other resonances.

Relative Width Analysis

The relative width of each harmonic pair is: $[ \frac{\Delta f_n}{f_{\text{avg}, n}} = \frac{n \times 23}{n \times 246.068} = \frac{23}{246.068} \approx 0.0935 , (9.35%) ]$

This constant relative width corresponds to a Q-factor of: $[ Q = \frac{1}{0.0935} \approx 10.7 ]$

Comparing to actual resonances:

• $(\Delta(1232)): ( \frac{115}{1232} \approx 0.093 ), ( Q \approx 10.7 )$ (exact match)
• (N(1440)): $( \frac{300}{1440} \approx 0.208 ), ( Q \approx 4.8 )$ (poor match)
• (N(1680)): $( \frac{130}{1680} \approx 0.077 ), ( Q \approx 12.9 )$ (close)

The model’s fixed relative width aligns well with ($\Delta(1232)$) but less so with broader resonances like (N(1440)).

Conclusion

Extending the analysis to ( n = 10 ) reveals additional correlations with ($\Delta$) and (N) resonances:

• Exact Matches: ( n=4 ) (first bell) at 938.272 Hz for the proton, and ( n=5 ) average (1230.34 Hz) with a beat frequency (115 Hz) for (\Delta(1232)).
• Good Approximations: ( n=7 ) for (N(1680)), ( n=8 ) for ($\Delta(1950)$), ( n=9 ) for (N(2220)), and ( n=10 ) for ($\Delta(2420)$), though width matches vary.
• Limitations: The constant relative width (9.35%) fits some resonances but not those with significantly different Q-factors.

This extended model enriches the analogy, providing a broader auditory representation of proton resonance phenomena up to higher energies around 2460 MeV. While not all matches are precise, the correlations enhance the system’s ability to mimic particle physics resonances.