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Chapter 2. The Wavelength of a Cubit: $\lambda = c/f$

5D Library › The Sacred Cubit and Resonant Geometry · 中文

Chapter 2. The Wavelength of a Cubit: λ=c/f\lambda = c/f

Every length resonates with something. A guitar string, an organ pipe, a radio aerial and a room all have frequencies at which waves fit them neatly and build up. This chapter asks which waves fit one royal cubit.

Wavelength, frequency and speed

A wave of frequency ff travelling at speed cc repeats itself every λ=cf.\lambda = \frac{c}{f}. The same formula covers sound and radio; only the speed changes. Sound in air at 20 °C travels at about 343m/s343\ \text{m/s}; radio waves in free space travel at the speed of light, 299792458m/s299\,792\,458\ \text{m/s}, almost a million times faster. So the same length fits a sound of a few hundred hertz and a radio wave of a few hundred megahertz.

Two kinds of fit

Evidence

A tube or wire resonates when its boundary conditions are met. A pipe open at both ends needs a pressure node at each end and resonates when its length is a whole number of half wavelengths, L=nλ/2L = n\lambda/2; a pipe closed at one end needs a node at the open end and an antinode at the closed end, and resonates when its length is an odd number of quarter wavelengths, L=(2n−1)λ/4L = (2n-1)\lambda/4 (Kinsler et al. 2000). A thin straight wire standing on a conducting ground plane (a monopole aerial) behaves like the closed pipe: its first resonance comes when it is about a quarter wavelength long (Balanis 2016).

What fits one cubit

Calculation

With L=0.5235mL = 0.5235\ \text{m}:

c_light = 299_792_458        # m/s
c_air = 343.0                # m/s at 20 °C
L = 0.5235                   # one royal cubit, m

print("Sound in air:")
print(f"  pipe closed at one end (L = λ/4): {c_air / (4 * L):7.1f} Hz")
print(f"  pipe open at both ends (L = λ/2): {c_air / (2 * L):7.1f} Hz")
print("Radio waves:")
print(f"  free-space wavelength = L:        {c_light / L / 1e6:7.1f} MHz")
print(f"  ideal quarter-wave monopole:      {c_light / (4 * L) / 1e6:7.1f} MHz")
print(f"  ... with a typical 0.95 factor:   {0.95 * c_light / (4 * L) / 1e6:7.1f} MHz")
# Sound in air:
#   pipe closed at one end (L = λ/4):   163.8 Hz
#   pipe open at both ends (L = λ/2):   327.6 Hz
# Radio waves:
#   free-space wavelength = L:          572.7 MHz
#   ideal quarter-wave monopole:        143.2 MHz
#   ... with a typical 0.95 factor:     136.0 MHz

Three of these numbers deserve comment.

163.8 Hz and 327.6 Hz. A pipe one cubit long, closed at one end, sounds close to the E below middle C (E3, 164.8 Hz); open at both ends, close to the E above it (E4, 329.6 Hz). Both are about 11 cents flat of equal temperament before any correction. In practice the air just beyond an open end also moves with the wave, which makes a pipe behave as if it were slightly longer; this end correction depends on the pipe’s width and lowers the frequency further (Kinsler et al. 2000). A cubit-long pipe is, in musical terms, an E.

143.2 MHz. A wire one cubit long standing on a ground plane would be an ideal quarter-wave aerial at 143.2 MHz, just below the 2-metre amateur radio band (144–148 MHz in most countries). Real wires have thickness, and their resonance falls a few percent lower than the ideal; a typical factor of about 0.95 brings it to about 136 MHz.

0.95. The factor is not a property of the cubit. It comes from the wire’s thickness relative to its length and from the effect of the wire’s ends. A thicker rod resonates lower; a thinner wire comes closer to the ideal. This is why the resonance of any real object has to be measured, not only calculated (Chapter 3).

Note

What this does and does not show. It shows that a cubit, like every other length, has resonances, and gives their values. It does not show that these frequencies are special. A length of 50 cm would resonate at 171.5 Hz and 149.9 MHz; a length of 60 cm at 142.9 Hz and 124.9 MHz. There is no physical sense in which one of these is more “harmonic” than another. Whether a resonance near E3 or near the 2-metre band carries any further meaning is a question for interpretation, and this book marks it as such.

Hypothesis · not established

The author proposes that cubit-length objects be studied as resonators in exactly this sense, by calculation first and by measurement second, and that any claim made about them be stated as a frequency, an inductance or a reflection coefficient that a reader can check. Chapter 3 applies this to the bifilar copper ring.

Summary

  • Wavelength, frequency and speed are linked by λ=c/f\lambda = c/f; the same length fits sound of a few hundred hertz and radio of a few hundred megahertz.
  • A one-cubit pipe resonates near 164 Hz (closed at one end) or 328 Hz (open at both ends), close to E3 and E4.
  • A one-cubit wire on a ground plane is an ideal quarter-wave aerial at 143.2 MHz, about 136 MHz in practice.
  • Every length has resonances; the calculation gives the cubit’s values but does not make them special.

References

Balanis, Constantine A. 2016. Antenna Theory: Analysis and Design. 4th ed. Wiley.
Kinsler, Lawrence E., Austin R. Frey, Alan B. Coppens, and James V. Sanders. 2000. Fundamentals of Acoustics. 4th ed. Wiley.

Fan, Y. (2026). The Sacred Cubit and Resonant Geometry: Measuring the Egyptian Royal Cubit, the Quarter Wave and the Bifilar Ring [神圣肘尺与谐振几何] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23666560 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).