Abstract
Sound is thought to carry because the night is quiet. This paper shows that the temperature aloft is bending the sound back—under a normal atmosphere it escapes upward; under an inversion it returns. The sign reverses. No new theorem or law is claimed.
First, the speed of sound at $0 ^\circ$C is the standard $\mathbf{331.3000000}$ m/s (Section 2).
Second, it runs from $325.1790$ at $-10$ to $\mathbf{349.0194}$ at $30$ (Section 2).
Third, this is the core of the paper. A ray leaving at $80$ degrees becomes $\mathbf{77.5204}$ under a normal atmosphere and $\mathbf{83.3000}$ under an inversion (Section 3).
Fourth, that the bending reverses is shown by the sign of the departure from $80$ degrees (Section 3).
Fifth, beyond the critical angle of $\mathbf{82.5593}$ degrees no refracted solution exists and the ray reflects totally (Section 3).
Sixth, the separator is whether this is a question of attenuation or of refraction (Section 4).
Keywords: sound refraction, speed of sound, temperature inversion, Snell’s law, sound propagation.
Scope (what is not claimed)
No new theorem or law is claimed—the temperature dependence of the speed of sound, Snell’s law and atmospheric refraction of sound are all standard. No weather is forecast—when and where inversions form is not the subject. No noise is assessed—how loud it seems and whether a limit is met are not the subject. Wind is excluded—wind gradients also bend rays, in practice as much as temperature or more; only temperature is treated. Two layers only—ground and aloft; the real atmosphere varies continuously and rays curve. Absorption and ground effect are excluded—atmospheric absorption and reflection or absorption at the ground are not the subject. Relation to earlier papers: Paper 473 showed that what separates is the gel and not the field—there too, the named side was not the working side. Paper 300 showed that common root and distinct root can be decided—“sound diminishing” and “sound bending” are distinct roots by that criterion. Paper 456 showed that doubling the distance does not halve the noise—it treated geometric spreading alone and excluded refraction; this paper takes the outside of that. What is added is giving the speed at four temperatures and matching known values, giving the refracted angle under a normal atmosphere and an inversion, showing the reversal by sign, giving the critical angle as $82.5593$ degrees and confirming that no solution exists beyond it, and placing the separator at attenuation versus refraction.
1. Notation and the Question
Setting: In dry air the speed of sound at Celsius temperature $T$ is
$$c(T)=331.3\sqrt{1+\frac{T}{273.15}}$$and a ray crossing between layers of different speed obeys Snell’s law,
$$\frac{\sin\theta_{1}}{c_{1}}=\frac{\sin\theta_{2}}{c_{2}}$$with $\theta$ measured from the vertical. Below, the ground is at $20 ^\circ$C, the normal air aloft at $15$, and the inversion aloft at $25$.
Observation 1.1 Sound is thought to carry because the night is quiet. Background noise falls after dark—so distant sounds are heard, runs the account.
Observation 1.2 But the speed of sound depends on temperature. Warmer is faster—and where the speed varies with place, rays bend.
Question: Which way, and by how much.
2. Forty Degrees Moves the Speed by 23.84 m/s
Computation 2.1: Take the speed at four temperatures.
| Temperature [$^\circ$C] | Speed [m/s] |
|---|---|
| $-10$ | $\mathbf{325.1790}$ |
| $0$ | $\mathbf{331.3000}$ |
| $15$ | $340.2751$ |
| $30$ | $\mathbf{349.0194}$ |
Observation 2.2 At $0 ^\circ$C it is the standard $\mathbf{331.3000000}$ m/s. Trivially, by construction—but the $340.2751$ at $15$ also matches the textbook “about $340$”, which is an independent check.
Observation 2.3 Forty degrees moves it by $23.84$ m/s. Seven point three per cent—large, spoken of as a refractive index.
Observation 2.4 And temperature varies with height. Normally colder going up; on a clear night the ground cools and it reverses—to sound, the atmosphere is a medium with a gradient of refractive index.
3. Normally Upward, Under an Inversion Downward
Computation 3.1: Take the angle aloft for a ray leaving the $20 ^\circ$C ground at $80$ degrees.
| Aloft | Speed [m/s] | Refracted [deg] | From $80$ |
|---|---|---|---|
| Normal ($15 ^\circ$C) | $340.2751$ | $\mathbf{77.5204}$ | $-2.4796$ |
| Inversion ($25 ^\circ$C) | $346.1292$ | $\mathbf{83.3000}$ | $+3.3000$ |
Observation 3.2 Under a normal atmosphere the angle shrinks. From $80$ to $77.5204$ degrees—towards the vertical, that is, upward.
Observation 3.3 Under an inversion it grows. From $80$ to $83.3000$ degrees—towards the horizontal, and eventually back down.
Observation 3.4 The negative product of the departures shows the reversal. $-2.4796$ and $+3.3000$—same air, same sound, only the temperature aloft differing.
Computation 3.5: Take the critical angle for total reflection under the inversion.
$\mathbf{82.5593}$ degrees.
Observation 3.6 Beyond it no refracted solution exists. At $83.5593$ degrees the sine exceeds $1$—the ray cannot escape upward and reflects totally back to the ground.
Observation 3.7 So an inversion becomes a waveguide holding sound at the ground. What should have escaped comes back—which is what a night of far-carrying sound really is.
4. The Separator
The separator: is this a question of attenuation, or of refraction.
Observation 4.1 Attenuation, and it is set by background noise and distance. Quiet means audible—a natural account of night.
Observation 4.2 Refraction, and it is set by the temperature aloft. The same quiet gives nothing if the air above is cold—the condition is in the sky.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “it carries well”—one a ratio against noise, one a property of the ray path.
Observation 4.4 This is the side Paper 456 excluded. There, geometric spreading alone was treated and refraction was set outside by the scope note—this paper takes the outside. The distance-decay formula holds only for straight rays.
Observation 4.5 It overlaps Paper 473 as well. There, what separated was the gel and not the field—here what carries the sound is the temperature aloft and not the quiet. In both, the conspicuous side is not the working one.
Observation 4.6 And an inversion forms on a clear, still night when the ground radiates its heat away. “A quiet night” and “a night with an inversion” are the same night—so the two accounts blend and are hard to tell apart.
Observation 4.7 To be honest—wind is not included. Wind gradients also bend rays, in practice as much as temperature or more—$82.5593$ degrees is a temperature-only figure, and downwind the critical angle is shallower still.
5. Roll Call and Closing
- $331.3000000$ m/s at $0 ^\circ$C (Observation 2.2): the standard value.
- $340.2751$ at $15$ (Computation 2.1): matching the textbook.
- $23.84$ m/s over forty degrees (Observation 2.3): seven per cent.
- $77.5204$ degrees normally (Computation 3.1): bending upward.
- $83.3000$ under an inversion (Computation 3.1): the core.
- A negative product (Observation 3.4): the reversal.
- Critical angle $82.5593$ degrees (Computation 3.5): total reflection.
- No solution beyond it (Observation 3.6): the sine exceeds $1$.
- The separator is attenuation or refraction (Observation 4.3): distinct roots by Paper 300.
- The side Paper 456 excluded (Observation 4.4): it assumed straight rays.
Closing: Sound is thought to carry because the night is quiet—background noise falls after dark, so distant sounds are heard. But the speed of sound depends on temperature, warmer being faster, and where the speed varies with place, rays bend. Temperature moves the speed considerably—$\mathbf{331.3000000}$ m/s at $0 ^\circ$C, $340.2751$ at $15$ matching the textbook “about $340$”, and from $325.1790$ at $-10$ to $\mathbf{349.0194}$ at $30$: $23.84$ m/s over forty degrees, seven point three per cent. And temperature varies with height—normally colder going up, reversing on a clear night when the ground cools. To sound, the atmosphere is a medium with a gradient of refractive index. The bending reverses with the temperature aloft—a ray leaving the $20 ^\circ$C ground at $80$ degrees becomes $\mathbf{77.5204}$ under a normal atmosphere and $\mathbf{83.3000}$ under an inversion. Normally the angle shrinks towards the vertical and the ray goes up; under an inversion it grows towards the horizontal and comes back down. The departures are $-2.4796$ and $+3.3000$, and their negative product shows the reversal: same air, same sound, only the temperature aloft differing. The critical angle for total reflection under the inversion is $\mathbf{82.5593}$ degrees, and beyond it no refracted solution exists—at $83.5593$ degrees the sine exceeds $1$, so the ray cannot escape and reflects totally back to the ground. So an inversion becomes a waveguide holding sound at the ground, and what should have escaped comes back. That is what a night of far-carrying sound really is. One thing separates them—is this a question of attenuation, or of refraction. Attenuation, and it is set by background noise and distance: quiet means audible, a natural account of night. Refraction, and it is set by the temperature aloft: the same quiet gives nothing if the air above is cold, because the condition is in the sky. They are distinct roots by the criterion of Paper 300, one a ratio against noise and one a property of the ray path. This is the side Paper 456 excluded, where geometric spreading alone was treated: the distance-decay formula holds only for straight rays. It overlaps Paper 473 too, where what separated was the gel and not the field: in both, the conspicuous side is not the working one. And an inversion forms on a clear, still night when the ground radiates its heat away, so “a quiet night” and “a night with an inversion” are the same night, and the two accounts blend. To be honest—wind is not included. Wind gradients also bend rays, in practice as much as temperature or more; $82.5593$ degrees is a temperature-only figure, and downwind the critical angle is shallower still.
References
- Snell, W. (1621). De Refractione. (posthumously described by Huygens.)
- Rayleigh, J. W. S. (1878). The Theory of Sound. Macmillan.
- Embleton, T. F. W. (1996). Tutorial on Sound Propagation Outdoors. Journal of the Acoustical Society of America.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Doubling the Distance Does Not Halve the Noise [Paper 456]. Zenodo. 10.5281/zenodo.22733038
- Yamagishi, Y. (2026). What Separates Them Is Not the Field but the Narrowness of the Path [Paper 473]. Zenodo. 10.5281/zenodo.22733552
Reference
Yamagishi, Y. (2026). On Nights When Distant Sounds Carry, the Sky Is Warm [Paper 474]. YOI: https://www.yamagishiyuuki.com/yoi/474
DOI: 10.5281/zenodo.22733583
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 474. This paper is a preprint and has not been peer reviewed.