Abstract
In gusty wind a cup anemometer turns more than the mean wind (Paper 890), and this overshoot adds to the friction undershoot (Paper 968). There, gusts were a single-frequency sine, and broadband turbulent fluctuation was not treated. This paper applies to the anemometer of Paper 890, with a distance constant of 3 m, broadband gusts of the same mean square as the sine (an Ornstein-Uhlenbeck process with set integral times), and counts how the overshoot changes and on what it collapses. No new theorem or law is claimed.
First, at the same mean square, broadband gusts overshoot less than a sine: at 5 m/s, against the overshoot $0.035459$ of a 0.5 Hz sine (amplitude 0.3), broadband gusts with standard deviation $0.212132$ gave $\mathbf{0.025784}$ at an integral time of 0.5 s (Section 2).
Second, the longer the integral time, the smaller the overshoot: at 5 m/s, integral times of 0.5, 2 and 10 s gave overshoots of $0.025784$, $0.011576$ and $0.003053$ (standard errors $0.000296$, $0.000186$, $0.000114$) (Section 2).
Third, with the same product of wind speed and integral time, the overshoot is the same: the three cases with gust length (wind speed times integral time) 2.5 m (5 m/s and 0.5 s, 10 m/s and 0.25 s, 2.5 m/s and 1 s) gave $0.025784$, $0.025762$ and $0.025899$, and the three with 10 m gave $0.011576$, $0.011515$ and $0.011623$ (Section 3).
Fourth, the collapse lies within the scatter of the random runs: for four pairs of equal length, the overshoot differences were $+0.000022$, $-0.000115$, $+0.000061$ and $-0.000047$, smaller than the sums of standard errors $0.000617$, $0.000556$, $0.000357$ and $0.000416$ (Section 3).
Fifth, the separator is whether gusts are counted by time or by length: with the same integral time of 1 s, the overshoot was $0.025899$ at 2.5 m/s (length 2.5 m) and $0.011515$ at 10 m/s (length 10 m). Counted by length (wind speed times time), the overshoot collapsed on its ratio to the distance constant (Section 4).
Keywords: cup anemometer, overspeeding, turbulence, integral time scale, distance constant, Ornstein-Uhlenbeck process.
Scope (what is not claimed)
No new theorem or law is claimed: that a cup anemometer is characterized by a distance constant, that gusty wind gives high readings (overspeeding), and that their size is set by gust intensity and by the ratio of gust length to distance constant, are all known in meteorological instruments and turbulence measurement. Gusts are simplified: broadband gusts are an Ornstein-Uhlenbeck process (exponentially decaying correlation, with set integral times of 0.25 to 10 s); the spectral shape of real turbulence, wind direction fluctuation and oblique wind are not included. Friction is not included: shaft friction (Paper 968) is left out, and only the response equation of Paper 890 is used. Results carry random scatter: the overshoot is the mean of three random runs of 3600 s, with standard errors from their scatter. No real anemometer or observation is represented. Relation to earlier papers: Paper 968, in its scope note, simplified gusts, taking them as in Paper 890 to be a sine at one frequency (0.5 Hz) and leaving out broadband turbulent fluctuation, wind direction fluctuation and oblique wind; this paper sits in the seat of broadband turbulent fluctuation (no earlier occupant). Paper 890, in its scope note, also took gusts as a single-frequency sine and left out broadband turbulent fluctuation; this paper sits in that seat as well. Paper 890 showed that in gusty wind a cup anemometer reads high because it catches up faster with faster wind. Here it is counted how the overshoot changes when gusts become broadband. Paper 968 showed that the friction undershoot and gust overshoot add. Here friction is removed and only the form of the gusts is changed. What is added: comparing with a sine of the same mean square, finding the overshoot for each integral time, confirming by varying wind speed and integral time separately that it collapses on length, comparing the collapse with standard errors, and placing the separator on “time, or length”.
1. Notation and the Question
Setting: as in Paper 890, the reading $V$ follows $dV/dt=U(U-V)/L$ with distance constant $L=3$ m. Wind is $U=U_0(1+\sigma \xi(t))$, where $\xi$ is an Ornstein-Uhlenbeck process with mean 0 and variance 1,
$$\xi_{i+1}=a\,\xi_i+\sqrt{1-a^2}\,\eta_i,\qquad a=e^{-\Delta t/T_I}$$($\eta_i$ standard normal, $T_I$ the integral time, step $\Delta t=0.01$ s). $\sigma=0.212132$ has the same mean square as a sine of amplitude 0.3. Runs of 3600 s were used, dropping the first 60 s, with overshoot defined as mean reading over mean wind minus 1 and averaged over three random runs. Gust length is $U_0T_I$.
Question: when gusts become broadband, how does the anemometer overshoot change, and on what does it collapse?
2. Sine and Broadband
Computation 2.1: the sine (5 m/s, amplitude 0.3, 0.5 Hz) gave an overshoot of $0.035459$ (as in Paper 890). Broadband gusts (5 m/s, standard deviation $0.212132$) gave:
| integral time | gust length | three runs | mean | standard error |
|---|---|---|---|---|
| $0.50$ s | $2.50$ m | $0.026186$, $0.025207$, $0.025958$ | $\mathbf{0.025784}$ | $0.000296$ |
| $2.00$ s | $10.00$ m | $0.011714$, $0.011208$, $0.011806$ | $0.011576$ | $0.000186$ |
| $10.00$ s | $50.00$ m | $0.003180$, $0.002826$, $0.003152$ | $0.003053$ | $0.000114$ |
Observation 2.2 with broadband gusts, the overshoot is smaller than with a sine and falls with integral time. For all three integral times, the overshoot was below the $0.035459$ of a 0.5 Hz sine of the same mean square: raising the integral time from 0.5 to 10 s lowered the overshoot from $0.025784$ to $0.003053$.
3. Gust Length
Computation 3.1: overshoot for wind speed and integral time varied separately (ratio to distance constant $L/(U_0T_I)$).
| wind speed | integral time | gust length | ratio to distance constant | mean overshoot (standard error) |
|---|---|---|---|---|
| $5.0$ m/s | $0.50$ s | $2.50$ m | $1.2000$ | $0.025784$ ($0.000296$) |
| $10.0$ m/s | $0.25$ s | $2.50$ m | $1.2000$ | $0.025762$ ($0.000322$) |
| $2.5$ m/s | $1.00$ s | $2.50$ m | $1.2000$ | $0.025899$ ($0.000260$) |
| $5.0$ m/s | $2.00$ s | $10.00$ m | $0.3000$ | $0.011576$ ($0.000186$) |
| $10.0$ m/s | $1.00$ s | $10.00$ m | $0.3000$ | $0.011515$ ($0.000171$) |
| $2.5$ m/s | $4.00$ s | $10.00$ m | $0.3000$ | $0.011623$ ($0.000230$) |
Differences for equal lengths were $+0.000022$ (5 and 10 m/s, sum of standard errors $0.000617$) and $-0.000115$ (5 and 2.5 m/s, $0.000556$) at 2.5 m, and $+0.000061$ ($0.000357$) and $-0.000047$ ($0.000416$) at 10 m, all four within three times the sum of standard errors.
Observation 3.2 with the same product of wind speed and integral time, the overshoot is the same. Even with wind speed varied fourfold from 2.5 to 10 m/s, differences at equal length stayed within the random scatter. What set the overshoot was not wind speed and integral time separately but their product, the gust length.
4. The Separator
Observation 4.1 the separator is whether gusts are counted by time or by length. Counted by time, the same integral time of 1 s gave overshoots of $0.025899$ at 2.5 m/s and $0.011515$ at 10 m/s: counted by length, these are 2.5 m and 10 m, and each matched the other cases of the same length. The distance constant of the anemometer is also a length, and the ratio of the two lengths set the overshoot.
Observation 4.2 what is not said. The spectral shape of real turbulence, wind direction fluctuation, oblique wind, friction, values for real anemometers and correction methods for observations are not treated. What is shown is that with exponentially correlated gusts, the overshoot is smaller than for a sine and collapses on gust length.
5. Roll Call and Closing
- Ornstein-Uhlenbeck wind (Section 1): same mean square as the sine.
- overshoot $\mathbf{0.025784}$ at 0.5 s (Computation 2.1): below the sine’s 0.035459.
- 0.025784, 0.011576, 0.003053 at 0.5, 2, 10 s (Computation 2.1): smaller for longer times.
- three cases at 2.5 m: 0.025784, 0.025762, 0.025899 (Computation 3.1): collapse on length.
- differences within summed standard errors (Computation 3.1): four pairs.
- time, or length (Observation 4.1): the separator; the scope notes of Papers 968 and 890.
Closing: With broadband gusts a cup anemometer overshoots less than with a sine, and the overshoot collapses on gust length. Applying broadband gusts of the same mean square as the sine to the anemometer of Paper 890, the overshoot at 5 m/s was $0.025784$, $0.011576$ and $0.003053$ for integral times of 0.5, 2 and 10 s, below the $0.035459$ of a 0.5 Hz sine: varying wind speed and integral time separately, the overshoot came together at $0.025762$ to $0.025899$ for a gust length of 2.5 m and $0.011515$ to $0.011623$ for 10 m. The separator is whether gusts are counted by time or by length. Placed among the earlier papers: the overshoot that Papers 890 and 968 counted with a sine was counted with the broadband gusts they did not treat. To be honest: the gusts are one form of exponentially decaying correlation, not real turbulence. When the form of the gusts is broadened, on what the overshoot collapses, and no further.
References
- Kaimal, J. C., & Finnigan, J. J. (1994). Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press.
- World Meteorological Organization (2008). Guide to Meteorological Instruments and Methods of Observation (7th ed.). WMO.
- Yamagishi, Y. (2026). In Gusty Wind a Cup Anemometer Reads High, Because It Catches Up Faster with Faster Wind [Paper 890]. YOI: https://www.yamagishiyuuki.com/yoi/890
- Yamagishi, Y. (2026). A Cup Anemometer’s Friction Undershoot and Gust Overshoot Add, and They Cancel at a Higher Wind the Weaker the Gusts [Paper 968]. YOI: https://www.yamagishiyuuki.com/yoi/968
Reference
Yamagishi, Y. (2026). With Broadband Gusts a Cup Anemometer Overshoots Less Than with a Sine, and the Overshoot Collapses on Gust Length [Paper 976]. YOI: https://www.yamagishiyuuki.com/yoi/976
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 976. This paper is a preprint and has not been peer reviewed.