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32-451: The Douglas sea scale, also called the "international sea and swell scale", was devised in 1921 by Captain H. P. Douglas, who later became vice admiral Sir Percy Douglas and hydrographer of the Royal Navy . Its purpose is to estimate the roughness of the sea for navigation . The scale has two codes: one code is for estimating the sea state , the other code is for describing the swell of

64-528: A NATO ranking code of OF-5. The rank is equivalent to a colonel in the British Army and Royal Marines , and to a group captain in the Royal Air Force . There are similarly named equivalent ranks in the navies of many other countries. In the Royal Navy, the officer in command of any warship of the rank of commander and below is informally referred to as "the captain" on board, even though holding

96-459: A continuous-time waveform ) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In the case of a set of n values { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS is The corresponding formula for

128-421: A population or a waveform x {\displaystyle x} is the RMS deviation of x {\displaystyle x} from its arithmetic mean x ¯ {\displaystyle {\bar {x}}} . They are related to the RMS value of x {\displaystyle x} by From this it is clear that the RMS value is always greater than or equal to

160-426: A continuous function (or waveform) f ( t ) defined over the interval T 1 ≤ t ≤ T 2 {\displaystyle T_{1}\leq t\leq T_{2}} is and the RMS for a function over all time is The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking

192-590: A decade after the Second World War . The title was probably used informally up until the abolition of frigate and destroyer squadrons with the Fleet FIRST reorganisation circa 2001. Ashore, the rank of captain is often verbally described as "captain RN" to distinguish it from the more junior Army and Royal Marines rank , and in naval contexts, as a "four-ring captain" (referring to the uniform lace) to avoid confusion with

224-508: A junior rank, but formally is titled "the commanding officer" (or CO). Until the nineteenth century, Royal Navy officers who were captains by rank and in command of a naval vessel were referred to as post-captains ; this practice is now defunct. Captain (D) or Captain Destroyers, afloat, was an operational appointment commanding a destroyer flotilla or squadron , and there was a corresponding administrative appointment ashore, until at least

256-402: A set x i {\displaystyle x_{i}} , its RMS is denoted as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} . The RMS is also known as the quadratic mean (denoted M 2 {\displaystyle M_{2}} ),

288-418: A special case of the generalized mean . The RMS of a continuous function is denoted f R M S {\displaystyle f_{\mathrm {RMS} }} and can be defined in terms of an integral of the square of the function. In estimation theory , the root-mean-square deviation of an estimator measures how far the estimator strays from the data. The RMS value of a set of values (or

320-420: Is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is: For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave: Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of

352-403: Is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation: So, the RMS value, I RMS , of the function I ( t ) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I ( t ). Average power can also be found using

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384-475: Is the optional rounding to the closest integer value. Without the rounding to integer, the root mean square error of this approximation is: R M S E ≤ 0.18 {\textstyle RMSE\leq 0.18} . Wavelength Wave height Captain (Royal Navy) Captain ( Capt ) is a senior officer rank of the Royal Navy . It ranks above commander and below commodore and has

416-585: Is the sampling period, where X [ m ] = DFT ⁡ { x [ n ] } {\displaystyle X[m]=\operatorname {DFT} \{x[n]\}} and N is the sample size, that is, the number of observations in the sample and DFT coefficients. In this case, the RMS computed in the time domain is the same as in the frequency domain: The standard deviation σ x = ( x − x ¯ ) rms {\displaystyle \sigma _{x}=(x-{\overline {x}})_{\text{rms}}} of

448-409: The power , P , dissipated by an electrical resistance , R . It is easy to do the calculation when there is a constant current , I , through the resistance. For a load of R ohms, power is given by: However, if the current is a time-varying function, I ( t ), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function

480-479: The RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus , as shown by Cartwright. In the case of the RMS statistic of a random process , the expected value is used instead of the mean. If the waveform is a pure sine wave , the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this

512-496: The above formula, which implies V P  =  V RMS  ×  √ 2 , assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 ×  √ 2 , or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and

544-506: The analogous equation for sinusoidal voltage: where I P represents the peak current and V P represents the peak voltage. Because of their usefulness in carrying out power calculations, listed voltages for power outlets (for example, 120   V in the US, or 230   V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from

576-417: The average, in that the RMS includes the squared deviation (error) as well. Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit. This is useful for electrical engineers in calculating the "AC only" RMS of

608-454: The component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself). Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly. The RMS of an alternating electric current equals the value of constant direct current that would dissipate

640-819: The high Height values ( λ H = 2.0872 , β H = 0.6091 {\textstyle \lambda _{H}=2.0872,\beta _{H}=0.6091} ). Then the Degree can be approximated as the average between the low and high estimations, i.e.: D ≃ [ 1 2 ( λ L H L + λ H H H ) + 1 2 ( β L + β H ) ] {\displaystyle D\simeq \left[{\tfrac {1}{2}}\left(\lambda _{L}{\sqrt {H_{L}}}+\lambda _{H}{\sqrt {H_{H}}}\right)+{\tfrac {1}{2}}\left(\beta _{L}+\beta _{H}\right)\right]} where [.]

672-408: The load, R , is purely resistive). Reactive loads (that is, loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power . In the common case of alternating current when I ( t ) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If I p is defined to be

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704-406: The long term. The term RMS power is sometimes erroneously used (e.g., in the audio industry) as a synonym for mean power or average power (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power . In the physics of gas molecules, the root-mean-square speed is defined as

736-401: The order of thousands of km/h, even though the average velocity of its molecules is zero. When two data sets — one set from theoretical prediction and the other from actual measurement of some physical variable, for instance — are compared, the RMS of the pairwise differences of the two data sets can serve as a measure of how far on average the error is from 0. The mean of the absolute values of

768-495: The pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae. The RMS can be computed in the frequency domain, using Parseval's theorem . For a sampled signal x [ n ] = x ( t = n T ) {\displaystyle x[n]=x(t=nT)} , where T {\displaystyle T}

800-459: The peak current, then: where t is time and ω is the angular frequency ( ω  = 2 π / T , where T is the period of the wave). Since I p is a positive constant and was to be squared within the integral: Using a trigonometric identity to eliminate squaring of trig function: but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving: A similar analysis leads to

832-428: The peak-to-peak mains voltage, about 650 volts. RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in

864-411: The rank of captain and above wear gold-laced trousers (the trousers are known as "tin trousers", and the gold lace stripes thereon are nicknamed "lightning conductors"), and may wear the undress tailcoat (without epaulettes). Root mean square In mathematics , the root mean square (abbrev. RMS , RMS or rms ) of a set of numbers is the square root of the set's mean square . Given

896-462: The same method that in the case of a time-varying voltage , V ( t ), with RMS value V RMS , This equation can be used for any periodic waveform , such as a sinusoidal or sawtooth waveform , allowing us to calculate the mean power delivered into a specified load. By taking the square root of both these equations and multiplying them together, the power is found to be: Both derivations depend on voltage and current being proportional (that is,

928-443: The same power in a resistive load . A special case of RMS of waveform combinations is: where V DC {\displaystyle {\text{V}}_{\text{DC}}} refers to the direct current (or average) component of the signal, and RMS AC {\displaystyle {\text{RMS}}_{\text{AC}}} is the alternating current component of the signal. Electrical engineers often need to know

960-538: The sea. The Degree (D) value has an almost linear dependence on the square root of the average wave Height (H) above, i.e., D ≃ β + λ H {\textstyle D\simeq \beta +\lambda {\sqrt {H}}} . Using linear regression on the table above, the coefficients can be calculated for the low Height values ( λ L = 2.3236 , β L = 1.2551 {\textstyle \lambda _{L}=2.3236,\beta _{L}=1.2551} ) and for

992-429: The square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation: where R represents the gas constant , 8.314 J/(mol·K), T is the temperature of the gas in kelvins , and M is the molar mass of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in

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1024-510: The title of a seagoing commanding officer. In the Ministry of Defence , and in joint service establishments, a captain may be referred to as a "DACOS" (standing for deputy assistant chief of staff) or an "AH" (assistant head), from the usual job title of OF5-ranked individuals who work with civil servants. The rank insignia features four rings of gold braid with an executive curl in the upper ring. When in mess dress or mess undress, officers of

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