In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.^{[1]}^{[2]}^{[3]}
The concept of physical dimension was introduced by Joseph Fourier in 1822.^{[4]} Physical quantities that are of the same kind (also called commensurable) have the same dimension (length, time, mass) and can be directly compared to each other, even if they are originally expressed in differing units of measure (such as yards and meters). If physical quantities have different dimensions (such as length vs. mass), they cannot be expressed in terms of similar units and cannot be compared in quantity (also called incommensurable). For example, asking whether a kilogram is larger than an hour is meaningless.
Any physically meaningful equation (and any inequality) will have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number – a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 miles per hour or 1.4 kilometers per second. Compound relations with "per" are expressed with division, e.g. 60 mi/1 h. Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m^{2} for square meters), or combinations thereof.
A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others, and in terms of which all the remaining units of the system can be expressed.^{[5]} For example, units for length and time are normally chosen as base units. Units for volume, however, can be factored into the base units of length (m^{3}), thus they are considered derived or compound units.
Sometimes the names of units obscure the fact that they are derived units. For example, a newton (N) is a unit of force, which will have units of mass (kg) times acceleration (m⋅s^{−2}). The newton is defined as 1 N = 1 kg⋅m⋅s^{−2}.
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100.
Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus:
In economics, one distinguishes between stocks and flows: a stock has units of "units" (say, widgets or dollars), while a flow is a derivative of a stock, and has units of "units/time" (say, dollars/year).
In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency) – but one may argue that in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance), and thus Debt-to-GDP should have units of years, which indicates that Debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa.
The most basic rule of dimensional analysis is that of dimensional homogeneity.^{[6]}
However, the dimensions form an abelian group under multiplication, so:
For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometer, as these have different dimensions, nor to add 1 hour to 1 kilometer. However, it makes perfect sense to ask whether 1 mile is more, the same, or less than 1 kilometer being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.
The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if m_{man}, m_{rat} and L_{man} denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression m_{man} + m_{rat} is meaningful, but the heterogeneous expression m_{man} + L_{man} is meaningless. However, m_{man}/L^{2}_{man} is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions.
This has the implication that most mathematical functions, particularly the transcendental functions must have a dimensionless quantity, a pure number, as the argument and must return a dimensionless number as a result. This is clear because many transcendental functions can be expressed as an infinite power series with dimensionless coefficients.
All powers of x must have the same dimension for the terms to be commensurable. But if x is not dimensionless, then the different powers of x will have different, incommensurable dimensions. However, power functions including root functions may have a dimensional argument and will return a result having dimension that is the same power applied to the argument dimension. This is because power functions and root functions are, loosely, just an expression of multiplication of quantities.
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension L^{2}MT^{−2}, they are fundamentally different physical quantities.
To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same units. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m.
A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.^{[7]} For example, Newton's laws of motion must hold true whether distance is measured in miles or kilometers. This principle gives rise to the form that conversion factors must take between units that measure the same dimension: multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in meters.
The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors as shown below:
Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units (or some intermediary unit), before being re-arranged to create a factor that cancels out the original unit. For example, as "mile" is the numerator in the original fraction and , "mile" will need to be the denominator in the conversion factor. Dividing both sides of the equation by 1 mile yields , which when simplified results in the dimensionless . Multiplying any quantity (physical quantity or not) by the dimensionless 1 does not change that quantity. Once this and the conversion factor for hours per second have been multiplied by the original fraction to cancel out the units mile and hour, 10 miles per hour converts to 4.4704 meters per second.
As a more complex example, the concentration of nitrogen oxides (i.e., ) in the flue gas from an industrial furnace can be converted to a mass flow rate expressed in grams per hour (i.e., g/h) of by using the following information as shown below:
After canceling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NO_{x} concentration of 10 ppm_{v} converts to mass flow rate of 24.63 grams per hour.
The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides (when expressed in terms of base units) of an equation implies that the equation is wrong.
For example, check the Universal Gas Law equation of PV = nRT, when:
As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units. Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal hitherto unknown or overlooked properties of matter, in the form of left-over dimensions — dimensional adjusters — that can then be assigned physical significance.^{[8]} It is important to point out that such ‘mathematical manipulation’ is neither without prior precedent, nor without considerable scientific significance, indeed, the Planck’s constant; a fundamental constant of the universe, was ‘discovered’ as a purely mathematical abstraction or representation that built on the Rayleigh-Jeans Equation for preventing the ultraviolet catastrophe. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment – not earlier.
The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins (or degrees Fahrenheit). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform (, rather than a linear transform ) between them.
For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, though this would yield the same formula at the end.
Hence, to convert the numerical quantity value of a temperature T[F] in degrees Fahrenheit to a numerical quantity value T[C] in degrees Celsius, this formula may be used:
To convert T[C] in degrees Celsius to T[F] in degrees Fahrenheit, this formula may be used:
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as while the surface area, being -dimensional, scales as Thus the volume of the n-ball in terms of the radius is for some constant Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios, economics ratios, and accounting ratios.
In fluid mechanics, dimensional analysis is performed in order to obtain dimensionless Pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable Pi terms or groups, it is possible to develop a similar set of Pi terms for a model that has the same dimensional relationships.^{[9]} In other words, Pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
The origins of dimensional analysis have been disputed by historians.^{[10]}^{[11]} The 19th-century French mathematician Joseph Fourier is generally credited with having made important contributions^{[12]} based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem. However, the first application of dimensional analysis has been credited to the Italian scholar François Daviet de Foncenex (1734–1799). It was published in 1761, 61 years before the publication of Fourier's work.^{[11]}
James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.^{[13]} Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity, thereby defining M = L^{3}T^{−2}.^{[14]} By assuming a form of Coulomb's law in which Coulomb's constant k_{e} is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = L^{3/2}M^{1/2}T^{−1},^{[15]} which, after substituting his M = L^{3}T^{−2} equation for mass, results in charge having the same dimensions as mass, viz. Q = L^{3}T^{−2}.
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time (Pesic 2005) in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book The Theory of Sound.^{[16]}
The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time.^{[17]} This was slightly changed by Maxwell, who said the dimensions of acceleration are LT^{−2}, instead of just the exponents.^{[18]}
The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
The dimension of a physical quantity can be expressed as a product of the basic physical dimensions such as length, mass and time, each raised to a rational power. The dimension of a physical quantity is more fundamental than some scale unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular scale unit chosen to express a quantity of mass. Except for natural units, the choice of scale is cultural and arbitrary.
There are many possible choices of basic physical dimensions. The SI standard recommends the usage of the following dimensions and corresponding symbols: length (L), mass (M), time (T), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J). The symbols are by convention usually written in roman sans serif typeface.^{[19]} Mathematically, the dimension of the quantity Q is given by
where a, b, c, d, e, f, g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis. For instance, one could replace the dimension of electrical current (I) of the SI basis with a dimension of electric charge (Q), since Q = IT.
As examples, the dimension of the physical quantity speed v is
and the dimension of the physical quantity force F is
The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have conversion factors that relate them. For example, 1 in = 2.54 cm; in this case (2.54 cm/in) is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.
There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,^{[20]} although this does not invalidate the usefulness of dimensional analysis.
The dimensions that can be formed from a given collection of basic physical dimensions, such as M, L, and T, form an abelian group: The identity is written as 1; L^{0} = 1, and the inverse to L is 1/L or L^{−1}. L raised to any rational power p is a member of the group, having an inverse of L^{−p} or 1/L^{p}. The operation of the group is multiplication, having the usual rules for handling exponents (L^{n} × L^{m} = L^{n+m}).
This group can be described as a vector space over the rational numbers, with for example dimensional symbol M^{i}L^{j}T^{k} corresponding to the vector (i, j, k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space. When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space.
A basis for such a vector space of dimensional symbols is called a set of base quantities, and all other vectors are called derived units. As in any vector space, one may choose different bases, which yields different systems of units (e.g., choosing whether the unit for charge is derived from the unit for current, or vice versa).
The group identity 1, the dimension of dimensionless quantities, corresponds to the origin in this vector space.
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The nullity describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {π_{1}, ..., π_{m}}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same units as some derived quantity X can be expressed in the general form
Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
The dimension of physical quantities of interest in mechanics can be expressed in terms of base dimensions M, L, and T – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a change of basis. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not arbitrary, because the dimensions must form a basis: they must span the space, and be linearly independent.
For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to M, L, T: the former can be expressed as [F = ML/T^{2}], L, M, while the latter can be expressed as M, L, [T = (ML/F)^{1/2}].
On the other hand, length, velocity and time (L, V, T) do not form a set of as base dimensions, for two reasons:
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents the dimension of electric charge. In thermodynamics, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ 6.02×10^{23} mol^{−1}) is defined as a base dimension, N, as well. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.
Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.
Similarly, while one can evaluate monomials (x^{n}) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x^{2}, the expression (3 m)^{2} = 9 m^{2} makes sense (as an area), while for x^{2} + x, the expression (3 m)^{2} + 3 m = 9 m^{2} + 3 m does not make sense.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
This is the height to which an object rises in time t if the acceleration of gravity is 32 feet per second per second and the initial upward speed is 500 feet per second. It is not even necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be
The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n.^{[21]}
When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A conversion factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
The factor is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to identical units so that their numerical values can be added or subtracted.
Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an origin. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).
Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:
This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement).
Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement.
Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.
This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,
but for temperature differences,
(Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C.
Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (This issue does not arise in 1 dimension, or rather is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional space, one also needs an orientation: they need to be compared to a frame of reference.
This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.
What is the period of oscillation T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g? That period is the solution for T of some dimensionless equation in the variables T, m, k, and g. The four quantities have the following dimensions: T [T]; m [M]; k [M/T^{2}]; and g [L/T^{2}]. From these we can form only one dimensionless product of powers of our chosen variables, = [T^{2} · M/T^{2} / M = 1], and putting for some dimensionless constant C gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical group. They are often called dimensionless numbers as well.
Note that the variable g does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k, m, and T, because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: , for some dimensionless constant κ (equal to from the original dimensionless equation).
When faced with a case where dimensional analysis rejects a variable (g, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.
When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as κ.
Consider the case of a vibrating wire of length ℓ (L) vibrating with an amplitude A (L). The wire has a linear density ρ (M/L) and is under tension s (ML/T^{2}), and we want to know the energy E (ML^{2}/T^{2}) in the wire. Let π_{1} and π_{2} be two dimensionless products of powers of the variables chosen, given by
The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
where F is some unknown function, or, equivalently as
where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ, and so infer that E = ℓs. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.
The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number, which may be interpreted by dimensional analysis.
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness t (L) and radius R (L). The disc has a density ρ (M/L^{3}), rotates at an angular velocity ω (T^{−1}) and this leads to a stress S (ML^{−1}T^{−2}) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:
Through the use of numerical experiments using, for example, the finite element method, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs^{[22]}
Huntley (Huntley 1967) has pointed out that it is sometimes productive to refine our concept of dimension. Two possible refinements are:
As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannonball travels when fired with a vertical velocity component and a horizontal velocity component , assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then , , both dimensioned as LT^{−1}, R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension LT^{−2}.
With these four quantities, we may conclude that the equation for the range R may be written:
Or dimensionally
from which we may deduce that and , which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions L and T, and four parameters, with one equation.
If, however, we use directed length dimensions, then will be dimensioned as L_{x}T^{−1}, as L_{y}T^{−1}, R as L_{x} and g as L_{y}T^{−2}. The dimensional equation becomes:
and we may solve completely as , and . The increase in deductive power gained by the use of directed length dimensions is apparent.
In a similar manner, it is sometimes found useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of quantity (substantial mass). For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may choose as the relevant variables
There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be and and we may express the dimensional equation as
where C and a are undetermined constants. If we draw a distinction between inertial mass with dimension and substantial mass with dimension , then mass flow rate and density will use substantial mass as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
where now only C is an undetermined constant (found to be equal to by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law.
Huntley's extension has some serious drawbacks:
It also is often quite difficult to assign the L, L_{x}, L_{y}, L_{z}, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?
Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's addition to real problems.
Angles are, by convention, considered to be dimensionless variables, and so the use of angles as physical variables in dimensional analysis can give less meaningful results. As an example, consider the projectile problem mentioned above. Suppose that, instead of the x- and y-components of the initial velocity, we had chosen the magnitude of the velocity v and the angle θ at which the projectile was fired. The angle is, by convention, considered to be dimensionless, and the magnitude of a vector has no directional quality, so that no dimensionless variable can be composed of the four variables g, v, R, and θ. Conventional analysis will correctly give the powers of g and v, but will give no information concerning the dimensionless angle θ.
Siano (1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1_{x} 1_{y} 1_{z} to denote vector directions, and an orientationless symbol 1_{0}. Thus, Huntley's L_{x} becomes L 1_{x} with L specifying the dimension of length, and 1_{x} specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1_{i}^{−1} = 1_{i}, the following multiplication table for the orientation symbols results:
Note that the orientational symbols form a group (the Klein four-group or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1_{z}. For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1_{x} and the side opposite has an orientation 1_{y}. Then, since tan(θ) = 1_{y}/1_{x} = θ + ... we conclude that an angle in the xy-plane must have an orientation 1_{y}/1_{x} = 1_{z}, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1_{z} while cos(θ) has orientation 1_{0}. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ), where a and b are real scalars. Note that an expression such as is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:
which for and yields . Physical quantities may be expressed as complex numbers (e.g. ) which imply that the complex quantity i has an orientation equal to that of the angle it is associated with (1_{z} in the above example).
The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives. Often the added information is that one of the powers of a certain variable is even or odd.
As an example, for the projectile problem, using orientational symbols, θ, being in the xy-plane will thus have dimension 1_{z} and the range of the projectile R will be of the form:
Dimensional homogeneity will now correctly yield a = −1 and b = 2, and orientational homogeneity requires that c be an odd integer. In fact the required function of theta will be sin(θ)cos(θ) which is a series of odd powers of θ.
It is seen that the Taylor series of sin(θ) and cos(θ) are orientationally homogeneous using the above multiplication table, while expressions like cos(θ) + sin(θ) and exp(θ) are not, and are (correctly) deemed unphysical.
It should be clear that the multiplication rule used for the orientational symbols is not the same as that for the cross product of two vectors. The cross product of two identical vectors is zero, while the product of two identical orientational symbols is the identity element.
The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the Ising model can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be where is the dimension of the lattice.
It has been argued by some physicists, e.g., M. J. Duff,^{[20]}^{[23]} that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: c, ħ, and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit , and . In problems involving a gravitational field the latter limit should be taken such that the field stays finite.
Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.^{[24]}^{[25]}^{[26]}
Energy, E
ML^{2}T^{−2} |
Expression | Nomenclature |
---|---|---|
Mechanical | F = force, d = distance | |
S = action, t = time, P = power | ||
m = mass, v = velocity, p = momentum | ||
L = angular momentum, I = moment of inertia, ω = angular velocity | ||
Thermal | p = pressure, T = temperature, S = entropy, k_{B} = boltzmann constant, R = gas constant | |
Waves | I = wave intensity, S = Poynting vector | |
Electromagnetic | q = electric charge, ϕ = electric potential (for changes this is voltage) | |
E = electric field, B = magnetic field, ε = permittivity, μ = permeability, V = 3d volume | ||
p = electric dipole moment, m = magnetic moment, A = area (bounded by a current loop), I = electric current in loop |
Momentum, p
MLT^{−1} |
Expression | Nomenclature |
---|---|---|
Mechanical | m = mass, v = velocity, F = force, t = time | |
S = action, L = angular momentum, r = displacement | ||
Thermal | = root mean square velocity, m = mass (of a molecule) | |
Waves | ρ = density, V = volume, v = phase velocity | |
Electromagnetic | A = magnetic vector potential |
Force, F
MLT^{−2} |
Expression | Nomenclature |
---|---|---|
Mechanical | m = mass, a = acceleration | |
Thermal | S = entropy, T = temperature, r = displacement (see entropic force) | |
Electromagnetic | E = electric field, B = magnetic field, v = velocity, q = charge |
If c = ħ = 1, where c is the speed of light and ħ is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of length L, mass M and time T can be expressed (dimensionally) as a power of energy E, because length, mass and time can be expressed using speed v, action S, and energy E:^{[26]}
though speed and action are dimensionless (v = c = 1 and S = ħ = 1) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions:
This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system.
However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the electron charge e though other choices are possible.
Quantity | p, q, r powers of energy | n power of energy | ||
---|---|---|---|---|
p | q | r | n | |
Action, S | 1 | 2 | –1 | 0 |
Speed, v | 0 | 1 | –1 | 0 |
Mass, M | 1 | 0 | 0 | 1 |
Length, L | 0 | 1 | 0 | –1 |
Time, t | 0 | 0 | 1 | –1 |
Momentum, p | 1 | 1 | –1 | 1 |
Energy, E | 1 | 2 | –2 | 1 |
Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.
Apache Kylin is an open source distributed analytics engine designed to provide a SQL interface and multi-dimensional analysis (OLAP) on Hadoop supporting extremely large datasets.
It was originally developed by eBay, and is now a project of the Apache Software Foundation.
Base unit (measurement)A base unit (also referred to as a fundamental unit) is a unit adopted for measurement of a base quantity. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the subset can be expressed in terms of the others.
A base unit is one that has been explicitly so designated; a secondary unit for the same quantity is a derived unit. For example, when used with the International System of Units, the gram is a derived unit, not a base unit.
In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, temperature, energy, and weight, and units are used to describe their magnitude or quantity. Many of these quantities are related to each other by various physical laws, and as a result the units of a quantities can be generally be expressed as a product of powers of other units; for example, momentum is mass multiplied by velocity, while velocity is measured in distance divided by time. These relationships are discussed in dimensional analysis. Those that can be expressed in this fashion in terms of the base units are called derived units.
Buckingham π theoremIn engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
Dimensionless quantityIn dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one and the corresponding unit of measurement in the SI is one (or 1) unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
Fermi problemIn physics or engineering education, a Fermi problem, Fermi quiz, Fermi question, Fermi estimate, or order estimation is an estimation problem designed to teach dimensional analysis or approximation, and such a problem is usually a back-of-the-envelope calculation. The estimation technique is named after physicist Enrico Fermi as he was known for his ability to make good approximate calculations with little or no actual data. Fermi problems typically involve making justified guesses about quantities and their variance or lower and upper bounds.
Functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.
Homogeneity (physics)In physics, a homogeneous material or system has the same properties at every point; it is uniform without irregularities. A uniform electric field (which has the same strength and the same direction at each point) would be compatible with homogeneity (all points experience the same physics). A material constructed with different constituents can be described as effectively homogeneous in the electromagnetic materials domain, when interacting with a directed radiation field (light, microwave frequencies, etc.).Mathematically, homogeneity has the connotation of invariance, as all components of the equation have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition. Cumulative distribution fits this description. "The state of having identical cumulative distribution function or values".
Infinite Dimensional Analysis, Quantum Probability and Related TopicsInfinite Dimensional Analysis, Quantum Probability and Related Topics is a quarterly peer-reviewed scientific journal published since 1998 by World Scientific. It covers the development of infinite dimensional analysis, quantum probability, and their applications to classical probability and other areas of physics.
NondimensionalizationNondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these values. The technique is especially useful for systems that can be described by differential equations. One important use is in the analysis of control systems.
One of the simplest characteristic units is the doubling time of a system experiencing exponential growth, or conversely the half-life of a system experiencing exponential decay; a more natural pair of characteristic units is mean age/mean lifetime, which correspond to base e rather than base 2.
Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following:
List of dynamical systems and differential equations topics
List of partial differential equation topics
Differential equations of mathematical physicsAlthough nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization in statistics.
Measuring devices are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.
On-balance volumeOn-balance volume (OBV) is a technical analysis indicator intended to relate price and volume in the stock market. OBV is based on a cumulative total volume.
Phase spaceIn dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Willard Gibbs.
Planck massIn physics, the Planck mass, denoted by m_{P}, is the unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams. Unlike some other Planck units, such as Planck length, Planck mass is not a fundamental lower or upper bound; instead, Planck mass is a unit of mass defined using only what Max Planck considered fundamental and universal units. One Planck mass is roughly the mass of a flea egg. For comparison, this value is of the order of 10^{15} (a quadrillion) times larger than the highest energy available to contemporary particle accelerators.
It is defined as:
where c is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant.
Substituting values for the various components in this definition gives the approximate equivalent value of this unit in terms of other units of mass:
For the Planck mass , the Schwarzschild radius () and the Compton wavelength () are of the same order as the Planck length .
Particle physicists and cosmologists often use an alternative normalization with the reduced Planck mass, which is
The factor of simplifies a number of equations in general relativity.
RadianThe radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at OEIS: A072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.Separately, the SI unit of solid angle measurement is the steradian.
The radian is most commonly represented by the symbol rad. An alternative symbol is c, the superscript letter c (for "circular measure"), the letter r, or a superscript R, but these symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r). So, for example, a value of 1.2 radians could be written as 1.2 rad, 1.2 r, 1.2rad, 1.2c, or 1.2R.
Rayleigh's method of dimensional analysisRayleigh's method of dimensional analysis is a conceptual tool used in physics, chemistry, and engineering. This form of dimensional analysis expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh.
The method involves the following steps:
Gather all the independent variables that are likely to influence the dependent variable.
If R is a variable that depends upon independent variables R1, R2, R3, ..., Rn, then the functional equation can be written as R = F(R1, R2, R3, ..., Rn).
Write the above equation in the form R = C R1a R2b R3c ... Rnm, where C is a dimensionless constant and a, b, c, ..., m are arbitrary exponents.
Express each of the quantities in the equation in some base units in which the solution is required.
By using dimensional homogeneity, obtain a set of simultaneous equations involving the exponents a, b, c, ..., m.
Solve these equations to obtain the value of exponents a, b, c, ..., m.
Substitute the values of exponents in the main equation, and form the non-dimensional parameters by grouping the variables with like exponents.Drawback – It doesn't provide any information regarding number of dimensionless groups to be obtained as a result of dimension analysis
SI base unitThe International System of Units (SI) defines seven units of measure as a basic set from which all other SI units can be derived. The SI base units and their physical quantities are the meter for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance.
The SI base units form a set of mutually independent dimensions as required by dimensional analysis commonly employed in science and technology.
The names and symbols of SI base units are written in lowercase, except the symbols of those named after a person, which are written with an initial capital letter. For example, the metre (US English: meter) has the symbol m, but the kelvin has symbol K, because it is named after Lord Kelvin and the ampere with symbol A is named after André-Marie Ampère.
Several other units, such as the litre (US English: liter), are formally not part of the SI, but are accepted for use with SI.
Similitude (model)Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. Similarity and similitude are interchangeable in this context.
The term dynamic similitude is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.
Similitude's main application is in hydraulic and aerospace engineering to test fluid flow conditions with scaled models. It is also the primary theory behind many textbook formulas in fluid mechanics.
The concept of similitude is strongly tied to dimensional analysis.
Transcendental functionA transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.
Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
WattThe watt (symbol: W) is a unit of power. In the International System of Units (SI) it is defined as a derived unit of 1 joule per second, and is used to quantify the rate of energy transfer. In dimensional analysis, power is described by .
Wave equation analysisWave equation analysis is a numerical method of analysis for the behavior of driven foundation piles. It predicts the pile capacity versus blow count relationship (bearing graph) and pile driving stress. The model mathematically represents the pile driving hammer and all its accessories (ram, cap, and cap block), as well as the pile, as a series of lumped masses and springs in a one-dimensional analysis. The soil response for each pile segment is modeled as viscoelastic-plastic. The method was first developed in the 1950s by E.A. Smith of the Raymond Pile Driving Company.
Wave equation analysis of piles has seen many improvements since the 1950s such as including a thermodynamic diesel hammer model and residual stress. Commercial software packages (such as AllWave-PDP and GRLWEAP) are now available to perform the analysis.
One of the principal uses of this method is the performance of a driveability analysis to select the parameters for safe pile installation, including recommendations on cushion stiffness, hammer stroke and other driving system parameters that optimize blow counts and pile stresses during pile driving. For example, when a soft or hard layer causes excessive stresses or unacceptable blow counts.
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