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89-411: S waves are transverse waves , meaning that the direction of particle movement of an S wave is perpendicular to the direction of wave propagation, and the main restoring force comes from shear stress . Therefore, S waves cannot propagate in liquids with zero (or very low) viscosity ; however, they may propagate in liquids with high viscosity. The name secondary wave comes from the fact that they are

178-511: A dual pair to show the underlying duality . This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation , the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application . In category theory and homological algebra , networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize

267-844: A function of the rest position x {\displaystyle {\boldsymbol {x}}} and time t {\displaystyle t} . The deformation of the medium at that point can be described by the strain tensor e {\displaystyle {\boldsymbol {e}}} , the 3×3 matrix whose elements are e i j = 1 2 ( ∂ i u j + ∂ j u i ) {\displaystyle e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)} where ∂ i {\displaystyle \partial _{i}} denotes partial derivative with respect to position coordinate x i {\displaystyle x_{i}} . The strain tensor

356-421: A longitudinal wave travels in the direction of its oscillations. All waves move energy from place to place without transporting the matter in the transmission medium if there is one. Electromagnetic waves are transverse without requiring a medium. The designation “transverse” indicates the direction of the wave is perpendicular to the displacement of the particles of the medium through which it passes, or in

445-432: A map or a mapping , but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds ). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H ). Some authors reserve

534-405: A roman type is customarily used instead, such as " sin " for the sine function , in contrast to italic font for single-letter symbols. The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let f ( x ) {\displaystyle f(x)} be a function". This is an abuse of notation that is useful for

623-409: A viscoelastic material, the speed of a shear wave is described by a similar relationship c ( ω ) = ω / k ( ω ) = μ ( ω ) / ρ {\displaystyle c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}} , however, here, μ {\displaystyle \mu }

712-449: A different amplitude.) In a homogeneous linear medium, complex oscillations (vibrations in a material or light flows) can be described as the superposition of many simple sinusoidal waves, either transverse or longitudinal. The vibrations of a violin string create standing waves , for example, which can be analyzed as the sum of many transverse waves of different frequencies moving in opposite directions to each other, that displace

801-549: A fixed point p → {\displaystyle {\vec {p}}} will see the particle there move in a simple harmonic (sinusoidal) motion with period T seconds, with maximum particle displacement A in each sense; that is, with a frequency of f = 1/ T full oscillation cycles every second. A snapshot of all particles at a fixed time t will show the same displacement for all particles on each plane perpendicular to d ^ {\displaystyle {\widehat {d}}} , with

890-429: A function defined by an integral with variable upper bound: x ↦ ∫ a x f ( u ) d u {\textstyle x\mapsto \int _{a}^{x}f(u)\,du} . There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis , linear forms and the vectors they act upon are denoted using

979-401: A function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details. A function f from a set X to a set Y is an assignment of one element of Y to each element of X . The set X is called the domain of the function and the set Y is called the codomain of

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1068-523: A function is commonly written as f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as f ( w , x , y ) {\displaystyle f(w,x,y)} , f ( w , x , y , z ) {\displaystyle f(w,x,y,z)} . A function may also be called

1157-513: A function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a real function , the determination of the domain of the function x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} requires knowing

1246-405: A function is then called a partial function . The range or image of a function is the set of the images of all elements in the domain. A function f on a set S means a function from the domain S , without specifying a codomain. However, some authors use it as shorthand for saying that the function is f  : S → S . The above definition of a function is essentially that of

1335-592: A function is uniquely represented by the set of all pairs ( x , f  ( x )) , called the graph of the function , a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. Functions are widely used in science , engineering , and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics. The concept of

1424-429: A high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory , and this greatly increased the possible applications of the concept. A function is often denoted by a letter such as f , g or h . The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x ) is denoted by f ( x ) ; for example,

1513-407: A linear combination (mixing) of those two waves. By combining two waves with same frequency, velocity, and direction of travel, but with different phases and independent displacement directions, one obtains a circularly or elliptically polarized wave. In such a wave the particles describe circular or elliptical trajectories, instead of moving back and forth. It may help understanding to revisit

1602-417: A multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain. where the domain U has the form If all the X i {\displaystyle X_{i}} are equal to the set R {\displaystyle \mathbb {R} } of the real numbers or to the set C {\displaystyle \mathbb {C} } of

1691-599: A plane linearly polarized sinusoidal light wave, except that the "displacement" S ( p → {\displaystyle {\vec {p}}} , t ) is the electric field at point p → {\displaystyle {\vec {p}}} and time t . (The magnetic field will be described by the same equation, but with a "displacement" direction that is perpendicular to both d ^ {\displaystyle {\widehat {d}}} and u ^ {\displaystyle {\widehat {u}}} , and

1780-450: A simpler formulation. Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced " maps to ". For example, x ↦ x + 1 {\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of R {\displaystyle \mathbb {R} }

1869-588: A solid medium is considered isotropic if its strain (deformation) in response to stress is the same in all directions. Let u = ( u 1 , u 2 , u 3 ) {\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})} be the displacement vector of a particle of such a medium from its "resting" position x = ( x 1 , x 2 , x 3 ) {\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})} due elastic vibrations, understood to be

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1958-425: A straight line are linearly polarized waves. But now imagine moving your hand in a circle. Your motion will launch a spiral wave on the string. You are moving your hand simultaneously both up and down and side to side. The maxima of the side to side motion occur a quarter wavelength (or a quarter of a way around the circle, that is 90 degrees or π/2 radians) from the maxima of the up and down motion. At any point along

2047-405: Is 1 2 μ A 2 ω 2 v x {\textstyle {\frac {1}{2}}\mu A^{2}\omega ^{2}v_{x}} Function (mathematics) In mathematics , a function from a set X to a set Y assigns to each element of X exactly one element of Y . The set X is called the domain of the function and the set Y is called

2136-604: Is a complex, frequency-dependent shear modulus and c ( ω ) {\displaystyle c(\omega )} is the frequency dependent phase velocity. One common approach to describing the shear modulus in viscoelastic materials is through the Voigt Model which states: μ ( ω ) = μ 0 + i ω η {\displaystyle \mu (\omega )=\mu _{0}+i\omega \eta } , where μ 0 {\displaystyle \mu _{0}}

2225-423: Is a function in two variables, and we want to refer to a partially applied function X → Y {\displaystyle X\to Y} produced by fixing the second argument to the value t 0 without introducing a new function name. The map in question could be denoted x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} using

2314-400: Is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. Formally, a function of n variables is a function whose domain is a set of n -tuples. For example, multiplication of integers is a function of two variables, or bivariate function , whose domain is

2403-428: Is called the Cartesian product of X and Y and denoted X × Y . {\displaystyle X\times Y.} Thus, the above definition may be formalized as follows. A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions: This definition may be rewritten more formally, without referring explicitly to

2492-438: Is implied. The domain and codomain can also be explicitly stated, for example: This defines a function sqr from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f : X × X → Y ; ( x , t ) ↦ f ( x , t ) {\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}

2581-432: Is in Y , or it is undefined. The set of the elements of X such that f ( x ) {\displaystyle f(x)} is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X , one often says that

2670-467: Is more than twice the speed β {\displaystyle \beta } of S waves. The steady state SH waves are defined by the Helmholtz equation ( ∇ 2 + k 2 ) u = 0 {\displaystyle \left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0} where k is the wave number. Similar to in an elastic medium, in

2759-454: Is often the letter f . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in The argument between the parentheses may be a variable , often x , that represents an arbitrary element of the domain of the function, a specific element of the domain ( 3 in

S wave - Misplaced Pages Continue

2848-485: Is related to the 3×3 stress tensor τ {\displaystyle {\boldsymbol {\tau }}} by the equation τ i j = λ δ i j ∑ k e k k + 2 μ e i j {\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}} Here δ i j {\displaystyle \delta _{ij}}

2937-1259: Is the Kronecker delta (1 if i = j {\displaystyle i=j} , 0 otherwise) and λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the Lamé parameters ( μ {\displaystyle \mu } being the material's shear modulus ). It follows that τ i j = λ δ i j ∑ k ∂ k u k + μ ( ∂ i u j + ∂ j u i ) {\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)} From Newton's law of inertia , one also gets ρ ∂ t 2 u i = ∑ j ∂ j τ i j {\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}} where ρ {\displaystyle \rho }

3026-904: Is the density (mass per unit volume) of the medium at that point, and ∂ t {\displaystyle \partial _{t}} denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media ρ ∂ t 2 u i = λ ∂ i ∑ k ∂ k u k + μ ∑ j ( ∂ i ∂ j u j + ∂ j ∂ j u i ) {\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}\sum _{k}\partial _{k}u_{k}+\mu \sum _{j}{\bigl (}\partial _{i}\partial _{j}u_{j}+\partial _{j}\partial _{j}u_{i}{\bigr )}} Using

3115-440: Is the value of the function at x , or the image of x under the function. A function f , its domain X , and its codomain Y are often specified by the notation f : X → Y . {\displaystyle f:X\to Y.} One may write x ↦ y {\displaystyle x\mapsto y} instead of y = f ( x ) {\displaystyle y=f(x)} , where

3204-506: Is the wave equation applied to the vector quantity ∇ × u {\displaystyle \nabla \times {\boldsymbol {u}}} , which is the material's shear strain. Its solutions, the S waves, are linear combinations of sinusoidal plane waves of various wavelengths and directions of propagation, but all with the same speed β = μ / ρ {\textstyle \beta ={\sqrt {\mu /\rho }}} . Assuming that

3293-436: Is the stiffness of the material and η {\displaystyle \eta } is the viscosity. Magnetic resonance elastography (MRE) is a method for studying the properties of biological materials in living organisms by propagating shear waves at desired frequencies throughout the desired organic tissue. This method uses a vibrator to send the shear waves into the tissue and magnetic resonance imaging to view

3382-415: Is the wave's amplitude or strength , T is its period , v is the speed of propagation, and ϕ {\displaystyle \phi } is its phase at t = 0 seconds at o → {\displaystyle {\vec {o}}} . All these parameters are real numbers . The symbol "•" denotes the inner product of two vectors. By this equation, the wave travels in

3471-469: Is typically the case for functions whose domain is the set of the natural numbers . Such a function is called a sequence , and, in this case the element f n {\displaystyle f_{n}} is called the n th element of the sequence. The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during

3560-472: The French Academy of Sciences an essay ("memoir") with a theory of the propagation of elastic waves in solids. In his memoir, he states that an earthquake would produce two different waves: one having a certain speed a {\displaystyle a} and the other having a speed a 3 {\displaystyle {\frac {a}{\sqrt {3}}}} . At a sufficient distance from

3649-692: The Riemann hypothesis . In computability theory , a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem ). A multivariate function , multivariable function , or function of several variables

S wave - Misplaced Pages Continue

3738-411: The codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically , the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had

3827-413: The complex numbers , one talks respectively of a function of several real variables or of a function of several complex variables . There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below. The functional notation requires that a name is given to the function, which, in the case of a unspecified function

3916-447: The curl of this equation and applying vector identities, one gets ∂ t 2 ( ∇ × u ) = μ ρ ∇ 2 ( ∇ × u ) {\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)} This formula

4005-537: The divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity ∇ ⋅ u {\displaystyle \nabla \cdot {\boldsymbol {u}}} , which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed α = ( λ + 2 μ ) / ρ {\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}} that

4094-832: The nabla operator notation of vector calculus , ∇ = ( ∂ 1 , ∂ 2 , ∂ 3 ) {\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})} , with some approximations, this equation can be written as ρ ∂ t 2 u = ( λ + 2 μ ) ∇ ( ∇ ⋅ u ) − μ ∇ × ( ∇ × u ) {\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla \left(\nabla \cdot {\boldsymbol {u}}\right)-\mu \nabla \times \left(\nabla \times {\boldsymbol {u}}\right)} Taking

4183-420: The zeros of f. This is one of the reasons for which, in mathematical analysis , "a function from X to Y " may refer to a function having a proper subset of X as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers , typically a subset that contains a non-empty open interval . Such

4272-550: The "total" condition removed. That is, a partial function from X to Y is a binary relation R between X and Y such that, for every x ∈ X , {\displaystyle x\in X,} there is at most one y in Y such that ( x , y ) ∈ R . {\displaystyle (x,y)\in R.} Using functional notation, this means that, given x ∈ X , {\displaystyle x\in X,} either f ( x ) {\displaystyle f(x)}

4361-452: The above example), or an expression that can be evaluated to an element of the domain ( x 2 + 1 {\displaystyle x^{2}+1} in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let f ( x ) = sin ⁡ ( x 2 + 1 ) {\displaystyle f(x)=\sin(x^{2}+1)} ". When

4450-432: The arrow notation for functions described above. In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers ( x , y ) {\displaystyle (x,y)} to the sum of their squares, x 2 + y 2 {\displaystyle x^{2}+y^{2}} . Such

4539-590: The arrow notation. The expression x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f ( x 0 , t 0 ) refers to the value of the function f at the point ( x 0 , t 0 ) . Index notation may be used instead of functional notation. That is, instead of writing f  ( x ) , one writes f x . {\displaystyle f_{x}.} This

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4628-469: The boundary of molten and solid cores at an oblique angle, S waves will form and propagate in the solid medium. When these S waves hit the boundary again at an oblique angle, they will in turn create P waves that propagate through the liquid medium. This property allows seismologists to determine some physical properties of the Earth's inner core. In 1830, the mathematician Siméon Denis Poisson presented to

4717-406: The case of EM waves, the oscillation is perpendicular to the direction of the wave. A simple example is given by the waves that can be created on a horizontal length of string by anchoring one end and moving the other end up and down. Another example is the waves that are created on the membrane of a drum . The waves propagate in directions that are parallel to the membrane plane, but each point in

4806-434: The concept of a relation, but using more notation (including set-builder notation ): A function is formed by three sets, the domain X , {\displaystyle X,} the codomain Y , {\displaystyle Y,} and the graph R {\displaystyle R} that satisfy the three following conditions. Partial functions are defined similarly to ordinary functions, with

4895-418: The direction d ^ {\displaystyle {\widehat {d}}} and the oscillations occur back and forth along the direction u ^ {\displaystyle {\widehat {u}}} . The wave is said to be linearly polarized in the direction u ^ {\displaystyle {\widehat {u}}} . An observer that looks at

4984-464: The direction of propagation. The motion of such a wave can be expressed mathematically as follows. Let d ^ {\displaystyle {\widehat {d}}} be the direction of propagation (a vector with unit length), and o → {\displaystyle {\vec {o}}} any reference point in the medium. Let u ^ {\displaystyle {\widehat {u}}} be

5073-796: The direction of the oscillations (another unit-length vector perpendicular to d ). The displacement of a particle at any point p → {\displaystyle {\vec {p}}} of the medium and any time t (seconds) will be S ( p → , t ) = A sin ⁡ ( ( 2 π ) t − ( p → − o → ) v ⋅ d ^ T + ϕ ) u ^ {\displaystyle S({\vec {p}},t)=A\sin \left((2\pi ){\frac {t-{\frac {({\vec {p}}-{\vec {o}})}{v}}\cdot {\widehat {d}}}{T}}+\phi \right){\widehat {u}}} where A

5162-553: The displacement of the solid particles away from their relaxed position, in directions perpendicular to the propagation of the wave. These displacements correspond to a local shear deformation of the material. Hence a transverse wave of this nature is called a shear wave . Since fluids cannot resist shear forces while at rest, propagation of transverse waves inside the bulk of fluids is not possible. In seismology , shear waves are also called secondary waves or S-waves . Transverse waves are contrasted with longitudinal waves , where

5251-405: The displacements in successive planes forming a sinusoidal pattern, with each full cycle extending along d ^ {\displaystyle {\widehat {d}}} by the wavelength λ = v T = v / f . The whole pattern moves in the direction d ^ {\displaystyle {\widehat {d}}} with speed V . The same equation describes

5340-479: The domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x\in X} and y ∈ Y . {\displaystyle y\in Y.} The set of all these pairs

5429-410: The domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function : the determination of the domain of definition of the function z ↦ 1 / ζ ( z ) {\displaystyle z\mapsto 1/\zeta (z)} is more or less equivalent to the proof or disproof of one of the major open problems in mathematics,

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5518-448: The domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set C {\displaystyle \mathbb {C} } of the complex numbers . The difficulty of determining

5607-405: The extreme of eccentricity your ellipse will become a straight line, producing linear polarization along the major axis of the ellipse. An elliptical motion can always be decomposed into two orthogonal linear motions of unequal amplitude and 90 degrees out of phase, with circular polarization being the special case where the two linear motions have the same amplitude. (Let the linear mass density of

5696-408: The founders of calculus , Leibniz , Newton and Euler . However, it cannot be formalized , since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory . This set-theoretic definition is based on the fact that a function establishes a relation between the elements of

5785-474: The function f  (⋅) from its value f  ( x ) at x . For example, a ( ⋅ ) 2 {\displaystyle a(\cdot )^{2}} may stand for the function x ↦ a x 2 {\displaystyle x\mapsto ax^{2}} , and ∫ a ( ⋅ ) f ( u ) d u {\textstyle \int _{a}^{\,(\cdot )}f(u)\,du} may stand for

5874-407: The function. If the element y in Y is assigned to x in X by the function f , one says that f maps x to y , and this is commonly written y = f ( x ) . {\displaystyle y=f(x).} In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable , and the corresponding element of Y

5963-499: The medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as μ = ρ β 2 = ρ ω 2 / k 2 {\displaystyle \mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}} where ω is the angular frequency and k is the wavenumber. Thus, β = ω / k {\displaystyle \beta =\omega /k} . Taking

6052-411: The membrane itself gets displaced up and down, perpendicular to that plane. Light is another example of a transverse wave, where the oscillations are the electric and magnetic fields , which point at right angles to the ideal light rays that describe the direction of propagation. Transverse waves commonly occur in elastic solids due to the shear stress generated; the oscillations in this case are

6141-436: The notation x ↦ f ( x ) , {\displaystyle x\mapsto f(x),} the symbol x does not represent any value; it is simply a placeholder , meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing

6230-427: The oscillations occur in the direction of the wave. The standard example of a longitudinal wave is a sound wave or "pressure wave" in gases, liquids, or solids, whose oscillations cause compression and expansion of the material through which the wave is propagating. Pressure waves are called "primary waves", or "P-waves" in geophysics. Water waves involve both longitudinal and transverse motions. Mathematically,

6319-404: The partial function is a total function . In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain. In calculus , a real-valued function of a real variable or real function is a partial function from

6408-851: The potential energy for one wavelength U = 1 2 μ A 2 ω 2 ∫ 0 λ sin 2 ⁡ ( 2 π x λ − ω t ) d x = 1 4 μ A 2 ω 2 λ {\displaystyle U={\frac {1}{2}}\mu A^{2}\omega ^{2}\int _{0}^{\lambda }\sin ^{2}\left({\frac {2\pi x}{\lambda }}-\omega t\right)dx={\frac {1}{4}}\mu A^{2}\omega ^{2}\lambda } So, total energy in one wavelength K + U = 1 2 μ A 2 ω 2 λ {\textstyle K+U={\frac {1}{2}}\mu A^{2}\omega ^{2}\lambda } Therefore average power

6497-565: The potential energy in mass element d U = 1 2   d m ω 2   y 2 = 1 2   μ d x ω 2   A 2 sin 2 ⁡ ( 2 π x λ − ω t ) {\displaystyle dU={\frac {1}{2}}\ dm\omega ^{2}\ y^{2}={\frac {1}{2}}\ \mu dx\omega ^{2}\ A^{2}\sin ^{2}\left({\frac {2\pi x}{\lambda }}-\omega t\right)} And

6586-405: The response in the tissue. The measured wave speed and wavelengths are then measured to determine elastic properties such as the shear modulus . MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues. Transverse wave In physics , a transverse wave is a wave that oscillates perpendicularly to the direction of the wave's advance. In contrast,

6675-402: The second type of wave to be detected by an earthquake seismograph , after the compressional primary wave, or P wave , because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the molten outer core of the Earth, and this causes a shadow zone for S waves opposite to their origin. They can still propagate through the solid inner core : when a P wave strikes

6764-408: The set R {\displaystyle \mathbb {R} } of the real numbers to itself. Given a real function f : x ↦ f ( x ) {\displaystyle f:x\mapsto f(x)} its multiplicative inverse x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} is also a real function. The determination of

6853-803: The set of all n -tuples ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} such that x 1 ∈ X 1 , … , x n ∈ X n {\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}} is called the Cartesian product of X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},} and denoted X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.} Therefore,

6942-865: The set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation . Commonly, an n -tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , … , n ) . {\displaystyle (1,2,\ldots ,n).} When using functional notation , one usually omits the parentheses surrounding tuples, writing f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} instead of f ( ( x 1 , … , x n ) ) . {\displaystyle f((x_{1},\ldots ,x_{n})).} Given n sets X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},}

7031-406: The simplest kind of transverse wave is a plane linearly polarized sinusoidal one. "Plane" here means that the direction of propagation is unchanging and the same over the whole medium; " linearly polarized " means that the direction of displacement too is unchanging and the same over the whole medium; and the magnitude of the displacement is a sinusoidal function only of time and of position along

7120-415: The source, when they can be considered plane waves in the region of interest, the first kind consists of expansions and compressions in the direction perpendicular to the wavefront (that is, parallel to the wave's direction of motion); while the second consists of stretching motions occurring in directions parallel to the front (perpendicular to the direction of motion). For the purpose of this explanation,

7209-1203: The string be μ.) The kinetic energy of a mass element in a transverse wave is given by: d K = 1 2   d m   v y 2 = 1 2   μ d x   A 2 ω 2 cos 2 ⁡ ( 2 π x λ − ω t ) {\displaystyle dK={\frac {1}{2}}\ dm\ v_{y}^{2}={\frac {1}{2}}\ \mu dx\ A^{2}\omega ^{2}\cos ^{2}\left({\frac {2\pi x}{\lambda }}-\omega t\right)} In one wavelength, kinetic energy K = 1 2 μ A 2 ω 2 ∫ 0 λ cos 2 ⁡ ( 2 π x λ − ω t ) d x = 1 4 μ A 2 ω 2 λ {\displaystyle K={\frac {1}{2}}\mu A^{2}\omega ^{2}\int _{0}^{\lambda }\cos ^{2}\left({\frac {2\pi x}{\lambda }}-\omega t\right)dx={\frac {1}{4}}\mu A^{2}\omega ^{2}\lambda } Using Hooke's law

7298-434: The string either up or down or left to right. The antinodes of the waves align in a superposition . If the medium is linear and allows multiple independent displacement directions for the same travel direction d ^ {\displaystyle {\widehat {d}}} , we can choose two mutually perpendicular directions of polarization, and express any wave linearly polarized in any other direction as

7387-453: The string, the displacement of the string will describe the same circle as your hand, but delayed by the propagation speed of the wave. Notice also that you can choose to move your hand in a clockwise circle or a counter-clockwise circle. These alternate circular motions produce right and left circularly polarized waves. To the extent your circle is imperfect, a regular motion will describe an ellipse, and produce elliptically polarized waves. At

7476-594: The study of a problem. For example, the map x ↦ f ( x , t ) {\displaystyle x\mapsto f(x,t)} (see above) would be denoted f t {\displaystyle f_{t}} using index notation, if we define the collection of maps f t {\displaystyle f_{t}} by the formula f t ( x ) = f ( x , t ) {\displaystyle f_{t}(x)=f(x,t)} for all x , t ∈ X {\displaystyle x,t\in X} . In

7565-443: The symbol ↦ {\displaystyle \mapsto } (read ' maps to ') is used to specify where a particular element x in the domain is mapped to by f . This allows the definition of a function without naming. For example, the square function is the function x ↦ x 2 . {\displaystyle x\mapsto x^{2}.} The domain and codomain are not always explicitly given when

7654-447: The symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin( x ) . Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case,

7743-431: The thought experiment with a taut string mentioned above. Notice that you can also launch waves on the string by moving your hand to the right and left instead of up and down. This is an important point. There are two independent (orthogonal) directions that the waves can move. (This is true for any two directions at right angles, up and down and right and left are chosen for clarity.) Any waves launched by moving your hand in

7832-691: The value of f at x = 4 is denoted by f (4) . Commonly, a specific function is defined by means of an expression depending on x , such as f ( x ) = x 2 + 1 ; {\displaystyle f(x)=x^{2}+1;} in this case, some computation, called function evaluation , may be needed for deducing the value of the function at a particular value; for example, if f ( x ) = x 2 + 1 , {\displaystyle f(x)=x^{2}+1,} then f ( 4 ) = 4 2 + 1 = 17. {\displaystyle f(4)=4^{2}+1=17.} Given its domain and its codomain,

7921-504: The word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang , use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions. In the theory of dynamical systems , a map denotes an evolution function used to create discrete dynamical systems . See also Poincaré map . Whichever definition of map

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