Function of a real variable

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Function_of_a_real_variable 

 

Function
x ↦ f (x)
Examples of domains and codomains
${\displaystyle X}$ → ${\displaystyle \mathbb {B} }$, ${\displaystyle \mathbb {B} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {B} ^{n}}$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {Z} }$, ${\displaystyle \mathbb {Z} }$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {R} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {R} ^{n}}$ → ${\displaystyle X}$ ${\displaystyle X}$ → ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {C} }$ → ${\displaystyle X}$, ${\displaystyle \mathbb {C} ^{n}}$ → ${\displaystyle X}$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Partial Multivalued Implicit space

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers ${\displaystyle \mathbb {R} }$, or a subset of ${\displaystyle \mathbb {R} }$ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of ${\displaystyle \mathbb {R} }$-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an ${\displaystyle \mathbb {R} }$-algebra, such as the complex numbers or the quaternions. The structure ${\displaystyle \mathbb {R} }$-vector space of the codomain induces a structure of ${\displaystyle \mathbb {R} }$-vector space on the functions. If the codomain has a structure of ${\displaystyle \mathbb {R} }$-algebra, the same is true for the functions.

The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve.

When the codomain of a function of a real variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.

Real function

The graph of a real function

A real function is a function from a subset of ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} ,}$ where ${\displaystyle \mathbb {R} }$ denotes as usual the set of real numbers. That is, the domain of a real function is a subset ${\displaystyle \mathbb {R} }$, and its codomain is ${\displaystyle \mathbb {R} .}$ It is generally assumed that the domain contains an interval of positive length.

Basic examples

For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous and differentiable at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:

• All polynomial functions, including constant functions and linear functions
• Sine and cosine functions
• Exponential function

Some functions are defined everywhere, but not continuous at some points. For example

• The Heaviside step function is defined everywhere, but not continuous at zero.

Some functions are defined and continuous everywhere, but not everywhere differentiable. For example

• The absolute value is defined and continuous everywhere, and is differentiable everywhere, except for zero.
• The cubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero.

Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:

• A rational function is a quotient of two polynomial functions, and is not defined at the zeros of the denominator.
• The tangent function is not defined for ${\displaystyle {\frac {\pi }{2}}+k\pi ,}$ where k is any integer.
• The logarithm function is defined only for positive values of the variable.

Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of:

• The square root is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).

General definition

A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.

Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subset X of ℝ, the domain of the function, which is always supposed to contain an interval of positive length. In other words, a real-valued function of a real variable is a function

${\displaystyle f:X\to \mathbb {R} }$

such that its domain X is a subset of ℝ that contains an interval of positive length.

A simple example of a function in one variable could be:

${\displaystyle f:X\to \mathbb {R} }$

${\displaystyle X=\{x\in \mathbb {R} \,:\,x\geq 0\}}$

${\displaystyle f(x)={\sqrt {x}}}$

which is the square root of x.

Image

Main article: Image (mathematics)

The image of a function ${\displaystyle f(x)}$ is the set of all values of f when the variable x runs in the whole domain of f. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.

The preimage of a given real number y is the set of the solutions of the equation y = f(x).

Domain

The domain of a function of several real variables is a subset of ℝ that is sometimes explicitly defined. In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted f_|Y. In practice, it is often not harmful to identify f and f_|Y, and to omit the subscript _|Y.

Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation. This means that it is not worthy to explicitly define the domain of a function of a real variable.

Algebraic structure

The arithmetic operations may be applied to the functions in the following way:

• For every real number r, the constant function ${\displaystyle (x)\mapsto r}$, is everywhere defined.
• For every real number r and every function f, the function ${\displaystyle rf:(x)\mapsto rf(x)}$ has the same domain as f (or is everywhere defined if r = 0).
• If f and g are two functions of respective domains X and Y such that X∩Y contains an open subset of ℝ, then ${\displaystyle f+g:(x)\mapsto f(x)+g(x)}$ and ${\displaystyle f\,g:(x)\mapsto f(x)\,g(x)}$ are functions that have a domain containing X∩Y.

It follows that the functions of n variables that are everywhere defined and the functions of n variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals (ℝ-algebras).

One may similarly define ${\displaystyle 1/f:(x)\mapsto 1/f(x),}$ which is a function only if the set of the points (x) in the domain of f such that f(x) ≠ 0 contains an open subset of ℝ. This constraint implies that the above two algebras are not fields.

Continuity and limit

Limit of a real function of a real variable.

Until the second part of 19th century, only continuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.

For defining the continuity, it is useful to consider the distance function of ℝ, which is an everywhere defined function of 2 real variables: ${\displaystyle d(x,y)=|x-y|}$

A function f is continuous at a point ${\displaystyle a}$ which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that ${\displaystyle |f(x)-f(a)|<\varepsilon }$ for all ${\displaystyle x}$ such that ${\displaystyle d(x,a)<\varphi .}$ In other words, φ may be chosen small enough for having the image by f of the interval of radius φ centered at ${\displaystyle a}$ contained in the interval of length 2ε centered at ${\displaystyle f(a).}$ A function is continuous if it is continuous at every point of its domain.

The limit of a real-valued function of a real variable is as follows.^[1] Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted

${\displaystyle L=\lim _{x\to a}f(x),}$

if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that

${\displaystyle |f(x)-L|<\varepsilon }$

for all x in the domain such that

${\displaystyle d(x,a)<\delta .}$

If the limit exists, it is unique. If a is in the interior of the domain, the limit exists if and only if the function is continuous at a. In this case, we have

${\displaystyle f(a)=\lim _{x\to a}f(x).}$

When a is in the boundary of the domain of f, and if f has a limit at a, the latter formula allows to "extend by continuity" the domain of f to a.

Calculus

One can collect a number of functions each of a real variable, say

${\displaystyle y_{1}=f_{1}(x)\,,\quad y_{2}=f_{2}(x)\,,\ldots ,y_{n}=f_{n}(x)}$

into a vector parametrized by x:

${\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})=[f_{1}(x),f_{2}(x),\ldots ,f_{n}(x)]}$

The derivative of the vector y is the vector derivatives of f_i(x) for i = 1, 2, ..., n:

${\displaystyle {\frac {d\mathbf {y} }{dx}}=\left({\frac {dy_{1}}{dx}},{\frac {dy_{2}}{dx}},\ldots ,{\frac {dy_{n}}{dx}}\right)}$

One can also perform line integrals along a space curve parametrized by x, with position vector r = r(x), by integrating with respect to the variable x:

${\displaystyle \int _{a}^{b}\mathbf {y} (x)\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {y} (x)\cdot {\frac {d\mathbf {r} (x)}{dx}}dx}$

where · is the dot product, and x = a and x = b are the start and endpoints of the curve.

Theorems

With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus integration by parts, and Taylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign.

Implicit functions

A real-valued implicit function of a real variable is not written in the form "y = f(x)". Instead, the mapping is from the space ℝ² to the zero element in ℝ (just the ordinary zero 0):

${\displaystyle \phi :\mathbb {R} ^{2}\to \{0\}}$

and

${\displaystyle \phi (x,y)=0}$

is an equation in the variables. Implicit functions are a more general way to represent functions, since if:

${\displaystyle y=f(x)}$

then we can always define:

${\displaystyle \phi (x,y)=y-f(x)=0}$

but the converse is not always possible, i.e. not all implicit functions have the form of this equation.

One-dimensional space curves in ℝⁿ

Space curve in 3d. The position vector r is parametrized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.

Formulation

Given the functions r₁ = r₁(t), r₂ = r₂(t), ..., r_n = r_n(t) all of a common variable t, so that:

${\displaystyle {\begin{aligned}r_{1}:\mathbb {R} \rightarrow \mathbb {R} &\quad r_{2}:\mathbb {R} \rightarrow \mathbb {R} &\cdots &\quad r_{n}:\mathbb {R} \rightarrow \mathbb {R} \\r_{1}=r_{1}(t)&\quad r_{2}=r_{2}(t)&\cdots &\quad r_{n}=r_{n}(t)\\\end{aligned}}}$

or taken together:

${\displaystyle \mathbf {r} :\mathbb {R} \rightarrow \mathbb {R} ^{n}\,,\quad \mathbf {r} =\mathbf {r} (t)}$

then the parametrized n-tuple,

${\displaystyle \mathbf {r} (t)=[r_{1}(t),r_{2}(t),\ldots ,r_{n}(t)]}$

describes a one-dimensional space curve.

Tangent line to curve

At a point r(t = c) = a = (a₁, a₂, ..., a_n) for some constant t = c, the equations of the one-dimensional tangent line to the curve at that point are given in terms of the ordinary derivatives of r₁(t), r₂(t), ..., r_n(t), and r with respect to t:

${\displaystyle {\frac {r_{1}(t)-a_{1}}{dr_{1}(t)/dt}}={\frac {r_{2}(t)-a_{2}}{dr_{2}(t)/dt}}=\cdots ={\frac {r_{n}(t)-a_{n}}{dr_{n}(t)/dt}}}$

Normal plane to curve

The equation of the n-dimensional hyperplane normal to the tangent line at r = a is:

${\displaystyle (p_{1}-a_{1}){\frac {dr_{1}(t)}{dt}}+(p_{2}-a_{2}){\frac {dr_{2}(t)}{dt}}+\cdots +(p_{n}-a_{n}){\frac {dr_{n}(t)}{dt}}=0}$

or in terms of the dot product:

${\displaystyle (\mathbf {p} -\mathbf {a} )\cdot {\frac {d\mathbf {r} (t)}{dt}}=0}$

where p = (p₁, p₂, ..., p_n) are points in the plane, not on the space curve.

Relation to kinematics

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

The physical and geometric interpretation of dr(t)/dt is the "velocity" of a point-like particle moving along the path r(t), treating r as the spatial position vector coordinates parametrized by time t, and is a vector tangent to the space curve for all t in the instantaneous direction of motion. At t = c, the space curve has a tangent vector dr(t)/dt|_{t = c}, and the hyperplane normal to the space curve at t = c is also normal to the tangent at t = c. Any vector in this plane (p − a) must be normal to dr(t)/dt|_{t = c}.

Similarly, d²r(t)/dt² is the "acceleration" of the particle, and is a vector normal to the curve directed along the radius of curvature.

Matrix valued functions

A matrix can also be a function of a single variable. For example, the rotation matrix in 2d:

${\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}$

is a matrix valued function of rotation angle of about the origin. Similarly, in special relativity, the Lorentz transformation matrix for a pure boost (without rotations):

${\displaystyle \Lambda (\beta )={\begin{bmatrix}{\frac {1}{\sqrt {1-\beta ^{2}}}}&-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&0&0\\-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}}$

is a function of the boost parameter β = v/c, in which v is the relative velocity between the frames of reference (a continuous variable), and c is the speed of light, a constant.

Banach and Hilbert spaces and quantum mechanics

Generalizing the previous section, the output of a function of a real variable can also lie in a Banach space or a Hilbert space. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of a ket or an operator. This occurs, for instance, in the general time-dependent Schrödinger equation:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$

where one takes the derivative of a wave function, which can be an element of several different Hilbert spaces.

Complex-valued function of a real variable

A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values.

If f(x) is such a complex valued function, it may be decomposed as

f(x) = g(x) + ih(x),

where g and h are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.

Cardinality of sets of functions of a real variable

The cardinality of the set of real-valued functions of a real variable, ${\displaystyle \mathbb {R} ^{\mathbb {R} }=\{f:\mathbb {R} \to \mathbb {R} \}}$, is ${\displaystyle \beth _{2}=2^{\mathfrak {c}}}$, which is strictly larger than the cardinality of the continuum (i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic:

$${\displaystyle \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=\mathrm {card} (\mathbb {R} )^{\mathrm {card} (\mathbb {R} )}={\mathfrak {c}}^{\mathfrak {c}}=(2^{\aleph _{0}})^{\mathfrak {c}}=2^{\aleph _{0}\cdot {\mathfrak {c}}}=2^{\mathfrak {c}}.}$$

Furthermore, if ${\displaystyle X}$ is a set such that ${\displaystyle 2\leq \mathrm {card} (X)\leq {\mathfrak {c}}}$, then the cardinality of the set ${\displaystyle X^{\mathbb {R} }=\{f:\mathbb {R} \to X\}}$ is also ${\displaystyle 2^{\mathfrak {c}}}$, since

$${\displaystyle 2^{\mathfrak {c}}=\mathrm {card} (2^{\mathbb {R} })\leq \mathrm {card} (X^{\mathbb {R} })\leq \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=2^{\mathfrak {c}}.}$$

However, the set of continuous functions ${\displaystyle C^{0}(\mathbb {R} )=\{f:\mathbb {R} \to \mathbb {R} :f\ \mathrm {continuous} \}}$ has a strictly smaller cardinality, the cardinality of the continuum, ${\displaystyle {\mathfrak {c}}}$. This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.^[2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:

$${\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\leq \mathrm {card} (\mathbb {R} ^{\mathbb {Q} })=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\cdot \aleph _{0}}=2^{\aleph _{0}}={\mathfrak {c}}.}$$

On the other hand, since there is a clear bijection between ${\displaystyle \mathbb {R} }$ and the set of constant functions ${\displaystyle \{f:\mathbb {R} \to \mathbb {R} :f(x)\equiv x_{0}\}}$, which forms a subset of ${\displaystyle C^{0}(\mathbb {R} )}$, ${\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\geq {\mathfrak {c}}}$ must also hold. Hence, ${\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))={\mathfrak {c}}}$.

Sound

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Sound 

A drum produces sound via a vibrating membrane

In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the reception of such waves and their perception by the brain. Only acoustic waves that have frequencies lying between about 20 Hz and 20 kHz, the audio frequency range, elicit an auditory percept in humans. In air at atmospheric pressure, these represent sound waves with wavelengths of 17 meters (56 ft) to 1.7 centimeters (0.67 in). Sound waves above 20 kHz are known as ultrasound and are not audible to humans. Sound waves below 20 Hz are known as infrasound. Different animal species have varying hearing ranges.

Acoustics

Main article: Acoustics

Acoustics is the interdisciplinary science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound, and infrasound. A scientist who works in the field of acoustics is an acoustician, while someone working in the field of acoustical engineering may be called an acoustical engineer. An audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound.

Applications of acoustics are found in almost all aspects of modern society, subdisciplines include aeroacoustics, audio signal processing, architectural acoustics, bioacoustics, electro-acoustics, environmental noise, musical acoustics, noise control, psychoacoustics, speech, ultrasound, underwater acoustics, and vibration.

Definition

Sound is defined as "(a) Oscillation in pressure, stress, particle displacement, particle velocity, etc., propagated in a medium with internal forces (e.g., elastic or viscous), or the superposition of such propagated oscillation. (b) Auditory sensation evoked by the oscillation described in (a)." Sound can be viewed as a wave motion in air or other elastic media. In this case, sound is a stimulus. Sound can also be viewed as an excitation of the hearing mechanism that results in the perception of sound. In this case, sound is a sensation.

Physics

Sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a sound source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium. As the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, and displacement of the medium vary in time. At an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave. This is intuitively obvious for a solid, and the same is true for liquids and gases (that is, the vibrations of particles in the gas or liquid transport the vibrations, while the average position of the particles over time does not change). During propagation, waves can be reflected, refracted, or attenuated by the medium.

The behavior of sound propagation is generally affected by three things:

• A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium.
• Motion of the medium itself. If the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the wind if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the sound wave will be decreased by the speed of the wind.
• The viscosity of the medium. Medium viscosity determines the rate at which sound is attenuated. For many media, such as air or water, attenuation due to viscosity is negligible.

When sound is moving through a medium that does not have constant physical properties, it may be refracted (either dispersed or focused).

Spherical compression (longitudinal) waves

The mechanical vibrations that can be interpreted as sound can travel through all forms of matter: gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium. Sound cannot travel through a vacuum.

Waves

Sound is transmitted through gases, plasma, and liquids as longitudinal waves, also called compression waves. It requires a medium to propagate. Through solids, however, it can be transmitted as both longitudinal waves and transverse waves. Longitudinal sound waves are waves of alternating pressure deviations from the equilibrium pressure, causing local regions of compression and rarefaction, while transverse waves (in solids) are waves of alternating shear stress at right angle to the direction of propagation.

Sound waves may be viewed using parabolic mirrors and objects that produce sound.

The energy carried by an oscillating sound wave converts back and forth between the potential energy of the extra compression (in case of longitudinal waves) or lateral displacement strain (in case of transverse waves) of the matter, and the kinetic energy of the displacement velocity of particles of the medium.

Longitudinal plane wave

 

Transverse plane wave

 

A 'pressure over time' graph of a 20 ms recording of a clarinet tone demonstrates the two fundamental elements of sound: Pressure and Time.

 

Sounds can be represented as a mixture of their component Sinusoidal waves of different frequencies. The bottom waves have higher frequencies than those above. The horizontal axis represents time.

Although there are many complexities relating to the transmission of sounds, at the point of reception (i.e. the ears), sound is readily dividable into two simple elements: pressure and time. These fundamental elements form the basis of all sound waves. They can be used to describe, in absolute terms, every sound we hear.

In order to understand the sound more fully, a complex wave such as the one shown in a blue background on the right of this text, is usually separated into its component parts, which are a combination of various sound wave frequencies (and noise).

Sound waves are often simplified to a description in terms of sinusoidal plane waves, which are characterized by these generic properties:

• Frequency, or its inverse, wavelength
• Amplitude, sound pressure or Intensity
• Speed of sound
• Direction

Sound that is perceptible by humans has frequencies from about 20 Hz to 20,000 Hz. In air at standard temperature and pressure, the corresponding wavelengths of sound waves range from 17 m (56 ft) to 17 mm (0.67 in). Sometimes speed and direction are combined as a velocity vector; wave number and direction are combined as a wave vector.

Transverse waves, also known as shear waves, have the additional property, polarization, and are not a characteristic of sound waves.

Speed

Main article: Speed of sound

 

U.S. Navy F/A-18 approaching the speed of sound. The white halo is formed by condensed water droplets thought to result from a drop in air pressure around the aircraft (see Prandtl–Glauert singularity).

The speed of sound depends on the medium the waves pass through, and is a fundamental property of the material. The first significant effort towards measurement of the speed of sound was made by Isaac Newton. He believed the speed of sound in a particular substance was equal to the square root of the pressure acting on it divided by its density:

${\displaystyle c={\sqrt {\frac {p}{\rho }}}.}$

This was later proven wrong and the French mathematician Laplace corrected the formula by deducing that the phenomenon of sound travelling is not isothermal, as believed by Newton, but adiabatic. He added another factor to the equation—gamma—and multiplied ${\displaystyle {\sqrt {\gamma }}}$ by ${\displaystyle {\sqrt {p/\rho }}}$, thus coming up with the equation ${\displaystyle c={\sqrt {\gamma \cdot p/\rho }}}$. Since ${\displaystyle K=\gamma \cdot p}$, the final equation came up to be ${\displaystyle c={\sqrt {K/\rho }}}$, which is also known as the Newton–Laplace equation. In this equation, K is the elastic bulk modulus, c is the velocity of sound, and ${\displaystyle \rho }$ is the density. Thus, the speed of sound is proportional to the square root of the ratio of the bulk modulus of the medium to its density.

Those physical properties and the speed of sound change with ambient conditions. For example, the speed of sound in gases depends on temperature. In 20 °C (68 °F) air at sea level, the speed of sound is approximately 343 m/s (1,230 km/h; 767 mph) using the formula v [m/s] = 331 + 0.6 T [°C]. The speed of sound is also slightly sensitive, being subject to a second-order anharmonic effect, to the sound amplitude, which means there are non-linear propagation effects, such as the production of harmonics and mixed tones not present in the original sound (see parametric array). If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations.

In fresh water the speed of sound is approximately 1,482 m/s (5,335 km/h; 3,315 mph). In steel, the speed of sound is about 5,960 m/s (21,460 km/h; 13,330 mph). Sound moves the fastest in solid atomic hydrogen at about 36,000 m/s (129,600 km/h; 80,530 mph).

Sound pressure level

Sound measurements
Characteristic	Symbols
Sound pressure	p, SPL,LPA
Particle velocity	v, SVL
Particle displacement	δ
Sound intensity	I, SIL
Sound power	P, SWL, LWA
Sound energy	W
Sound energy density	w
Sound exposure	E, SEL
Acoustic impedance	Z
Audio frequency	AF
Transmission loss	TL

Sound pressure is the difference, in a given medium, between average local pressure and the pressure in the sound wave. A square of this difference (i.e., a square of the deviation from the equilibrium pressure) is usually averaged over time and/or space, and a square root of this average provides a root mean square (RMS) value. For example, 1 Pa RMS sound pressure (94 dBSPL) in atmospheric air implies that the actual pressure in the sound wave oscillates between (1 atm ${\displaystyle -{\sqrt {2}}}$ Pa) and (1 atm ${\displaystyle +{\sqrt {2}}}$ Pa), that is between 101323.6 and 101326.4 Pa. As the human ear can detect sounds with a wide range of amplitudes, sound pressure is often measured as a level on a logarithmic decibel scale. The sound pressure level (SPL) or L_p is defined as

${\displaystyle L_{\mathrm {p} }=10\,\log _{10}\left({\frac {{p}^{2}}{{p_{\mathrm {ref} }}^{2}}}\right)=20\,\log _{10}\left({\frac {p}{p_{\mathrm {ref} }}}\right){\mbox{ dB}}\,}$

where p is the root-mean-square sound pressure and ${\displaystyle p_{\mathrm {ref} }}$ is a reference sound pressure. Commonly used reference sound pressures, defined in the standard ANSI S1.1-1994, are 20 µPa in air and 1 µPa in water. Without a specified reference sound pressure, a value expressed in decibels cannot represent a sound pressure level.

Since the human ear does not have a flat spectral response, sound pressures are often frequency weighted so that the measured level matches perceived levels more closely. The International Electrotechnical Commission (IEC) has defined several weighting schemes. A-weighting attempts to match the response of the human ear to noise and A-weighted sound pressure levels are labeled dBA. C-weighting is used to measure peak levels.

Perception

Main article: Psychoacoustics

A distinct use of the term sound from its use in physics is that in physiology and psychology, where the term refers to the subject of perception by the brain. The field of psychoacoustics is dedicated to such studies. Webster's 1936 dictionary defined sound as: "1. The sensation of hearing, that which is heard; specif.: a. Psychophysics. Sensation due to stimulation of the auditory nerves and auditory centers of the brain, usually by vibrations transmitted in a material medium, commonly air, affecting the organ of hearing. b. Physics. Vibrational energy which occasions such a sensation. Sound is propagated by progressive longitudinal vibratory disturbances (sound waves)." This means that the correct response to the question: "if a tree falls in the forest with no one to hear it fall, does it make a sound?" is "yes", and "no", dependent on whether being answered using the physical, or the psychophysical definition, respectively.

The physical reception of sound in any hearing organism is limited to a range of frequencies. Humans normally hear sound frequencies between approximately 20 Hz and 20,000 Hz (20 kHz), The upper limit decreases with age. Sometimes sound refers to only those vibrations with frequencies that are within the hearing range for humans or sometimes it relates to a particular animal. Other species have different ranges of hearing. For example, dogs can perceive vibrations higher than 20 kHz.

As a signal perceived by one of the major senses, sound is used by many species for detecting danger, navigation, predation, and communication. Earth's atmosphere, water, and virtually any physical phenomenon, such as fire, rain, wind, surf, or earthquake, produces (and is characterized by) its unique sounds. Many species, such as frogs, birds, marine and terrestrial mammals, have also developed special organs to produce sound. In some species, these produce song and speech. Furthermore, humans have developed culture and technology (such as music, telephone and radio) that allows them to generate, record, transmit, and broadcast sound.

Noise is a term often used to refer to an unwanted sound. In science and engineering, noise is an undesirable component that obscures a wanted signal. However, in sound perception it can often be used to identify the source of a sound and is an important component of timbre perception (see above).

Soundscape is the component of the acoustic environment that can be perceived by humans. The acoustic environment is the combination of all sounds (whether audible to humans or not) within a given area as modified by the environment and understood by people, in context of the surrounding environment.

There are, historically, six experimentally separable ways in which sound waves are analysed. They are: pitch, duration, loudness, timbre, sonic texture and spatial location. Some of these terms have a standardised definition (for instance in the ANSI Acoustical Terminology ANSI/ASA S1.1-2013). More recent approaches have also considered temporal envelope and temporal fine structure as perceptually relevant analyses.

Pitch

Figure 1. Pitch perception

Pitch is perceived as how "low" or "high" a sound is and represents the cyclic, repetitive nature of the vibrations that make up sound. For simple sounds, pitch relates to the frequency of the slowest vibration in the sound (called the fundamental harmonic). In the case of complex sounds, pitch perception can vary. Sometimes individuals identify different pitches for the same sound, based on their personal experience of particular sound patterns. Selection of a particular pitch is determined by pre-conscious examination of vibrations, including their frequencies and the balance between them. Specific attention is given to recognising potential harmonics. Every sound is placed on a pitch continuum from low to high. For example: white noise (random noise spread evenly across all frequencies) sounds higher in pitch than pink noise (random noise spread evenly across octaves) as white noise has more high frequency content. Figure 1 shows an example of pitch recognition. During the listening process, each sound is analysed for a repeating pattern (See Figure 1: orange arrows) and the results forwarded to the auditory cortex as a single pitch of a certain height (octave) and chroma (note name).

Duration

Figure 2. Duration perception

Duration is perceived as how "long" or "short" a sound is and relates to onset and offset signals created by nerve responses to sounds. The duration of a sound usually lasts from the time the sound is first noticed until the sound is identified as having changed or ceased. Sometimes this is not directly related to the physical duration of a sound. For example; in a noisy environment, gapped sounds (sounds that stop and start) can sound as if they are continuous because the offset messages are missed owing to disruptions from noises in the same general bandwidth. This can be of great benefit in understanding distorted messages such as radio signals that suffer from interference, as (owing to this effect) the message is heard as if it was continuous. Figure 2 gives an example of duration identification. When a new sound is noticed (see Figure 2, Green arrows), a sound onset message is sent to the auditory cortex. When the repeating pattern is missed, a sound offset messages is sent.

Loudness

Figure 3. Loudness perception

Loudness is perceived as how "loud" or "soft" a sound is and relates to the totalled number of auditory nerve stimulations over short cyclic time periods, most likely over the duration of theta wave cycles.  This means that at short durations, a very short sound can sound softer than a longer sound even though they are presented at the same intensity level. Past around 200 ms this is no longer the case and the duration of the sound no longer affects the apparent loudness of the sound. Figure 3 gives an impression of how loudness information is summed over a period of about 200 ms before being sent to the auditory cortex. Louder signals create a greater 'push' on the Basilar membrane and thus stimulate more nerves, creating a stronger loudness signal. A more complex signal also creates more nerve firings and so sounds louder (for the same wave amplitude) than a simpler sound, such as a sine wave.

Timbre

Figure 4. Timbre perception

Timbre is perceived as the quality of different sounds (e.g. the thud of a fallen rock, the whir of a drill, the tone of a musical instrument or the quality of a voice) and represents the pre-conscious allocation of a sonic identity to a sound (e.g. “it's an oboe!"). This identity is based on information gained from frequency transients, noisiness, unsteadiness, perceived pitch and the spread and intensity of overtones in the sound over an extended time frame. The way a sound changes over time (see figure 4) provides most of the information for timbre identification. Even though a small section of the wave form from each instrument looks very similar (see the expanded sections indicated by the orange arrows in figure 4), differences in changes over time between the clarinet and the piano are evident in both loudness and harmonic content. Less noticeable are the different noises heard, such as air hisses for the clarinet and hammer strikes for the piano.

Texture

Sonic texture relates to the number of sound sources and the interaction between them. The word texture, in this context, relates to the cognitive separation of auditory objects. In music, texture is often referred to as the difference between unison, polyphony and homophony, but it can also relate (for example) to a busy cafe; a sound which might be referred to as cacophony.

Spatial location

Main article: Sound localization

Spatial location represents the cognitive placement of a sound in an environmental context; including the placement of a sound on both the horizontal and vertical plane, the distance from the sound source and the characteristics of the sonic environment. In a thick texture, it is possible to identify multiple sound sources using a combination of spatial location and timbre identification.

Frequency

See also: Audio frequency

Ultrasound

Approximate frequency ranges corresponding to ultrasound, with rough guide of some applications

Ultrasound is sound waves with frequencies higher than 20,000 Hz. Ultrasound is not different from audible sound in its physical properties it just cannot be heard by humans. Ultrasound devices operate with frequencies from 20 kHz up to several gigahertz.

Medical ultrasound is commonly used for diagnostics and treatment.

Infrasound

See also: Perception of infrasound

Infrasound is sound waves with frequencies lower than 20 Hz. Although sounds of such low frequency are too low for humans to hear, whales, elephants and other animals can detect infrasound and use it to communicate. It can be used to detect volcanic eruptions and is used in some types of music.