Swing period

Representation of a senoidal movement in which the oscillation period is increasing.

In physics, the period of an oscillation or wave (T) is the time elapsed between two equivalent points of the wave. The concept appears both in mathematics and in physics and other areas of knowledge.

Definition

A simple pendulum executes a periodic movement oscillation period is given approximately T≈ ≈ 2π π l l g{displaystyle Tapprox 2pi {sqrt {ell over g}}}} When the oscillations don't get far from the vertical.

It is the minimum lapse that separates two instants in which the system is in exactly the same state: same positions, same speeds, same amplitudes. Thus the period of oscillation of a wave is the time it takes to complete a wavelength. In brief terms, it is the time it takes for a cycle of the wave to start over. For example, in a wave, the period is the time between two successive crests or troughs. The period (T) is inverse to the frequency (f):

T=1frequency=2π π ω ω {displaystyle T={frac {1}{mbox{frequency}}}}{frac {2pi }{omega }}}}

Since period is always the inverse of frequency, wavelength is also related to period, using the propagation velocity formula. In this case the speed of propagation will be the quotient between the wavelength and the period.

In physics, a periodic motion is always a bounded motion, that is, it is confined to a finite region of space from which the particles never leave.

a particle by the action of a conservative force      U ( x )    {displaystyle scriptstyle U(x)}  is the potential associated with the conservation force, for energies slightly higher than a minimum of energy E_{0}}" xmlns="http://www.w3.org/1998/Math/MathML">     E ▪  E  0      {displaystyle scriptstyle E/2002/E_{0}} E_{0}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2ecf19aa09ac9a519939efe8fa2386d773cad1" style="vertical-align: -0.671ex; width:4.579ex; height:2.009ex;"/> the particle will make a swing motion around the balance position given by the local minimum of energy. The oscillation period depends on the energy and is given by the expression:

TE=2m∫ ∫ x1(E)x2(E)dxE− − U(x){displaystyle T_{E}={sqrt {2m}}int _{x_{1}(E)}{x_{2}(E)}{frac {dx}{sqrt {E-U(x}}}}}}}}}{frac {dx}}}}}}{

Stop. (E− − E0){displaystyle scriptstyle (E-E_{0})} small enough the movement can be represented by a quasi-armonic movement of the form:

{xE(t)=x0+AEwithout (ω ω E(t)t+φ φ 0)=xE(t)=x0+A(t)without (ω ω 0t+φ φ 0)+B(t)# (ω ω 0t+φ φ 0){displaystyle {begin{cases}x_{E}(t)=x_{0}+A_{E}sin(omega _{E}(t)t+varphi _{0})}{x_{E}(t)=x_{0}+A(t)sin(omega _{0}{0}{0cos}{A(t)=AE(1+t4α α (t))B(t)=AE(1+t2β β (t)){displaystyle {begin{cases}A(t)=A_{E}(1+t^{4}alpha (t))B(t)=A_{E}(1+t^{2}beta (t))end{cases}}}}}}

The term ω ω E(t)t+φ φ 0{displaystyle scriptstyle omega _{E}(t)t+varphi _{0} is the phase, being φ φ 0{displaystyle scriptstyle varphi _{0}} is the initial phase, ω ω E(t){displaystyle scriptstyle omega _{E}(t)} is the angular frequency given the approximate relationship:

ω ω E(0)=ω ω 0≈ ≈ 2π π TE,AE=日本語x2(E)− − x1(E)日本語{displaystyle omega _{E}(0)=omega _{0}approx {frac {2pi }{T_{E}}}}},qquad A_{E}=Δx_{2}(E)-x_{1}(E)IND}

Depending on the degree of approximation of how close the energy is to the minimum, for energies slightly above the minimum the motion is very close to the harmonic motion given by:

xE(t)≈ ≈ x0+AEwithout (ω ω 0t+φ φ 0)=x0+AEwithout (2π π tTE+φ φ 0){displaystyle x_{E}(t)approx x_{0}+A_{Esin(omega _{0}t+varphi _{0})=x_{0}+A_{E}sin left({frac {2pi t}{T_{E}}}}}}

Mathematical definition

A period of a real function f is a number such that for all t the following is true:

f(t+T)=f(t),Русский Русский t:[chuckles]t,t+T] Df{displaystyle f(t+T)=f(t),qquad forall t:[t,t+T]subset {mathcal {D}}}_{f}}}}

Note that there is generally an infinity of values T which satisfy the previous condition, in fact the set of periods of a function forms an additive subgroup of R{displaystyle mathbb {R} }. For example f(t) = sen t has as set of periods a 2πZthe multiples of 2yuya

• If the subgroup is discreet, it is called the period of f a his less positive element not null. In the previous example, the period of the sinus function is 2π. Other periodic functions, i.e. admitting a period, are the cosine, tangent and function x - E(xWhere E(x) is the whole part of x.
• If the subgroup is continuous, the period cannot be defined. For example, the constant function g(t) = k admits all real as a period, but none receives the name of period of g. A more esoteric example: The characteristic function χ χ Q{displaystyle chi _{mathbf {Q}}} of Q{displaystyle mathbf {Q} }the whole of rationals is as follows: x It's rational, then. χ χ Q(x)=1{displaystyle chi _{mathbf {Q}(x)=1}And if x It's not rational. χ χ Q(x)=0{displaystyle chi _{mathbf {Q}(x)=0}. The period group χ χ Q{displaystyle chi _{mathbf {Q}}} That's it. Q{displaystyle mathbf {Q} } that has no less positive element not null; therefore there is no the period of this function.

A sum of periodic functions is not necessarily periodic, as seen in the following figure with the function cos t + cos(√2·t):

To be so, the quotient of the periods must be rational, when this last condition is not fulfilled, the resulting function is said to be quasiperiodic.