Register to reply

Spatial and temporal periods and periodic functions

by Jhenrique
Tags: functions, periodic, periods, spatial, temporal
Share this thread:
Jhenrique
#1
Apr19-14, 08:18 PM
P: 686
A periodic function is one that ##f(\theta) = f(\theta + nT)##, by definition. However, the argument ##\theta## can be function of space and time ( ##\theta(x, t)## ), so exist 2 lines of development, one spatial and other temporal: $$f(\theta) = f(kx + \varphi) = f(2 \pi \xi x + \varphi) = f\left(\frac{2 \pi x}{\lambda} + \varphi \right)$$ $$f(\theta) = f(\omega t + \varphi) = f(2 \pi \nu t + \varphi) = f\left(\frac{2 \pi t}{T} + \varphi \right)$$ or the both together: $$f(\theta) = f(kx + \omega t + \varphi) = f(2 \pi \xi x + 2 \pi \nu t + \varphi) = f\left(\frac{2 \pi x}{\lambda} + \frac{2 \pi t}{T} + \varphi \right)$$ so, becomes obvius that ##\lambda## is the analogus of ##T##, thus the correct wound't be say that a periodic function is one that ##f(\theta) = f(\theta + nT + m\lambda)## ?
Phys.Org News Partner Mathematics news on Phys.org
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
Iranian is first woman to win 'Nobel Prize of maths' (Update)
gopher_p
#2
Apr19-14, 09:38 PM
P: 455
Quote Quote by Jhenrique View Post
A periodic function is one that ##f(\theta) = f(\theta + nT)##, by definition. However, the argument ##\theta## can be function of space and time ( ##\theta(x, t)## ), so exist 2 lines of development, one spatial and other temporal: $$f(\theta) = f(kx + \varphi) = f(2 \pi \xi x + \varphi) = f\left(\frac{2 \pi x}{\lambda} + \varphi \right)$$ $$f(\theta) = f(\omega t + \varphi) = f(2 \pi \nu t + \varphi) = f\left(\frac{2 \pi t}{T} + \varphi \right)$$ or the both together: $$f(\theta) = f(kx + \omega t + \varphi) = f(2 \pi \xi x + 2 \pi \nu t + \varphi) = f\left(\frac{2 \pi x}{\lambda} + \frac{2 \pi t}{T} + \varphi \right)$$ so, becomes obvius that ##\lambda## is the analogus of ##T##, thus the correct wound't be say that a periodic function is one that ##f(\theta) = f(\theta + nT + m\lambda)## ?
If you want to generalize the notion of a periodic function of a single variable to a multivariable situation, the most natural way to do that is to say that ##f:\mathbb{R}^n\rightarrow \mathbb{R}^m## is periodic with period ##\vec{T}\in\mathbb{R}^n## iff ##f(\vec{x})=f(\vec{x}+n\vec{T})## for all ##\vec{x}\in\mathbb{R}^n## and ##n\in\mathbb{Z}##; i.e. the period is a vector/##n##-tuple rather than a scalar. Unlike the single variable case, the mutivariate case can have multiple linearly independent periods, although we wouldn't necessitate that to be true. There are some fun theorems that can be proven regarding the number of ##\mathbb{Q}##-linearly independent periods that a continuous function can have.
Jhenrique
#3
Apr20-14, 03:24 PM
P: 686
happens that when I ploted in the geogebra the wave ##f(\theta(x,t)) = \cos(kx - wt)## happend that ##f(\theta) = f(\theta + m \lambda)## but ##f(\theta) \neq f(\theta + n T)## Why???

gopher_p
#4
Apr20-14, 08:33 PM
P: 455
Spatial and temporal periods and periodic functions

Quote Quote by Jhenrique View Post
happens that when I ploted in the geogebra the wave ##f(\theta(x,t)) = \cos(kx - wt)## happend that ##f(\theta) = f(\theta + m \lambda)## but ##f(\theta) \neq f(\theta + n T)## Why???
That's weird. When I plot ##f(\Phi(\sigma,\tau)) = \cos(\alpha\sigma - \beta\tau)## in GnuPlot, I get both ##f(\Phi)=f(\Phi+m\nu)## and ##f(\Phi)=f(\Phi+n\mu)##.


Register to reply

Related Discussions
What is Spatial and Temporal Coherence. Classical Physics 4
Temporal and Spatial Coherence Classical Physics 1
Can there be other spatial dimensions?Can there be other temporal General Physics 2
Spatial and temporal Atomic, Solid State, Comp. Physics 1
Spatial and temporal interrelationships General Discussion 2