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Spatial and temporal periods and periodic functions

by Jhenrique
Tags: functions, periodic, periods, spatial, temporal
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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)## ?
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gopher_p
#2
Apr19-14, 09:38 PM
P: 435
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: 435
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)##.


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