Spatial and temporal periods and periodic functions

In summary, a periodic function is one that satisfies the condition ##f(\theta) = f(\theta + nT)##, but this can also be extended to multivariable functions where the period is a vector or ##n##-tuple. Multiple linearly independent periods are possible, but not always necessary. In some cases, such as when plotting ##f(\theta(x,t)) = \cos(kx - wt)##, we may see that ##f(\theta) = f(\theta + m \lambda)## but not ##f(\theta) = f(\theta + n T)##, which may be due to the specific values chosen for the parameters.
  • #1
Jhenrique
685
4
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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  • #2
Jhenrique said:
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.
 
  • #3
happens that when I ploted in the geogebra the wave ##f(\theta(x,t)) = \cos(kx - wt)## happened that ##f(\theta) = f(\theta + m \lambda)## but ##f(\theta) \neq f(\theta + n T)## Why?
 
  • #4
Jhenrique said:
happens that when I ploted in the geogebra the wave ##f(\theta(x,t)) = \cos(kx - wt)## happened 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)##.
 
  • #5


Yes, you are correct. When discussing periodic functions, it is important to consider both spatial and temporal periods. In the first two equations, you have shown that the spatial period (represented by ##\lambda##) and the temporal period (represented by ##T##) are analogous. This means that a periodic function can be written as ##f(\theta) = f(\theta + nT + m\lambda)##, where ##n## and ##m## are integers. This represents a function that repeats itself both in space and in time.

In the third equation, you have combined both spatial and temporal periods into one equation. This is also a valid representation of a periodic function. However, it may not always be necessary to include both periods in one equation. It depends on the specific context and application of the function.

Overall, your understanding of periodic functions and their representation in terms of spatial and temporal periods is correct. This is an important concept in many fields of science, including physics, mathematics, and engineering.
 

1. What is the difference between spatial and temporal periods?

The spatial period refers to the distance or length of one complete cycle of a periodic function in physical space. This is often represented by the wavelength of a wave. On the other hand, the temporal period refers to the time it takes for one complete cycle of a periodic function to occur. This is often represented by the period of a wave.

2. How are spatial and temporal periods related?

Spatial and temporal periods are related through the speed of the wave. The spatial period is equal to the wavelength of the wave, while the temporal period is equal to the wavelength divided by the speed of the wave.

3. What is a periodic function?

A periodic function is a mathematical function that repeats itself at regular intervals. This means that the function has a specific pattern that repeats over and over again. Examples of periodic functions include sine and cosine waves.

4. How are periodic functions used in science?

Periodic functions are used in various scientific fields such as physics, chemistry, and biology to describe and analyze natural phenomena. For example, periodic functions are used to model the movement of waves, the oscillation of a pendulum, and the growth of biological populations.

5. Can periodic functions have different periods?

Yes, periodic functions can have different periods. The period of a periodic function can be adjusted by changing the frequency or amplitude of the function. This means that the pattern of the function will repeat at a faster or slower rate, resulting in a different period.

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