Function describing a moving waveform

[fog37]

Hello Forum,

• I am first wondering about the possibility to factor a function $f(x,t)$ into a product of two functions, i.e. $g(x) p(t)$. Is there any general rule that tells us if this decomposition is possible based on the characteristics of the function $f(x,t)$?

• If a function $f(x,t)$ is to represent a traveling wave, the same waveform $f(x,t_0)$ at time $t_0$ must be translated along the x-axis at later times $t$: $$f(x-vt)$$

The function $f(x,t)$ and $f(x-vt)$ are the same waveforms, just translated in space by a factor $vt$.

• A standing wave is a wave that does not move or travel. For real-valued functions, the function describing a standing wave ca be factored: $$f(x,t)=g(x)p(t)$$ This means that the spatial function $g(x)$ is not translated but just modulated by the temporal function $p(t)$. However, if the function $g(t)$ is not periodic, I think the product $f(x,t)=g(x)p(t)$ does not represent a standing wave but a traveling wave! Is that correct? could anyone provide further insight into this?

Thanks!

 

[fresh_42]

Why don't you simply take a standing wave and then make position time dependent instead of a translation?

 

[fog37]

Sorry fresh_42, I am not sure what you exactly mean. So, let's say we have function $$f(x,t)= [sin(x)] [e^{3t}]$$

where $g(x)=sin(x)$ and $p(t)=e^{3t}$. What do you exactly suggest doing at this point?

 

[fresh_42]

Sorry fresh_42, I am not sure what you exactly mean. So, let's say we have function $$f(x,t)= [sin(x)] [e^{3t}]$$

where $g(x)=sin(x)$ and $p(t)=e^{3t}$. What do you exactly suggest doing at this point?

This is a wave with an increasing amplitude, not a moving wave. I thought, a wave is $f(x)=A(x)\cdot \sin(\omega x)$ if you want to have a variable amplitude. Now just make the position time dependent, $x=x(t)$, e.g. $x=v \cdot t$ which in this case with a constant velocity is the same as considering the variable as time flow.

 

[fog37]

Actually, the function I gave seems to not be a moving wave: the spatial structure is a sine wave and the amplitude at each point, as you mention, increases exponentially with time. I don't see any oscillatory behavior or translational behavior.

A function like $f(x,t)= sin(x) cos(t)$ is instead a standing wave: the wave does not move but the amplitude at each point oscillates harmonically in time since $cos(t)$ is periodic.

I am still wondering if, given an arbitrary spatial function $g(x)$, we can multiply it by a temporal function $p(t)$ and obtain a traveling wave $f(x,t)=g(x) p(t)$...