I Wave conventions: E(x,t) and E(t), E(x) confusion

[betelgeuse91]

I understand that waves are function of space and time in nature, so E(x,t) will be fundamental description of a wave. I notice that often people denote a wave as E(t) for instance, an envelop function of a pulse. For this case, E is an oscillation at a fixed spatial point x? So that the point x moves up and down as the wave passes through it in time?
And for E(x) this is a snap shot picture of the wave at some time t? This is easier to see (although i don't know if I understand it correctly)
Well then can I treat E(x) and E(t) as like... same quantity in some sense?
Like for example, when I read a Gaussian envelope E(t), then I image the pulse to be Gaussian in space at some point in time...

Thanks for help!

 

[Nugatory]

So that the point x moves up and down as the wave passes through it in time?

No, when we write $E(t)$ we're looking at the value of $E$ over time at some fixed point $x$. Water waves (with $E$ being the depth of the water) are an example: we can describe the wave with the function $E(x,t)$ which tells us what depth we'd find if we drop a measuring buoy into the water at point $x$ and time $t$; or we can use the function $E(t)$ which tells us how the depth reported by a measuring buoy tethered at some fixed location will vary over time.

And for E(x) this is a snap shot picture of the wave at some time t? This is easier to see (although i don't know if I understand it correctly)

That is correct. Just as $E(t)$ tells us how $E$ varies over time at some fixed point, $E(x)$ tells us how $E$ varies with position at some fixed time.

 

[betelgeuse91]

No, when we write $E(t)$ we're looking at the value of $E$ over time at some fixed point $x$. Water waves (with $E$ being the depth of the water) are an example: we can describe the wave with the function $E(x,t)$ which tells us what depth we'd find if we drop a measuring buoy into the water at point $x$ and time $t$; or we can use the function $E(t)$ which tells us how the depth reported by a measuring buoy tethered at some fixed location will vary over time.
That is correct. Just as $E(t)$ tells us how $E$ varies over time at some fixed point, $E(x)$ tells us how $E$ varies with position at some fixed time.

Ahh... right. Thank you for clarification. In literature, people confusingly use E(x) and E(t) and take derivatives with respect x and t which drives me crazy...