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

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!
 
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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.
 
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....
 

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