Understanding the Wave Function: Time, Distance, and Confusion Explained

[Sobhan]

i don't understand why wave is a function of both time and distance and i think one is enough.can some one explain that to me?

 

[Geofleur]

A function of distance only is static - it just sits there. If you graph it, it looks the same no matter at what time you graph it. But suppose you want a function that looks different at different times. A traveling wave is like that. It's helpful to plot, say, ## \cos(x-2t) ## at times ## t = 0 ##, ## t = 1 ##, etc. You will get a cosine wave that travels toward the right.

Also, remember that you can shift a function ## f ## to the right by ## c ## if you stuff ## x - c ## in as the argument. So ## f(x-c) ## looks just like ## f(x) ##, but shifted over to the right by ## c ##. Similarly, ## \cos(x-2t) ## is the same as ## \cos(x) ## but shifted to the right by ## 2t ##, and the amount of the shift depends on how much time has passed in this case.

 

[jtbell]

You can see a couple of examples of how x and t "interact" in a wave equation in this post:

What exactly does 'x-vt' mean in the wave equation?

(You'll have to draw your own graphs to illustrate it. It's a good exercise.)

 

[ZapperZ]

i don't understand why wave is a function of both time and distance and i think one is enough.can some one explain that to me?

That doesn't make any sense, and you offered zero explanation why you think it is so.

Let's say I can see this wave and I take a snapshot of it at a particular time. If I see "wavy pattern", that is a pattern at a FIXED time (i.e. no time variation), but instead, a wave pattern in SPACE, i.e. as a function of position.

But what if I sit at a particular location, and simply detect this wave, as it I hold an antenna? In this case, I am at a fixed location (i.e. no space variation), but instead, I am detecting the wave pattern IN TIME , i.e. as a function of time!

So now, explain to me why you think it is enough to simply have only one dependent variable here.

Zz.

 

[Windadct]

Well it does not need to be TIME, but it does need to be 2 (or more) variables.

 

[blue_leaf77]

i don't understand why wave is a function of both time and distance and i think one is enough.can some one explain that to me?

You solve time-dependent Schroedinger equation to get eigenfunctions, so, eigenfunctions is a function of space and time. Arbitrary wavefunction may be constructed from those time and space dependent eigenfunctions, so, arbitrary wavefunction must also be a function of space and time. Conclusion is, it is all caused by the Schroedinger equation.

Well it does not need to be TIME, but it does need to be 2 (or more) variables.

These two or more variables (at most 4) must include time.

 

[Windadct]

I am reading this as a much simpler question, a wave is a pattern, and can occur in two (non time) dimensions. y=sin(x) - haha - but did not realize which forum it was. Still - something like electron probability is essentially a static evaluation, regardless of time.

 

[blue_leaf77]

I am reading this as a much simpler question, a wave is a pattern, and can occur in two (non time) dimensions. y=sin(x)

You must be confusing "oscillation" and "wave".

 

[Sobhan]

What I have been told is not to imagine it as a snake moving and consider it as repeating snapshots.i don't understand what can the length of the snapshot.

 

[ZapperZ]

Well it does not need to be TIME, but it does need to be 2 (or more) variables.

This is the known wave equation:

[tex]\

What I have been told is not to imagine it as a snake moving and consider it as repeating snapshots.i don't understand what can the length of the snapshot.

1. Who told you this?

2. "repeating snapshots" is EXACTLY an example of a TIME-DEPENDENT operation, or are you not seeing this?

Zz.

 

[Sobhan]

My question is that the length of this snapshot is 1 or 2 or ...

 

[ZapperZ]

My question is that the length of this snapshot is 1 or 2 or ...

What is this "length of this snapshot is 1 or 2..."? What do you define as "length of a snapshot"? The length of exposure? Or the length BETWEEN snapshots? What is "1 or 2"? Do those have units? Can you spend some time and effort (i.e more than 1 dimension) to provide a clear and more complete explanation?

What about my point that repeated snapshots, regardless of the "length of the snapshot" is an EXPLICIT indication of a time dependent operation? Why have you ignored that?

Zz.

 

[Sobhan]

I know it can be a function of location or time but I can not get it been function of location and time together.
I meant length of the snapshot and 1,2 can be in meters or any other length unit.

 

[ZapperZ]

I know it can be a function of location or time but I can not get it been function of location and time together.

What do you mean that you "cannot get" that it is a function of location and time? You SOLVE the differential equation, and there you have it! What exactly is it that you do not get?

This is the unambiguous definition of the "wave equation":

http://mathworld.wolfram.com/WaveEquation.html

It is not negotiable. When you walk up to a mathematician, an engineer, or a physicist, and you mention the wave equation, this is the FORM that everyone agrees upon! Now, how do you propose to solve this equation using one variable but without the other? There's no "getting it"! It just is!

Is this really an issue of solving a differential equation with multiple variables? Because if it is, then you'd better bail out now, because you'll see differential equation with MORE than just 2 dependent variables. If this is the source of your problem, then you have a problem with the mathematics, not physics.

Zz.