Is f(x,t)=exp[-i(ax+bt)^2] a harmonic wave?

[reedc15]

Homework Statement

Dear Guys,

Does f(x,t)=exp[-i(ax+bt)^2] qualify as a harmonic waves? Please help!

Manish
Germany

Homework Equations

The Attempt at a Solution

it is of the form g(ax+bt). which is the general form for harmonic wave. but what bothers me is the quadratic exponent. would my equation qualify as the harmonic wave equation? please help

 

[Simon Bridge]

Does f(x,t)=exp[-i(ax+bt)^2] qualify as a harmonic waves? Please help!

Depends - the expression represents a phasor. It expands out into:
$f(x,t)=\cos(ax+bt)^2+i\sin(ax+bt)^2$
... so how would that be related to a "harmonic wave"?

it is of the form g(ax+bt). which is the general form for harmonic wave. but what bothers me is the quadratic exponent. would my equation qualify as the harmonic wave equation? please help

Well let's see...

What does: $f(x,y)=1-(ax+bt)^2$ represent? (What is the shape, and what is it doing?)

 

[reedc15]

f(x,y)=1−(ax+bt)^2 is a parabola in 3d shifted upward by 1 and inverted.

If g(ax+bt) is general form for harmonic wave, my equation should represent a harmonic wave as it has argument ax+bt. is it true?

 

[Simon Bridge]

You described the shape - but what is it doing?
I'm trying to get you to figure out the answer to your question yourself - if you don't, then you'll have to ask someone next time you get stuck on this sort of thing too.

It may also help to consider the difference between a "harmonic" wave and a "wave" ... what is it that makes it "harmonic"? Don't look at the equation - look at what it does.

 

[reedc15]

It is oscillating up and down. The 3d parabola!

Also f(x,t)=exp[-i(ax+bt)^2] will have a certain velocity given by sqrt(b/a). This implies omega = b, which is constant. So the the wave is constantly oscillating.

 

[Simon Bridge]

It can help if you know what the equations are telling you:

It is oscillating up and down. The 3d parabola!

It's only a 2D parabola ... at t=0, it is centered on the origin and $f(x,0)=1-x^2$. For t>0, it is traveling in the +x direction but keeps exactly the same shape.

In general, if y=f(x) is an arbitrary shape at t=0, and it is traveling in the +x direction with a speed v, then at t>0, $y(x,t)=f(x-vt)$.

What is the definition of "harmonic wave"?
Is this a harmonic wave?

You function is a little more complex than that - in fact, it is complex valued.
At x=0, the function is $f(0,t)=\exp -ib^2t^2$ which is a phasor with unit amplitude... what is happening with time?

BTW: I have a feeling that your class is using the term "harmonic wave" differently to me.

 

[reedc15]

Yes, I got your point. But what is your definition of Harmonic Wave? Is it defined as any linear combination of sin(kx+wt) and cos(kx+wt)? Can the argument of these function take powers of 2 and so on.

In my college, all sins and cosines with argument (kx+wt)^2 is considered as harmonic wave. That is why some of the textbooks generalizes the harmonic wave equation as f(kx+wt). What is your definition of harmonic wave function?

 

[Simon Bridge]

what is your definition of Harmonic Wave?

If you google "harmonic wave" you will see how many people view the term to refer to the wave of the fundamental harmonics in a system. General waves can be expressed as a linear sum of harmonic waves. (technically - any function can be) though I'd usually think of harmonic waves as being "made by" harmonic oscillators.

some of the textbooks generalizes the harmonic wave equation as f(kx+wt)

Which manages to cover pulses and solutions to the Schrödinger equation as "harmonic waves" ... kinda makes me wonder what's left. Would this definition include standing waves - the sum of at least two harmonic waves?

Is it defined as any linear combination of sin(kx+wt) and cos(kx+wt)?

That won't work because any function can be described as a linear combination of those and, presumably, some functions are not waves and some waves are not harmonic waves - or: why bother with the specific terms?

Basically, what your school is calling a "harmonic wave" is so general that most people would just call it a "wave". Compare, for eg.
http://en.wikipedia.org/wiki/Wave

 

[reedc15]

f(kx+wt) would of course including standing waves. I understand your point about linear combination of sin and cosine not working as harmonic waves in many cases.

f(x,t)=exp[-i(ax+bt)^2] has some velocity given by sqrt(b/a), with b as omega. this implies it is in certain harmonics for 'b' as omega. I would think, such a wave would also pass for harmonic wave. even f(x,t)=exp[-i(ax+bt)^n], n belongs to integers.

But if I understand you correctly, you don't think exp[-i(ax+bt)^2] to be a harmonic wave. is it?

 

[Simon Bridge]

I would not have thought to call it a "harmonic" wave because it is not a harmonic, nor is it harmonic (to do with harmony or pleasant sounds). I've been trying to find a reference to this use of the term online to no avail ... you wouldn't help me out and supply one? I may just be out of date.

A standing wave would have a function like $y(x,t)=\sin(kx+\omega t)+\sin(kx-\omega t)$ ... how does that fit the general form of $y(x,t)=f(ax+bt)$ i.e. what is a and b in this case? Or, is the definition: "able to be expressed as a sum: $y(x,t)=\sum f_i(a_ix+b_it)$" ? But then - as you've seen, the trick would be finding a function that is not a harmonic wave by that definition[*].

Anyway - this is quite aside from the point: you have to do your work in the context of the course you are actualy in right now.

You will need to come up with a list of properties that will identify a function as a "harmonic wave" - write them down - and then see if the function in question is, in fact, one.

The main wrinkle seems to be that the function $f(t)=e^{it^2}$ is a phasor in the complex plane - so the real and imaginary components of $f(x,t)$ are traveling sinusoids of a form you've already met - so does the fact that one has an imaginary amplitude make a difference as far as the definition is concerned?

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[*] i.e. any f(x) would be harmonic, by that definition, because it is f(ax+bt) with a=1 and b=0.
Presumably not every f(x) is a harmonic wave?
I just think it would help you to pin down the definition of the term a bit more.