Wave Packet: What is It & What Does NLS Say?

[hanson]

Hello.
What is actually a wave packet?
I am looking at the derivation of the nonlinear Schrödinger equation in hydrodynamics, which seemingly says that the envelop of a wave packet obeys the NLS.
But, in the first place, why would a wave packet be produced?
Is the wave-number a constant within a wave packet? I mean, does a wave packet has a only One wave number? Actually how many waves are these in a wave packet?

 

[jtbell]

All "real" waves in nature are actually wave packets!

The monochromatic ("pure") waves that we describe with a single wavenumber $k$ and frequency $\omega$ are mathematical idealizations. They can't actually exist in nature because they extend to infinity in both directions.

Real waves that we observe in nature always have a finite length. Nevertheless, we can analyze them mathematically as superpositions of (in general) an infinite number of monochromatic waves, with different wavenumbers and frequencies. Many times we don't have to worry about this because the wave is very long, and one wavenumber and frequency dominate overwhelmingly. In this case it's often a sufficient approximation to analyze it as if it were an infinitely long monochromatic wave.

But there are other situations where we have to deal with the fact that the wave actually contains components with different wavenumbers and frequencies.

 

[hanson]

All "real" waves in nature are actually wave packets!

The monochromatic ("pure") waves that we describe with a single wavenumber $k$ and frequency $\omega$ are mathematical idealizations. They can't actually exist in nature because they extend to infinity in both directions.

Real waves that we observe in nature always have a finite length. Nevertheless, we can analyze them mathematically as superpositions of (in general) an infinite number of monochromatic waves, with different wavenumbers and frequencies. Many times we don't have to worry about this because the wave is very long, and one wavenumber and frequency dominate overwhelmingly. In this case it's often a sufficient approximation to analyze it as if it were an infinitely long monochromatic wave.

But there are other situations where we have to deal with the fact that the wave actually contains components with different wavenumbers and frequencies.

Thanks for your reply.
Is a wave packet necessarily a pulse or something?
I am rather confused by this:
Amplitude = A(x,t) exp (ikx), so A(x,t) is a function that will vary with x and t. So, A(x,t) is the envelope, right? So is this a wave packet? but how come there is just ONE wave number k in this expression?

Please kindly explain

 

[olgranpappy]

Thanks for your reply.
Is a wave packet necessarily a pulse or something?
I am rather confused by this:
Amplitude = A(x,t) exp (ikx), so A(x,t) is a function that will vary with x and t. So, A(x,t) is the envelope, right? So is this a wave packet? but how come there is just ONE wave number k in this expression?

Please kindly explain

If you think about writing A(x,t) as a Fourier transform you will see that it is indeed the sum of many different waves of many different wave-numbers, thus so is "Amplitude."

 

[meopemuk]

Thanks for your reply.
Is a wave packet necessarily a pulse or something?
I am rather confused by this:
Amplitude = A(x,t) exp (ikx), so A(x,t) is a function that will vary with x and t. So, A(x,t) is the envelope, right? So is this a wave packet? but how come there is just ONE wave number k in this expression?

Please kindly explain

Note that any function of x can be expressed in the form A(x) exp (ikx) (here I omit the time dependence, i.e., set t=0, because the time dependence should be derived from the wave equation). So, when we speak about wave packets we usually assume some additional requirements on the form of the amplitude function A(x). For example, it is common to assume that A(x) is smooth, i.e., it doesn't change much on the scale of one period of the oscillating factor exp (ikx). Then, it is easy to see that the dominant frequency in the Fourier decomposition of A(x) exp (ikx) is k.

Eugene.

 

[hanson]

Um...please refer to my new thread. Thanks
https://www.physicsforums.com/showthread.php?t=179569