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

In summary, the conversation discusses the concept of wave packets and how they differ from ideal monochromatic waves. It is explained that real waves in nature always have a finite length and can be analyzed as superpositions of different monochromatic waves. The concept of amplitude is also discussed and how it can be expressed as a Fourier transform. The conversation ends with a request for further explanation on the topic.
  • #1
hanson
319
0
Hello.
What is actually a wave packet?
I am looking at the derivation of the nonlinear schrodinger 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?
 
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  • #2
All "real" waves in nature are actually wave packets!

The monochromatic ("pure") waves that we describe with a single wavenumber [itex]k[/itex] and frequency [itex]\omega[/itex] 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.
 
  • #3
jtbell said:
All "real" waves in nature are actually wave packets!

The monochromatic ("pure") waves that we describe with a single wavenumber [itex]k[/itex] and frequency [itex]\omega[/itex] 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
 
  • #4
hanson said:
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."
 
  • #5
hanson said:
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.
 

1. What is a wave packet?

A wave packet is a localized, transient disturbance that travels through a medium as a wave. It is usually a combination of multiple waves with different frequencies and amplitudes, resulting in a specific shape and size.

2. How is a wave packet different from a single wave?

A single wave has a constant frequency and amplitude, while a wave packet is a combination of multiple waves with varying frequencies and amplitudes. Additionally, a wave packet has a finite size and duration, whereas a single wave extends infinitely in both space and time.

3. What does NLS stand for?

NLS stands for Nonlinear Schrödinger equation. It is a mathematical model that describes the propagation of wave packets in nonlinear media, taking into account the effects of dispersion and nonlinearity.

4. How does NLS explain the behavior of wave packets?

NLS takes into account the nonlinear interactions between the different frequencies and amplitudes within a wave packet, as well as the effects of dispersion. This allows for a more accurate description of the behavior and propagation of wave packets in nonlinear media.

5. What are some practical applications of understanding wave packets and NLS?

Understanding wave packets and NLS is crucial in fields such as optics, acoustics, and quantum mechanics. It allows for the design and analysis of advanced technologies such as laser systems, fiber optics, and quantum computers. Additionally, it is also important in the study of natural phenomena such as ocean waves and seismic waves.

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