Superposition of two waves and infinitely many waves?

[hanson]

Hi. I am struggling with wave packets.
I know wave packets are generated when two or more waves of slightly different frequencies are superposed together.
When considering only two or finite number of waves superposed together, the resulted wave shall be still periodic? I mean the "peak" of wave packet will actually repeat itself periodically, right? Just see the superposition of two sine waves of slightly different frequencies.

But are wave packets referred in nonlinear Schrödinger equation correspond to "non-periodic" wave packets? I mean, the wave packet shall have just ONE single peak?

How do we produce that ONE single peak wave packet? Is it realistic in nature?
I am guessing that we will have ONE single peak wave packet as long as we have infinitely many waves superposed together (rather than finite number of waves). Is this the sufficient condition for having "one peak wave packet"?

Let's say I superpose waves of frequencies from 2Hz to 3Hz, there will be infinitely many waves. If the range of frequency is now 2Hz to 2.000001Hz, there will be still infinitely many waves, right? So, are they going to produce "one peak wave packet" anyway?

Please kindly help. THANKS.

 

[Claude Bile]

Hi. I am struggling with wave packets.
I know wave packets are generated when two or more waves of slightly different frequencies are superposed together.
When considering only two or finite number of waves superposed together, the resulted wave shall be still periodic? I mean the "peak" of wave packet will actually repeat itself periodically, right? Just see the superposition of two sine waves of slightly different frequencies.

Yes, if the number of frequency components is finite, then the resultant wave will still be periodic.

But are wave packets referred in nonlinear Schrödinger equation correspond to "non-periodic" wave packets? I mean, the wave packet shall have just ONE single peak?

The term "Wave-packet" usually refers to non-periodic waveforms.

How do we produce that ONE single peak wave packet? Is it realistic in nature? I am guessing that we will have ONE single peak wave packet as long as we have infinitely many waves superposed together (rather than finite number of waves). Is this the sufficient condition for having "one peak wave packet"?

A single peak is produced by using a continuum of frequencies rather than a series of discrete frequencies

Let's say I superpose waves of frequencies from 2Hz to 3Hz, there will be infinitely many waves. If the range of frequency is now 2Hz to 2.000001Hz, there will be still infinitely many waves, right? So, are they going to produce "one peak wave packet" anyway?

Yes, this is correct. The effect of increasing the range of frequencies is to reduce the temporal width of the wave-packet. It is a manifestation of the uncertainty principle because the spectral and temporal widths are inversely dependent on one another, and their product can never go below a certain amount.

Please kindly help. THANKS.

You're welcome.

Claude.

 

[Loren Booda]

The (nonlinear) Gaussian (one packet) wavefunction is given by the equation:

S(x,t)/A=exp(-(x-x₀)²/4a²) exp(ip₀x/h) exp(-iw₀t)

Where S is Psi, h is Planck's constant divided by 2(pi), and w is omega