Circular Wave Equations: Pebble Dropped in a Pond

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    Circular Pond Wave
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Discussion Overview

The discussion revolves around the equations governing circular waves, specifically in the context of a pebble dropped in a pond. Participants explore mathematical representations, particularly focusing on Bessel functions and their properties, as well as related equations that may describe similar wave phenomena.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the equations for circular waves, specifically referencing the scenario of a pebble dropped in a pond.
  • Another participant suggests that Bessel functions of the First Kind are relevant to the discussion of circular waves.
  • A different participant notes that while Bessel functions are applicable, they observe that the amplitude and wavelength appear to decrease in graphs of these functions.
  • One participant mentions that the wavelength approaches a constant value when examining Bessel functions in different regions, referencing the asymptotic behavior of the governing differential equation.
  • A participant discusses the equation y=sinkx/x^2, suggesting it leads to a second-order differential equation that resembles Bessel functions but is not identical, noting a discrepancy in the first derivative term.
  • Another participant questions the connection between the equation y=sinkx/x^2 and circular water waves, proposing an alternative equation related to Fraunhofer diffraction.
  • One participant asserts that graphing sinx/x^2 results in a damped sine curve with a constant wavelength, attempting to connect this to the behavior of ripples on a pond.
  • A later reply challenges the assumption of constant wavelengths without a proper differential equation setup, mentioning that Bessel functions arise in solutions to drum-membrane problems with appropriate boundary conditions.

Areas of Agreement / Disagreement

Participants express differing views on the relevance and application of Bessel functions and other equations to the topic of circular waves. There is no clear consensus on the connection between the discussed equations and the behavior of circular waves in a pond.

Contextual Notes

Some participants highlight the need for proper boundary conditions and differential equations to accurately describe the wave phenomena, indicating potential limitations in the assumptions made during the discussion.

rlduncan
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What are the equation(s) for circular waves such as pebble dropped in a pond.
 
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OK, I'll bite. Bessel function of the First Kind?

Zz.
 
Circular Wave

Yes, a Bessel function. How about a circular wave in which the wavelength is constant. I have looked at graphs of Bessel functions in a plane and they appear to decrease in amplitude and wavelength.
 
If you look at the Bessel functions carefully you will see that the wavelength approaches a constant value as you go from the near field to the far field regions. You also infer that behavior from the asymptotic behavior of the governing differential equation (wave equation) for the Bessel functions.
 
I am studing the equation y=sinkx/x^2 and find that the second order differential equation found for this equation is Bessel like, but not identical to Bessel functions. The first derivative term, for example, is off by a constant.
 
Last edited:
rlduncan said:
I am studing the equation y=sinkx/x^2 and find that the second order differential equation found for this equation is Bessel like, but not identical to Bessel functions. The first derivative term, for example, is off by a constant.

Are you sure it is not

[tex]y=\frac{sin^2(kx)}{(kx)^2}[/tex]

which is the Fraunhofer diffraction equation? If it is, then I don't see the connection with asking for circular water waves.

Zz.
 
ZapperZ said:
Are you sure it is not

[tex]y=\frac{sin^2(kx)}{(kx)^2}[/tex]

which is the Fraunhofer diffraction equation? If it is, then I don't see the connection with asking for circular water waves.

Zz.

First time I've seen you use Latex!
 
If you graph just sinx/x^2 where k=1 then you get a damped sine curve in which the wavelength is constant. I assume for ripples on a pond the wavelengths are constant and I was trying to make a connection if any.
 
rlduncan said:
If you graph just sinx/x^2 where k=1 then you get a damped sine curve in which the wavelength is constant. I assume for ripples on a pond the wavelengths are constant and I was trying to make a connection if any.

I'm not sure how you can assume that when you haven't set up the diff. equation to solve for such a problem. Note that for a drum-membrane problem, you do have bessel functions as the solution to the diff. equation with the proper boundary conditions.

Zz.
 

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