Deriving equation of wave motion

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SUMMARY

The discussion centers on the derivation of the wave motion equation, specifically referencing equation (2.9) from Eugene Hecht's "Optics," 5th edition. Participants critique the derivation of the D'Alembert equation, emphasizing that while it is a linear differential equation, it does not adequately represent wave phenomena. The consensus is that a second-order differential equation is necessary to fully describe waves, as it requires two parameters: amplitude and frequency.

PREREQUISITES
  • Understanding of linear differential equations
  • Familiarity with wave phenomena and their characteristics
  • Knowledge of the D'Alembert equation
  • Basic concepts from optics, particularly from Eugene Hecht's "Optics"
NEXT STEPS
  • Study the derivation and applications of the D'Alembert equation
  • Explore the characteristics of wave motion in various media
  • Learn about second-order differential equations and their significance in physics
  • Review the content of Eugene Hecht's "Optics," focusing on wave equations
USEFUL FOR

Students and professionals in physics, particularly those studying wave motion, optics, and differential equations. This discussion is beneficial for anyone seeking to deepen their understanding of wave equations and their derivations.

Pushoam
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The equation below (2.9) is also a linear differential equation.
This equation also describes the wave phenomena.
So, why is this equation not considered as wave equation?
I have taken it from the optics book by Chapter two Eugene Hecht,5th edition ,Pearson.
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391fdd9f-40f6-4648-874e-37d04b73169a
 
Last edited:
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I must say that this is the worst derivation of the D'Alambert equation I've ever seen. It's simple but seems to describe any differentiable function. It is true that "wave" is a really general concept, in fact there are many different kind of waves and many different equation describing them, but it's not like anything is a wave!
Anyway, as the text say on page 21, you need at least a second order differential equation to describe a wave since it has in the simplest case an amplitude and a frequency, so two parameters which requires two conditions.
 

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