Deriving equation of wave motion

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The discussion centers on the classification of a linear differential equation presented in Eugene Hecht's optics book, questioning why it is not recognized as a wave equation despite describing wave phenomena. The equation is criticized for its poor derivation of the D'Alembert equation, which is deemed overly simplistic and applicable to any differentiable function. It is emphasized that the concept of "wave" is broad, with various types and corresponding equations. The consensus is that a second-order differential equation is necessary to accurately describe waves, as it accounts for both amplitude and frequency, requiring two conditions. This highlights the importance of proper mathematical representation in wave motion studies.
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
 
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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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