What Are the Different Dimensions of Sound and How Can They Be Distinguished?

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Sound can be understood across various dimensions, with each dimension representing different characteristics and behaviors of sound waves. In one-dimensional sound, waves propagate along a single line, while two-dimensional sound involves waves spreading out in a plane, allowing for more complex interactions. Three-dimensional sound encompasses spatial distribution, enabling the perception of direction and distance. Distinguishing these dimensions involves examining how sound waves travel, interact, and are perceived in different environments. Understanding these distinctions enhances the comprehension of sound's nature and its applications in various fields.
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What is the nature of sound in each dimension e.d 1D, 2D, etc and how can one be differentiated from the other. I do not seek formulas just a straight forward explanation.
 
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Thanks jedishrfu, I did stumble upon that article but what I require is someone to explain that to me. :)
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...

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