What is a wave exactly? And why does it move like the way it does instead of following any random like pattern of motion, I mean why it moves in a semicircular pattern instead of semi square pattern?
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Where have you seen a wave move in a semi-circular pattern? Are you asking about a particular situation, such as diffraction? – sammy gerbil Feb 17 '18 at 10:30
2 Answers
Roughly speaking, a wave is any function that travels uniformly along space.
It implies that energy is transferred without net movement of the particles involved.
Transverse waves are easier to see. Imagine a stadium full of people. What is "making the wave"? Everybody stands out and sits down again. Nobody has lost their seat (they have moved but not displaced), and the "perturbation" has travelled along the stadium. That's a wave.
But waves can have any shape as long as they fulfill those definitions above (again, rougly speaking). There exists semicircle waves, semisquare waves, and so on.
Then why do we represent waves always following a "sine function" (harmonic waves). Because of two reasons:
- Because they are the simplest and most beautiful case.
- Because ANY OTHER shape can be formed by summing those sine waves.
(I think this is what you wanted to understand, so I've sacrified some rigour to explain this, assuming you don't know wave math enough, hope I helped).

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Your response appears to correctly address the spatial shape in the direction of propagation of the disturbance. I think, but am not certain, that the OP is asking about the shape of the surfaces of constant phase. E.g., plane wave, spherical wave, etc. (transverse to the direction of propagation). – garyp Feb 17 '18 at 21:13
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Hmm okay I think I understand. Thanks for the comment. Should we explain that too? I don't know if the asker is satisfied. I understood my version because "the typical image" of a wave is an harmonic one. – FGSUZ Feb 17 '18 at 21:19
shaswat Your doubt is about the movement of transverse waves . In transverse waves, energy is transfer in up down movement perpendicular to axis and with moving forward. In medium, any particle carry energy by moving perpendicular to axis . It moves up then down and if you draw a line joining the positions of all particles at same time then you get graph like you got when you join 2 parabolic shapes which are opens in opposite direction. It is simple. You have to imagine it. You can take help of simulation / videos also. Thanks for asking question.