Let Ω be a domain in ℝ³ in which Ω^{τ}=Ω×[0,τ]⊂ℝ³⁺¹ is a spatio-tempral domain upon which a non trivial solution of a sound wave exists.
Let us consider the following cross sections of Ω^{τ}
Σ¹={(x,y,z,t₀):(x,y,z)∈Ω and t₀∈(0,τ)} - a temporal cross section of Ω^{τ}
Σ²={(x,y,z₀,t):(x,y,z₀,t)=(x,y,z₀,t)_{∣t=t₀}∈Ω^{τ} and t∈(0,τ)} - a spatial cross section of Ω^{τ}
On which cross section do sound waves exist to be heard? and therefore can flat landers communicate with voices as we see in cartoons?
This question emanates from an idea of the fact that worldly natural phenomena exist in 3+1 space-time and nature (the world) is 3-D. Thus any natural worldly phenomenon such as sound should exist in a 3-D space evolving in time.
References
[1] A Simple Proof that the World is Three-Dimensional
Tom Morley
SIAM Review, Vol. 27, No. 1. (Mar., 1985), pp. 69-71.
[2] Where are the hidden dimensions of space? https://www.researchgate.net/post/Where_are_the_hidden_dimensions_of_space
I appreciate your answers, insightful comments and discussions ?
Regards,
Dear Mr. Lakew,
the probelm is deeper: Do you use Huygen's principle to describe waves, or do you use the Laplace operator, which will lead to fallacies in many cases? The older Euler operator would work, which transforms the term \frac{\partial^2 f}{\partial x^2} to \frac{d^2 f]{dx^2}, using the Leibniz notation and its properties.
If you like to use Huygen's principle, you should know, that the mathematical discussion of this principle will lead to the fractional Riemann operator, if sound propagation is discussed in a fractal dimension of space, for example caused by partial signal reflections within a rope, etc. If you use the Euler operator instead of the Laplace operator, you will be able to transform the time fractional Riemann operator to the correct fractional space operator, which demonstrates, that both time and space fractional derivatives in an equation seem to be double description of a physical phenomenon.
So far I do not know your kind of notation, but please check carefully, whether you want to discuss properties of a formalism or want to describe real world phenomena.
Since you are discussing a linear differential equation, you should start by Dirac's delta function as initial value problem, make use of Fourier convolution and discuss the boundary problems like Zernicke by discussing reflection laws.
If there are questions or comments, please feel free to contact me.
With kind regards,
Norbert Südland
http://www.Norbert-Suedland.info
Dear Mr. Norbert,
Thanks for your comments. Indeed Huygens principle clearly stipulates the presence or absence of sharp signals of waves in spaces depending on the parity of the dimension of the space. However, from the references I put, my question specifically is in regards to real world phenomena. I would like to know in which cross sections of space time (real world in time) can we hear sound?. Cartoons are good examples for my questions as they are flat landers but we hear them communicate on television. In which surface they might be if they indeed communicate? Can it be in both?
Regards,
Maybe I'm oversimplifying matters, but how can sound travel in flat space?
It is true that sound can be described mathematically as a Fourier series, along a time axis only. But when you say "sound wave exist and heard," the phenomenon requires compression and rarefaction of air, or some other medium. Consequently, mass. You won't get that in a 2D world.
Huygen's principle exists too, but why is it necessary to answer the actual question? I think it only reinforces the point.
Dear Albert,
The question does not simply asks if sound travels in flat surfaces per se, but asks whether it is possible to encounter a sound wave on flat surfaces that are cross section of a particular kind. Temporal cross sections of waves are from existing waves in 3-D, intercepted by a plane that is perpendicular to the time axis, while spatial cross sections are planes that are isomorphic to a plane in 2-D that extends along the time axis and they both are different in existence.
Dear collegues,
thank you for your answers. To my knowledge, Huygens' principle is the best idea to describe waves. In 1992 I found by this the spatial shape of the hologramme of a single point, which is a special Fresnel's zone plate and much more precise. If you are looking for this, you should discuss sin(r)/r for 3-dimensional waves.
Due to Euler, waves can exist in any space, his total derivatives are more realistic than the variants of Laplace and Riesz operators. Of course, Huygens's principle can be used both in time and in space version. We have understood the mathematical description, if we are able to transform from one representation to the other and back again. By this we will be able to overcome with old calculation mistakes of Theoretical Physics. To my knowledge the transform between both representations is not possible by use of Laplace operators in more than 1 dimension. The oldest wave equation is d'Alembert's 1+1-dimensional wave equation of a playing string from 1748. This might have been the first differential equation in 2 coordinates.
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