Behavior of Waves

[thadman]

Can we qualitatively define the behavior of waves?

With regards to a point charge (electricity), the vector defining amplitude decreases at a rate of 1/(r^2), where r=radius. This can be demonstrated by observing the surface area of a sphere (4*pi*r^2).

Let's assume we have a sphere with a uniform distribution of energy (~100) over its surface at a radius of 1. Units are arbitrary, we are making a relative analysis. Energy at a finite point can be determined by dividing the energy by the surface of the sphere. E=100/(4*Pi*(1^2)), E=100/(4*PI), E=7.9577.

If we double the radius, our surface area increases by a square factor multiplied by a constant. E=100/(4*PI*(2^2)), E=100/(16*PI), E=1.9894.

1.9894/7.9577 = 1/4, 1/(2^2)=1/4

I believe this is the reason the point charge's intensity falls off at a rate defined by 1/(r^2). We observe the same phenomenon in acoustics, I believe. It is because the wave (independent of the type) is expanding in 3 dimensions simultaneously

If we observe a uniformly charged rod at a distance where the length of the rod >> observed distance, the vector defining amplitude decreases at a rate of 1/r. This is because the x and z axis (assuming f(x,y,z)) are symmetric about the y axis and as a result, the wave expands in only 2 dimensions.

Geometry will define the rate of change. Physics will dictate the constants that the equation is defined by.

The behavior of a function (whether it equals a value or approaches a value), is defined by the rate of change. Multiplying the function by a constant does not affect this property of a function. I believe if a function equals a value, it will always a equal a value regardless if it is multiplied by an arbitrary constant. The same applies to functions which approach a value.

Assuming this is true, Geometry will dictate whether or not a function equals or approaches a value.

Are any of these assumptions/conclusions fallacious, inaccurate, or incomplete?