Why Does Cylindrical Wave Amplitude Vary Inversely with Radius?

[dibiz116]

does anyone know why, in order to conserve flux, the amplitude of a cylindrical wave varies inversely with its radius?

I know the equation for a cylindrical wave is $$\frac{A}{\rho^{1/2}}e^{i(k\rho\pm\omega t)}$$ , but how does this relate to conserving the flux?

The main reason for my question is that I do not know which flux this is referring to, the question only says "flux" and no type of flux is referred to in the chapter I'm working on.

So .. what type of flux is conserved because of a cylindrical waves' amplitude's inverse proportionality with the square root of the radius?

Thanks so much if anyone knows what I'm talking about.

 

[BigJDubs]

The flux referred to in this case is the energy flux of the cylindrical wave. The inverse proportionality between the amplitude and the radius of the wave is needed to ensure that the energy flux remains constant (i.e. conserved). This can be seen by calculating the total energy flux passing through a cylinder of radius r:E_flux = A^2*k*r*2*pi*r Since the amplitude A is inversely proportional to the square root of the radius, r^(1/2), the energy flux will remain constant as the radius changes.