Velocity of 2-dimensional and 3-dimensional waves

[atat1tata]

My book (an old copy of Halliday-Resnick) gives a proof for the fact that the wave velocity is constant in 1-dimensional transversal elastic waves, but it says nothing about other types of waves. Basically it makes a tacit assumption that all waves have constant velocity.
However it proves that the amplitude of a circular wave (a ripple in water) decreases proportionally to $$\frac{1}{r^2}$$. I think that it assumes that the wave velocity is constant.
From another point of view if one uses cowishly the relation $$v^2 = \frac{T}{\mu}$$ one could say that, at least for an elastic circular wave, $$\mu$$ is proportional to $$r$$ and the wave velocity should vary.
As you can see I'm a bit confused and I would like to ask if someone could at least provide me with a proof of why the wave velocity is constant in water waves, 2- and 3-dimensional elastic waves and acoustic waves.

PS: I would be extremely grateful if someone could correct my English where I made mistakes in the language

 

[Born2bwire]

Why would \mu be proportional to r? These are homogeneous wave equations (I assume) meaning that the material properties in regards to the wave characteristics are constant throughout the medium.

 

[atat1tata]

Why would \mu be proportional to r? These are homogeneous wave equations (I assume) meaning that the material properties in regards to the wave characteristics are constant throughout the medium.

If you think of the medium as a giant rope that starts at a points and gradually broadens to cover concentric increasing circles (you can think of a circular sector which constitutes the rope in the limiting case of an angle of 2pi) then the cross-section increases lineary and so does the mass per unit "length" of the rope. I understand this is a bit nonphysical, but I'd like to see a proof for the fact that velocity is constant for 2- and 3-dimensional waves

 

[Born2bwire]

No, that's not how it would work. Like I said, the medium is homogeneous, the inherent characteristics of the medium, like the mass density, would remain constant. Your first problem I think is using mass per unit length, that is a 1 dimensional characteristic. You need to do mass per unit area for two dimensions and volume for three dimensions. A 2D surface is a sheet, which you can deconstruct into an infinite number of ropes that radiate out of a single point, or in any other configuration should you desire I guess but ropes are a bad way of thinking because it restricts the propagation along the ropes. In bulk materials you can get shear and plane waves which would not exist together in your given rope configuration.