I imagine throwing a stone in a pond of water and watching waves emanating out in circles from around the spot where the stone hit the water's surface. I thought that all waves are two-dimensional (flat) with amplitude (height) and frequency (period or whatever). So would such a captured picture of the circular-looking waves be a sideways (from the center, where the stone hit the water, outwards) look along a radius of the circular waves? Would a complete mathematical capture of the waves involve calculus, thereby capturing all radii?

"Looking outward from the center, where the stone hit the water" seems to imply a view that would be the same along any one radius as along every other one, wouldn't it? When you've seen, captured, or calculated the frequency, amplitude, attenuation, etc. along one radius, wouldn't it seem to suffice ? If we're talking about a point source of light, for instance, there is an infinity of radii in three dimensional space. We might know how many foot/candles of light fall on a foot square surface one foot from the source, calculate the surface area of a sphere one foot in radius, and thus arrive at the number of lumens that particular light source puts out.

frankvan: It seems that the two-dimensional (flat) picture of the wave along a radius will surfice until it comes time to bring in the sphere. I did not know about the part of bringing in the sphere.