In the process of overhauling and expanding the code’s most formal expression, I have come to appreciate just how much space is devoted to abstract reasoning behind the applications, and how far the code may seem removed from reality.
But in reviewing the math for wave-formed contouring recently, I found the connection easier to make with the volume of DIRT needed to realize a landscaping plan. Before showing how useful the formulas are, here’s a brief description of how they were derived:
For half-waves spun about their crests or troughs to shape mounds or round berms and embankments, I used the thin shell integration method on the wave equation Y = M cos X, with Y being wave height, M the maximum slope, X the radius and dx the thickness of the cylinder. I treated a quarter wave section at a time, and spun it from both directions to account for concave rounding circumstances.
After integrating the convex crest quarter wave section, I divided this generalized volume by that of the cylinder bounding it to obtain a proportionality constant. For the concave trough volume, the convex crest volume is subtracted from the same cylindrical volume and divided by same to obtain its proportionality constant.
The same procedure was used for the concave crest and the convex trough, but as these are separated from the pivot, the integrating limits are shifted from π/2 to π and proportioned to the volume of the (thick) cylindrical shell bounding them. I then I found I could multiply the proportionality constants by the volume of a cylindrical shell bounding the wave in a specified situation to obtain a real waveform’s volume.
I am not absolutely certain that this is a valid approach for volumes of waves spun about axes, but it seems reasonable and the relative size of the constants derived are consistent with what one would expect. For the derived constants and formulas, check out the new PDF when it is published (with the other 6) in a few weeks on this website.
What I am certain of is taking the area of the wave cross-section and multiplying it by the length of a straight berm or embankment section to determine its volume. What seems strange is that constants do not depend on slope. However, the more one scrutinizes a range of slopes, the more this makes sense.
Volume is important for two simple related reasons. Obviously, when one lays out a landscaping plan, quantity of dirt required is a must. The second value comes at the end of the embanking procedure which commences with determining and marking the wave’s height and width, with their relationship keyed to the maximum slope chosen.
The next task is to stake out a string to signify the maximum slope which would typically situated half way between both the height and width. The next thing is to drive pointed rods every few feet with flat pieces angled to the maximum slope secured to the rod at the height of the string and extending upward.
Next the dirt is piled in such a way that the crest and trough are horizontal and the steepening curvature of slope from the crest jives with the stakes’ board slope along the string. Upon satisfying these basic conditions, any remaining dirt is placed at the biggest deficits eyeballed. If dirt is consumed before satisfying these conditions, material shift from excess to deficits is in order.
In my mind, this procedure was the most one could do without getting into sophisticated instruments. But then I explained what I was doing to my father (whose livelihood once entailed working with road construction engineers), and he quickly pointed out the identical mirror image nature of the top and bottom wave portions.
So enlightened, I saw that the dirt volume in its entirety could be piled up in a straight slope from top to the outward boundary – first. Dirt could then be sluffed away from the midline and placed directly above the string with full knowledge that the two volumes are equal as should be. Much better way. To perfect the slopes, matching guides can be cut from a sheet of plywood.
This would work well for a symmetric wave; and even though this approach is unnecessary or even impractical in many situations, it still poses the simplest, most elegant way of dealing with the default waveform. All that remains is to plant the ground cover selected and wait for the waveform to unite the abode with the earth.