First off, sorry about the size of the graphic… its large, perhaps too large. But I’m lazy and wanted it all in one illustration. In my little part of the world, its all beach break. I can travel a bit and surf reef break, but I simply hate driving around and more often than not, I go with the beach break, or if I’m lucky maybe some outside or slightly offshore bar is going off. So, more often than not after arriving at the beach my thoughts move along the following lines. “I can make that!!” “I can make that???” “Ahhh, what the hell, won’t know if I don’t try.” Principle One. There’s a general rule out there that waves break in a water depth of approximately ¾ of their height. Is this gospel? Hardly, it is really crude and doesn’t account for a lot of factors, but its handy and allows one to get a feel for the way things might be working. (I think Blair Kinsman, in his classic “Wind Waves” gives something similar, and I also think he got it from somebody else. Kinsman’s book is not the best one for breaking waves, I don’t recommend it, but its got a nice Introduction and its likely to be available at a major library.) I don’t want to get into a discussion about height – I’m fine with the current region terminology, however, for the moment I will use the term in the way that it is used in the technical literature; wave height is the vertical distance from trough to crest. For example, a scientist might say that the wave is 6 foot, but your more likely to call as being something less, probably only around 3 foot, or maybe waist to chest. An Hawaiian may not even see it. (By the way, the difference is not a ‘doubling’ as might be suggested by the example.) If you want to have a more accurate (technical, at least) wave height value, view the wave height from the lowest point in the water ahead of the wave. Sort of the way surf photographers do, at least the ones that paddle around with fins and camera. Principle Two. We also know that shoaling waves travel with a speed of the square root of gD, where g is the acceleration due to gravity at the surface of the planet, and D is the water depth. (Ref. Open University Press, Waves, Tides and Shallow-Water Processes, ISBN 0-08-036371-7) Now waves change form as they shoal, but for purposes here lets deal with that height that you would call peak height. So, if H is the height of the wave, the shoaling speed of a wave which is just about to break is the square root of g(3/4H). That is, this is roughly the speed at which the wave is moving towards the beach when its just about to break. Here’s a table, (actually you can’t make tables very easily on Swaylocks, so its in the form of ordered pairs.) (Wave Height H ft,Beachward Speed mph) (3 ft,6 mph), (6 ft,8 mph), (9 ft,10 mph), (12 ft,11.5 mph), (15 ft,13 mph) Conversion Notes. 1 foot per sec = 0.6818 mph g = 32 ft/sec sec (A problem with this simplistic approach is that as the wave shoals it height will increase, but it may not be that bad if H is thought of the height at the peak. I looking for crude numbers here, a feel for the kinds of speeds involved. Nothing all that precise.) Also, not all waves break the same way, some don’t seem to jack very much at all but just start crumbling, or barreling. This has a lot to do with the bottom contours and the wave’s approach. So, please accept my apologies for this crude analysis.) Now the real problem is not really beachward speed, at least where I surf, its down-the-line speed, which is more of a function the way the wave is hitting the bar, see diagram. As you can see, or maybe guess, the more acute the angle of approach, the faster the down the line speed. Summarizing the results in the diagram we have, Beachward Speed, Vb = [g(3/4)H]^(1/2) Along-the-beach Speed, Va = Vb / Tan(A) Down-the-line Speed, Vd = Vb / Sin(A) (See diagram for direction references.) Checking the formulas. For A = 0, Sin(0) = Tan(0) = 0, that is the down-the-line speed and along the beach speed are infinite and we have close out conditions. (That is, its when only me and a few other idiots can be seen in the water.) For very small A, we got some pretty fast waves, in terms of down-the-line speed that is. Here’s a table of sorts for a 6 foot wave. The first is for angle of approach of 11.25, the second 22.5 degrees, third 45 degrees. and the fifth for 67.5 degrees (the angles the big hand of a clock makes at 12:45 1:30, 3:00 and 4:30.) So, for a 6 foot wave, Vb = [32 (3/4) 6]^1/2 (.6818) mph = 8.2 mph ~ 8 mph So the following formula was used to generate the tables, Vd = 8 / Sin(A) mph Va = 8 / Tan(A) mph (A deg, Vb mph, Vd mph, Va mph) (11.25 deg,8 mph,41 mph,40 mph), (22.5 deg,8 mph,21 mph,19 mph), (45 deg,8 mph,11 mph,8 mph), (67.5 deg ,8 mph,9 mph,3 mph) You can see the trend – the lower the angle of approach the faster the down-the-line speed (and the along-the-beach speed.) Can you make a down-the-line speed of 41 mph, perhaps not, especially on a 6 foot wave. A 21 mph, or maybe 11 mph? Perhaps, but you’ve got to get the energy from somewhere, gravity perhaps? Yes, and no. Gravity is only part of the solution, the rest comes from the flow interacting with the surfboard. This can best be understood by doing a similarly crude energy analysis, using only kinetic (the energy inherent in motion) and potential energy (the energy inherent in position in some position sensitive force field like Gravity) considerations. So, it seems I get to continue this nonsense in my next post. Closing Note. A spreadsheet or applet would have been nice at this point; you could plug in some angles and wave heights and get a feel for the speeds involved. Hopefully, soon I will be in a position to put them up on a different site, but I would much rather wait for Swaylocks new improvements to this site (I love the hypertext based Web, but its still best not to break things up, if there’s no real need to.) Anyway, if I made my own site, it would just wind up joining the zillions of unvisited ‘Here’s my 15 minutes of fame’ sites. Here at least if somebody gets really bored they may read this stuff. 
We also know that shoaling waves travel with a speed of the square root of > gD, where g is the acceleration due to gravity at the surface of the > planet, and D is the water depth. (Ref. Open University Press, Waves, > Tides and Shallow-Water Processes, ISBN 0-08-036371-7)>>> Now waves change form as they shoal, but for purposes here lets deal with > that height that you would call peak height.>>> So, if H is the height of the wave, the shoaling speed of a wave which is > just about to break is the square root of g(3/4H). That is, this is > roughly the speed at which the wave is moving towards the beach when its > just about to break. Heres a table, (actually you can’t make tables very > easily on Swaylocks, so its in the form of ordered pairs.)>>> (Wave Height H ft,Beachward Speed mph)>>> (3 ft,6 mph), (6 ft,8 mph), (9 ft,10 mph), (12 ft,11.5 mph), (15 ft,13 > mph)>>> Conversion Notes.>>> 1 foot per sec = 0.6818 mph>>> g = 32 ft/sec sec>>> (A problem with this simplistic approach is that as the wave shoals it > height will increase, but it may not be that bad if H is thought of the > height at the peak. I looking for crude numbers here, a feel for the kinds > of speeds involved. Nothing all that precise.)>>> Also, not all waves break the same way, some dont seem to jack very much > at all but just start crumbling, or barreling. This has a lot to do with > the bottom contours and the waves approach. So, please accept my > apologies for this crude analysis.)>>> Now the real problem is not really beachward speed, at least where I surf, > its down-the-line speed, which is more of a function the way the wave is > hitting the bar, see diagram. As you can see, or maybe guess, the more > acute the angle of approach, the faster the down the line speed.>>> Summarizing the results in the diagram we have,>>> Beachward Speed, Vb = [g(3/4)H]^(1/2)>>> Along-the-beach Speed, Va = Vb / Tan(A)>>> Down-the-line Speed, Vd = Vb / Sin(A)>>> (See diagram for direction references.)>>> Checking the formulas.>>> For A = 0, Sin(0) = Tan(0) = 0, that is the down-the-line speed and along > the beach speed are infinite and we have close out conditions. (That is, > its when only me and a few other idiots can be seen in the water.)>>> For very small A, we got some pretty fast waves, in terms of down-the-line > speed that is.>>> Heres a table of sorts for a 6 foot wave. The first is for angle of > approach of 11.25, the second 22.5 degrees, third 45 degrees. and the > fifth for 67.5 degrees (the angles the big hand of a clock makes at 12:45 > 1:30, 3:00 and 4:30.)>>> So, for a 6 foot wave,>>> Vb = [32 (3/4) 6]^1/2 (.6818) mph = 8.2 mph ~ 8 mph>>> So the following formula was used to generate the tables,>>> Vd = 8 / Sin(A) mph>>> Va = 8 / Tan(A) mph>>> (A deg, Vb mph, Vd mph, Va mph)>>> (11.25 deg,8 mph,41 mph,40 mph), (22.5 deg,8 mph,21 mph,19 mph), (45 deg,8 > mph,11 mph,8 mph), (67.5 deg ,8 mph,9 mph,3 mph)>>> You can see the trend the lower the angle of approach the faster the > down-the-line speed (and the along-the-beach speed.)>>> Can you make a down-the-line speed of 41 mph, perhaps not, especially on a > 6 foot wave. A 21 mph, or maybe 11 mph? Perhaps, but youve got to get the > energy from somewhere, gravity perhaps? Yes, and no.>>> Gravity is only part of the solution, the rest comes from the flow > interacting with the surfboard. This can best be understood by doing a > similarly crude energy analysis, using only kinetic (the energy inherent > in motion) and potential energy (the energy inherent in position in some > position sensitive force field like Gravity) considerations.>>> So, it seems I get to continue this nonsense in my next post. Kevin - you just need to build an HMF so you can measure these speeds instead of guessing. All you’d need is radar and a couple of lasers. See “Nonintrusive Multiple Point Measurements of Water Surface Slope, Elevation, and Velocity” 18th Symposium on Naval Hydrodynamic (1991)at http://books.nap.edu/books/0309045754/html/349.html#pagetop JUST KIDDING. Actually I have a hard time believing that a waist high wave hits the beach at 1/2 the speed of a double-overhead one. Despite what the formulas say…
Kevin - you just need to build an HMF so you can measure these speeds > instead of guessing. All you’d need is radar and a couple of lasers. See > “Nonintrusive Multiple Point Measurements of Water Surface Slope, > Elevation, and Velocity” 18th Symposium on Naval Hydrodynamic > (1991)at http://books.nap.edu/books/0309045754/html/349.html#pagetop>>> JUST KIDDING.>>> Actually I have a hard time believing that a waist high wave hits the > beach at 1/2 the speed of a double-overhead one. Despite what the formulas > say… I understand, but in a way, that was the point. Kevin