WAVE HEIGHT FROM PLANING AND SEMI-PLANING SMALL BOATS y

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1 RIVER RESEARCH AND APPLICATIONS River Res. Applic. 21: 1 17 (2005) Published online in Wiley InterScience ( DOI: /rra.803 WAVE HEIGHT FROM PLANING AND SEMI-PLANING SMALL BOATS y STEPHEN T. MAYNORD* US Army Corps of Engineers, Engineering Research and Development Center, Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180, USA ABSTRACT The increasing number of small boats has raised concerns about their effects on the environment, particularly their waves. Bank erosion is one of the foremost concerns of boat waves but disruption of habitat, resuspension of bottom sediments, and damage to aquatic plants are other areas of concern. A large programme of field measurement of boat waves was conducted on Johnson Lake in Alaska to evaluate boats typically used on the Kenai River. The boat wave study compared wave characteristics of four boats under a variety of loadings, speeds, distances and motor powers. Over 400 tests were run on Johnson Lake with each test providing wave measurement at four locations. Two measures of waves and two types of tests were used in the study. MAX- POW was the wave height at the maximum power of the motor. MAXWAV was the maximum wave height produced by the boat which required runs at a range of speeds to determine the MAXWAV. While the MAXWAV data herein have considerable scatter in magnitude, the conditions at which MAXWAVoccurs are consistent from boat to boat. To prevent generation of maximum wave heights, small boats should operate as far as possible either above or below length Froude number of 0.6, displacement Froude number of 1.3, or beam Froude number of 1.0. A general boat wave height equation was developed for the four boats based on boat speed, volume displaced by the boat and distance from the boat, and are applicable to semi-planing and planing boats based on MAXPOW and MAXWAV data. The predictive equation for V-hull boats was compared to independent data not used in the development and was found to be in agreement with the data. The predictive equation is limited to depth/boat length greater than Published in 2005 by John Wiley & Sons, Ltd. key words: boats; waves; bank erosion; navigation; environmental effects INTRODUCTION Small boating in some areas of the world occurs at high levels and has raised concerns about the environmental effects of small boats, particularly their waves. While bank erosion is likely the foremost concern regarding use of small boats, other concerns include disruption of habitat for fish and birds, resuspension of sediments and resultant turbidity, damage to aquatic vegetation, and impact on moored boats. A bibliography of studies addressing recreational boating disturbances is given in York (1994) and a workshop on the environmental impact of boating is documented in Crawford et al. (1998). Concerning high levels of small boats, boat passages per day at a site can be up to 704 on the Upper Mississippi River (Bhowmik et al., 1991) and 1071 on the Kenai River in Alaska (Dorava and Moore, 1997). A small boat is not well-defined but the data used herein are from boats having a length of 8.5 m or less. According to Crawford et al. (1998), throughout the United States, 78% of boats have a length of 5.5 m or less and 95% have a length of 7.9 m or less. The objective of this study is to develop techniques for estimating the wave height of small boats in the semi-planing to planing mode for the purpose of assessing environmental effects. While a huge number of factors affect boat wave height, this study attempts to relate wave height to the dominant and known parameters. Where more precise wave height estimates are needed, field measurement or, in some cases, physical or numerical modelling are alternatives. * Correspondence to: Stephen T. Maynord, US Army Corps of Engineers, Engineering Research and Development Center, Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180, USA. stephen.t.maynord@erdc.usace.army.mil y This article is U.S. Government work and is in the public domain in the U.S.A. Published in 2005 by John Wiley & Sons, Ltd. Accepted 8 January 2004

2 2 S. T. MAYNORD BOAT WAVE CHARACTERISTICS Waves are primarily characterized by their length, period and height. The length L w of a wave is the distance from one point on a wave to the same point on the next wave. The period of a wave T is the time it takes for two successive wave crests to pass a given point. The height of a wave H is the vertical distance between a trough, or low point in the wave profile, and the following crest, which is the high point in the wave profile. The speed of the wave front is the celerity C ¼ L w /T. Water depth h, if low enough, can have a significant impact on wave characteristics and depth effects are grouped by depth/wave length (h/l w ). For h/l w greater than 0.5, water depth has little impact on wave characteristics and this classification is referred to as deep water. For h/l w < 0.04, wave speed is completely determined by water depth and this classification is referred to as shallow water. The region having h/l w ¼ 0.04 to 0.5 is referred to as transitional water. Data presented subsequently will show that depth effects on wave height were small. The movement of a boat hull across the surface of a water body creates a variable pressure distribution along the length of the hull. The pressure variation generates a set of waves that move out away from the boat. The wave pattern consists of diverging and transverse waves. Diverging waves form at the bow and stern of the boat at an angle that depends on the vessel length Froude number F L based on boat waterline length L, defined as F L ¼ p V ffiffiffiffiffi gl ð1þ where V is the boat speed relative to the water, and g is the gravitational constant. The transverse waves are normal to the axis of the vessel and tend to disappear at larger F L. For all ships, the intersections of the diverging and transverse waves are called cusp points and are generally the locations of largest wave height. The angle of the cusp points with the axis of the vessel is about Figure 1 shows the general trend of maximum wave height versus boat speed (equivalent to applied power). Every boat hull and weight has its own curve like Figure 1 but the shape and trends are similar for the boats used in this investigation. Planing boats operate in three modes as shown in Figure 1. Figure 1. Wave height versus speed trends in small planing boats

3 WAVE HEIGHT FROM SMALL BOATS 3 (1) displacement mode boat weight offset by buoyant force of water. (2) semi-planing lift force on bottom of boat causes boat to partially rise out of water. Also called transition, semidisplacement, ploughing, and mushing. Bow generally high in water and semi-planing mode is characterized by large waves. Speed or Froude number limit of semi-planing is not well defined and depends on weight, length, beam, hull shape, etc. Some boats have a narrow speed range in which semi-planing occurs whereas others stay in this range for a relatively wide range of speeds. (3) Planing lift force maintains hull position with little contribution from buoyant force. In the field data used in this study, two wave measures were the focus of the investigation. MAXWAV is the maximum wave height that can be produced by the boat and is determined by running the boat at a range of speeds. MAXWAV is located at point A in Figure 1 (a point known as the hump), within the semi-planing mode. As will be shown subsequently, MAXWAV is difficult to measure and MAXWAV data have considerable scatter. The second measure of boat wave, MAXPOW, is the wave height at the maximum power of the motor. MAXPOW (points B E in Figure 1) are located somewhere to the right of point A and may be within the semi-planing or planing mode depending on the applied power (or speed) and loading of the boat. Boat waves decay with distance from the boat as shown in Figure 2. Note that close to the boat, a few large waves are present. As distance increases from the boat, maximum wave height decreases and the number of waves increases. The change in wave period throughout a boat wave event is also shown in Figure 2. The numbers along the lower part of each plot of water level are half of the wave period measured in seconds. Figure 2. Time histories of boat waves

4 4 S. T. MAYNORD Wave height data studies PREVIOUS STUDIES Sorensen (1967) presented wave height data for various vessels. Only the 7 m long Ladd-built cabin cruiser with V-bottom hull had a significant number of data points in the semi-planing to planing mode of operation. At rest draft was 0.51 m, beam was 2.5 m, and displacement was 3 tons approximate. Attempts were made to find out more about this boat but were unsuccessful. These data will be used later in this report. Zabawa and Ostrom (1980) present data from a 5.1 m three-point planing hull (Boston Whaler) and a 8.5 m deep-v planing hull (Uniflite Cruiser). Maximum wave height from Zabawa and Ostrom (1980) is plotted against boat speed in Figure 3 for various distances from the wave gauge. The plot shows a general trend of decreasing wave height with increasing boat speed typical of the semi-planing to planing mode. The plot also shows decay of wave height with distance but the scatter prevents any conclusions about variation of rate of decay with distance. The Zabawa and Ostrom (1980) data will be used subsequently. Sudol skii (1986) reported studies on small motorboats whose size was not given but all results point to boats similar to the sizes addressed in this paper. Sudol skii observed that maximum heights of diverging boat waves formed by moving motorboats increase from near-zero values when the speed of the boat is about 1.0 m/sec to m with speeds of about 3 4 m/sec. For boat speeds greater than 4ms 1, Sudol skii observed that wave height decreases with increasing speed. Sudol skii presented a plot of wave height versus boat speed for about eight different boats that was similar to the plot of the Zabawa and Ostrom (1980) data in Figure 3. The Sudol skii data could not be used further because the data were not presented. Wave height prediction studies While the focus of this paper is semi-planing or planing mode of small boats, studies of wave height prediction for displacement boats are presented because many of the same parameters in displacement wave equations are used herein. Sorensen and Weggel (1984) developed wave height prediction techniques limited to displacement boats. The functional relationship for wave height is h i ¼ f V; g; h; r 1=3 ; x ð2þ Figure 3. Maximum wave height versus boat speed from Zabawa and Ostrom (1980)

5 where ¼ maximum wave height, r¼volume of water displaced by the boat ¼ total boat weight/unit weight of water, and x ¼ lateral distance from the centreline of the boat. Bhowmik (1975) developed an equation for boat wave height based on field data as H 2 max ¼ 0:0345 VMPH 1:174 x 0:915 ð3þ d s L where d s ¼ boat draft and VMPH ¼ boat speed in miles per hour. This equation shows wave height to increase with boat speed indicating operation in the displacement mode. Bhowmik et al. (1991) developed another equation based on field data for boats that were travelling at speeds indicating semi-planing and planing operation. The equation was presented in a non-dimensional and dimensional form. The dimensional form in metric units is used herein to examine the individual parameters and is given by ¼ 0:537 L0:56 ds 0:355 V 0:346 x 0:345 ð4þ where all variables are in metres except V which is in m s 1. The exponent of boat speed shows decreasing wave height with increasing speed that is consistent with the semi-planing to planing mode. The Bhowmik et al. (1991) data tables did not include which prevented further use of their data. Boat design studies WAVE HEIGHT FROM SMALL BOATS 5 While the number of studies providing data sets or planing boat wave height prediction are limited, studies of planing boat design and performance are frequently found in the literature. Parameters affecting the performance of planing hulls are likely the same parameters important in the wave-making characteristics of a hull. Blount (1993) states that planing technology is both speed and hull loading dependent. Savitsky and Gore (1980) state that the smooth-water drag is predominantly dependent on hull deadrise, trim angle, and length/beam ratio. The various speed and hull parameters are discussed in the following paragraphs. To characterize speed, various Froude numbers have been used to characterize the three different modes of operation of a planing boat shown in Figure 1. Limits based on length Froude number from Savitsky (1985) are shown in Table I. Blount (1993) uses two different Froude numbers having characteristic length based on displacement and based on beam. The volume or displacement Froude number is defined as V F r ¼ ð5þ ðgr 1=3 Þ 1=2 The beam Froude number which is also referred to as the speed coefficient C v is defined as C v ¼ V ð6þ ðgbþ 1=2 where B is the maximum bottom width of the boat. Blount (1993) gives ranges of planing mode as shown in Table I based on displacement Froude number. Calkins (1983) and Savitsky and Brown (1976) use the speed coefficient to define ranges of planing mode also shown in Table I. Savitsky (1985) states that above semi-planing speed, use of boat length as the characteristic length in the Froude number is not very useful and that the volume Froude number is often used. Depth is also used as the characteristic length in Froude number in many wave investigations of Table I. Planing boat operation versus various Froude numbers Operating range F L F r C V Displacement <0.4 <1.3 <0.5 Semi-planing 0.4 to to to 1.5 Planing >0.9 >2.3 >1.5

6 6 S. T. MAYNORD displacement vessels and it is generally assumed that wave height reaches a maximum at a Froude number based on depth near unity. Calkins (1983) quantifies hull geometry with form ratios of length/beam ratio (L/B), slenderness ratio L/r 1/3, beam loading coefficient r/b 3. Savitsky and Gore (1980) note that many recreational craft are built with L/B of 3 to 4. The boats used in this investigation fall in this range of L/B and do not vary enough to use L/B in a wave prediction equation. Hull geometry is also defined by the deadrise angle which is a measure of the bottom angle from horizontal for V-hull boats. Flat-bottomed boats have a deadrise angle of 0. Deadrise for V-hull boats generally varies from large at the bow to small at the stern and is employed in boat design to prevent wave pounding that occurs with flat-bottomed boats. Flat-bottomed boats have the lowest drag ratios but are limited to smoothwater conditions. V-hull boats produced larger waves than flat-bottomed boats in Maynord (2001). Trim angle is a critical parameter in defining planing boat performance and thus wave making. Minimum dragto-lift ratios occur at trim angles of about 4. Blount (1993) states that there is no widely accepted definition of the reference plane from which zero trim is measured and provides four different methods of defining the reference plane. In the small boats used in this investigation, passenger and gear location can have a significant effect on trim angle. Accumulated bilge water can significantly affect values of trim in small boats. The position/ alignment of the motor in boats with manual trim or power trim and the presence of trim tabs have an affect on the trim. Trim is a critical parameter but it cannot be controlled or defined in small boats. A possible mitigating factor is that, in the author s small-boat experience, the trim in most small boats is adjusted to provide acceptable performance that is primarily related to achieving maximum speed. This adjustment may result in a relatively small range of actual trim angles for planing boats but probably not for boats at MAXWAV conditions in the semi-planing mode. Summary of relevant parameters Based on the review of previous wave studies and boat design studies, the relevant parameters in boat wave prediction for semi-planing and planing boats include some or all of the following parameters ¼ f ðv; r; g; h; d s ; L; B; x; trim ; deadrise Þ ð7þ where trim is the trim angle and deadrise is the deadrise angle. FIELD DATA COLLECTION General The Kenai River is Alaska s most popular salmon sport fishery and is located on the Kenai Peninsula south of Anchorage. The State of Alaska and the US Army Corps of Engineers (USACE) sponsored the wave measurements in Maynord (2001) as part of a study to address bank erosion from the combined effects of the high summer flows and the high volume of boat traffic related to the salmon. Field studies were conducted to determine wave characteristics from various boat/motor combinations that are used on the Kenai River. The boats used in this study were m (16 20 ft) in length, had motors ranging from 35 to 50 horsepower (1 hp ¼ W), and boat length/chine beam of 3.1 to 4.1. The objectives of this study were to (1) compare waves from different boats, (2) compare the effects of different loads, motors and distances, and (3) define the absolute magnitude of boat waves. One of the issues being addressed is the present regulation that limits motors used on the Kenai River to a maximum of 35 hp for safety reasons. The study compared waves from boats having 35 hp to the same boat having either a 40 or 50 hp motor under different load configurations, different distances, and different speeds. While Kenai River boat hulls are designed for planing at maximum speeds, the 35 hp limitation may result in some of the heavier boats operating at speeds in the semi-planing mode. The field studies were conducted at Johnson Lake from 23 to 28 July 2000 and on the Kenai River from 1 to 3 August 2000, both sites near Soldotna, AK. The Johnson Lake studies provided a semi-controlled environment where waves could be measured without significant influence from other effects. The Kenai River wave measurements were significantly affected by other factors, primarily large river velocities (2 m s 1 ) and stage variations in the river, but provided wave measurements in the river of interest. Only the Johnson Lake measurements are used in this study.

7 Table II. Characteristics of boats used in Johnson Lake tests Boat Overall Hull Chine Weight, 3 Weight, 4 Weight, 5 or Rated hull Actual length (m) shape beam (m) people (kg) people (kg) 6 people (kg) power (hp)* power (hp) Willie Predator (WP) 6.1 V (6) , 50 Koeffler (KF) 6.1 Flat (6) , 50 Klamath (KL) 4.9 V (5) 70 35, 40 Lowe (LW) 4.9 Flat (5) 35 35, 40 *1hp¼ W. Description of boats, motors and loadings WAVE HEIGHT FROM SMALL BOATS 7 Boat characteristics used in the tests are shown in Table II. In order to test the Willie Predator (WP) and Koeffler (KF) at 50 and 35 hp and the Klamath (KL) and Lowe (LW) at 40 and 35 hp, the changeover was made by detuning the engine by carburetor adjustment for all boat engines. The pattern of loading used in the tests was considered to be the most likely for that number of passengers. Prior to running the wave tests, the boat was run with different trim settings until the maximum speed was found. This trim setting was used in both the MAXPOW and MAXWAV tests. Site description, ambient conditions, and gauge orientation Johnson Lake is located south of Soldotna, Alaska, and is 34 ha in area. The lake had several features that were desirable for boat wave measurement in a controlled environment. First, the lake was small enough to minimize wind wave problems. Second, the lake was large enough to avoid problems from reflection of boat waves off shorelines. Third, the lake perimeter was lined with aquatic vegetation that dampened wave activity and minimized waiting times between tests. Fourth, no other motorized boats were allowed on the lake. The gauge layout, sailing line layout, and water depths are shown in Figure 4. The wave gauges at Johnson Lake were placed in water depths where depth/boat length exceeded 0.5 to minimize depth effects. Data acquisition The wave gauge system was based on capacitive sensor techniques and has been used in scale model testing for years. These gauges have demonstrated repeatable calibrations, stability with temperature, and are accurate enough to use in scale model tests. The wave staff consists of an insulated wire, 0.76 m long, drawn taut in a stainless steel tube bow. The data were recorded at 50 samples per second per channel for the duration of the test. The capacitance gauges were calibrated at numerous times throughout the field measurements. Experimental procedure Due to concerns about variability in the data, five replicates were run at Johnson Lake for each combination of boat, motor, loading and sailing line. Speed was measured by GPS receiver that was on board the test vessel and speed was recorded to the nearest 0.16 km h 1. The MAXWAV tests were run by running single tests at speeds in roughly 1 km h 1 increments. Maximum wave height was determined for all four gauges for the single test at each speed. The speed producing the maximum wave height was used for four more tests (to provide five replicates). Each combination of boat, power and load was tested at nine distances from the wave gauges ranging from 9 to 45 m. Over 400 tests were run for the various combinations, with each test providing wave measurement at four locations. BOAT FROUDE NUMBERS Various Froude numbers characterizing the field tests are shown in Table III. Length Froude number is based on waterline length, which is assumed equal to 0.85 times the overall length of the boat. As reported by Stumbo et al. (1999), boats having F L greater than 1 are dominated by diverging waves with little influence of transverse waves.

8 8 S. T. MAYNORD Figure 4. Schematic of wave gauges Table III. Boat speeds and Froude numbers Boat/power* Wave type Loading Average Speed F L F r C V (km h 1 ) WP50 MAXPOW All WP MAXWAV All KF35 MAXPOW All KF50 MAXPOW All KF MAXWAV All KL35 MAXPOW All KL40 MAXPOW All KL MAXWAV All LW35 MAXPOW All LW40 MAXPOW All LW MAXWAV All * Power in hp (1 hp ¼ W). All MAXPOW tests were dominated by the diverging waves and had large length Froude numbers. All MAXPOW tests were in the planing mode based on F L, F r and C V ranges shown in Table I. The MAXWAV tests all had F L of about 0.6 where transverse waves are present. MAXWAV data are all in the semi-planing mode and occurred at average values of F L ¼ 0.6, F r ¼ 1.3, and C V ¼ 1.05.

9 SUMMARY OF RESULTS FOR WAVE HEIGHT Results from an analysis of the field data, presented in Maynord (2001), for wave magnitude are as follows. (1) MAXPOW and MAXWAV for all conditions ranged from 0.04 to 0.28 m. (2) MAXPOW from the four boats are different. The WP produces the largest MAXPOW, KF and KL are next, and LW is the smallest. Differences between boats are largest at the lower engine power. (3) MAXPOW and MAXWAV decrease with increasing distance from the boat. (4) MAXPOW decreases with decreasing load. (5) MAXPOW decreases with increasing power for the WP, KF and KL. Differences in MAXPOW due to power effects are not significant for the LW. (6) V-hull boats (WP and KL) caused larger MAXPOW than flat-bottomed boats (KF and LW) because of their greater weight and hull shape. Wave height equations for each boat WAVE HEIGHT FROM SMALL BOATS 9 GENERAL EQUATION FOR BOAT WAVE HEIGHT The boat wave equations developed herein are applicable to the portion of Figure 1 to the right of the hump and include both MAXPOW and MAXWAV data. The equations are limited to the range of experimental data of F L ¼ , F r ¼ , and C V ¼ In the parameters shown in Equation 7, vessel draught is eliminated because it varies with speed and with position along the boat and is reflected in r. Trim angle is eliminated because of being unknown as discussed previously. Water depth is omitted and the equation is limited to depth/boat length greater than a limit that will be defined subsequently. Planing boat performance data show limited effects above a certain depth. Blount and Hankley (1976) evaluated the effects of water depth on effective power and speed of planing boats and found that changes in speed or effective power were less than 10% at depths greater than or equal to 0.4 times the overall length of the boat. Based on repeating variables of r and g, dimensional analysis results in r 1=3 ¼ f " # V x p ffiffiffiffiffiffiffiffiffiffiffiffi ; gr 1=3 r ; L 1=3 r ; r 1=3 B 3 ; deadrise Note that the first, third and fourth parameters on the right-hand side of the equation are the displacement Froude number, slenderness ratio, and the beam loading coefficient, respectively. The slenderness ratio and beam loading coefficient are eliminated because the equations will be developed for each boat, and L and B are constant for an individual boat. In Equation 8, deadrise angle is a parameter that is difficult to determine for any given V-hull boat because it varies from bow to stern and the most pertinent location along the boat for wave height prediction is unknown. For this reason, the effects of deadrise angle were handled by developing an equation for each boat. The functional relationship of wave height parameters for an individual boat becomes r 1=3 ¼ f " # V x p ffiffiffiffiffiffiffiffiffiffiffiffi ; gr 1=3 r 1=3 The Equation 9 parameters were used in a least-squares regression. The regression equation for the flat-bottomed Lowe boat is r ¼ 1:08 ð F 1=3 rþ 0:795 x 0:41 ð10þ r 1=3 Equation 10 was based on 480 wave measurements and has an adjusted R 2 ¼ The regression equation for the flat-bottomed Koeffler boat is r ¼ 0:71 ð F 1=3 rþ 0:51 x 0:43 ð11þ r 1=3 ð8þ ð9þ

10 10 S. T. MAYNORD Figure 5. Scatter plot of observed versus computed maximum wave height for flat-bottomed Lowe boat Equation 11 is based on 480 measurements and has an adjusted R 2 ¼ The regression equation for the V-hull Willie Predator boat is r ¼ 0:90 ð F 1=3 rþ 0:46 x 0:44 ð12þ r 1=3 Equation 12 is based on 320 measurements and has an adjusted R 2 ¼ The regression equation for the V-hull Klamath boat is r ¼ 0:93 ð F 1=3 rþ 0:58 x 0:39 ð13þ r 1=3 Equation 13 is based on 480 measurements and has an adjusted R 2 ¼ Using the Kolmogorov Smirnov onesample test and ¼ 0.05, there is no evidence to reject the null hypothesis of a normal distribution of residuals of predicted minus observed for Equations Based on normality of residuals, the least-squares regression equations are accepted as valid. As with any empirical relations, the limits of applicability must be followed. Limits on Equations are: /r 1/3 ¼ 0.05 to 0.30, F r ¼ 1.2 to 4.1, x/r 1/3 ¼ 8 to 56, r/b 3 ¼ 0.25 to 0.39, and L/B ¼ 3.1 to 4.1. Scatter plots of observed versus computed wave height from Equations are shown in Figures 5 8. The scatter plots show the difficulty when measuring MAXWAV having the lower values of the displacement Froude number. The largest waves in the scatter plots are the MAXWAV measurements that show much greater scatter when compared to the smaller waves that are the MAXPOW measurements. The greater scatter occurs for three reasons. (1) MAXWAV measurements are extremely sensitive to speed because the wave height speed relationship is steep on both sides of point A in Figure 1. In addition, the MAXWAV speed is not stable in some boats because the boat is on the verge of coming up on a full plane. (2) As stated previously, MAXWAV occurs at vessel Froude numbers where both transverse and diverging waves are present. The intersections of the transverse and diverging waves are the cusp points where peak wave height occurs. Between the cusp points, the diverging and transverse waves are out of phase and much lower wave heights occur. The spacing of cusp points is a function of the square of the boat speed. Consequently, small changes in boat speed between two replicate tests will result in the cusp point falling near the wave gauge in one test and between the cusp points for the second test, leading to considerable scatter in the data. MAXPOW tests are not bothered with these pesky cusp points because they are not present at high Froude numbers.

11 WAVE HEIGHT FROM SMALL BOATS 11 Figure 6. Scatter plot of observed versus computed maximum wave height for flat-bottomed Koeffler boat Figure 7. Scatter plot of observed versus computed maximum wave height for V-hull Willie Predator boat (3) While trim may not vary significantly for boats operating at MAXPOW, operation at MAXWAV is a very different situation. In some boats, MAXWAV operation requires the operator to stand up to see over the bow. In other boats, the rise of the bow at MAXWAV conditions is not nearly as severe. These variations in trim at MAXWAV conditions likely have a significant effect on the magnitude of MAXWAV. These are some of the reasons why it is difficult to measure and predict MAXWAV conditions. General wave height equation One of the objectives of this study was to develop a general equation applicable to a wide range of boats and conditions to the right of the hump in Figure 1. Figure 9 shows the individual boat Equations for a common

12 12 S. T. MAYNORD Figure 8. Scatter plot of observed versus computed maximum wave height for V-hull Klamath boat Figure 9. Comparison of Kenai River and Boston Whaler boat wave equations versus displacement Froude number at same distance from boat and same total boat weight distance (27.4 m) and a common total boat weight (909 kg). This weight is larger than any weight tested for the Lowe and Klamath boats and smaller than any weight tested for the Willie Predator and the Koeffler and is only used to compare boats. The Zabawa and Ostrom (1980) data were also analysed using the Equation 9 parameters. The regression equation for the tri-hull Boston Whaler is r ¼ 1:39 ð F 1=3 rþ 0:57 x r 1=3 0:52 ð14þ

13 WAVE HEIGHT FROM SMALL BOATS 13 Equation 14 is based on 23 measurements and has an adjusted R 2 ¼ The equation for the Boston Whaler is also shown on Figure 9 and is consistent with the V-hull data. The regression equation for the deep V-hull Uniflite Cruiser is r ¼ 0:79 ð F 1=3 rþ 0:59 x 0:36 ð15þ r 1=3 Equation 15 is based on 14 measurements and has an adjusted R 2 ¼ The uniflite Cruiser equation was not plotted on Figure 9 because the 909 kg weight is far from the actual weight on the Uniflite Cruiser. Figure 9 demonstrates the difficulty of finding a general equation applicable to the right-hand side of Figure 1. For large F r where boats are in the planing mode, the hull form is the dominant effect and the V-hull boats produce the largest waves. Boat length has a lesser effect than hull shape on planing boats, with the 6.1 m boats producing slightly smaller waves than the 4.9 m long boats. For MAXWAV conditions at the hump in Figure 1, the dominant effect is likely how the boat trims when operated at MAXWAV conditions. The V-hull and tri-hull boats appear to behave in a similar manner at MAXWAV conditions based on Figure 9. The Lowe and Koeffler flat-bottomed boats respond very differently at MAXWAV conditions based on Figure 9. The Koeffler produces a relatively small MAXWAV whereas the Lowe produces a relatively large MAXWAV. Any general wave equation is going to be approximate in the vicinity of MAXWAV until the trim angle is incorporated into the equation. The general equation for wave height is developed using Equation 8 but the third and fourth parameters on the right-hand side were not used because B and L vary over a small range in the Kenai River data. If additional data become available for a larger range of B and L, these parameters may become significant in the general equation, but B and L tend to correlate with r which may limit their significance. The Kenai River boat Equations have average exponents of for F r and 0.42 for x/r 1/3. This agrees with the average exponents from the Zabawa and Ostrom boats in Equations 14 and 15 of 0.58 for F r and 0.44 for x/r 1/3. The adopted general wave equation is r ¼ CF ð 1=3 rþ 0:58 x 0:42 ð16þ r 1=3 The coefficient C is used to account for the effects of deadrise angle in Equation 8. The Kenai River data were evaluated to determine the C that results in an average difference between observed and computed wave height of zero. The average of the Lowe and Koeffler flat-bottomed boats results in a C of The average of the Willie Predator and Klamath V-hull boats results in a C of Since Equation 16 and the C values were not based on regression, correlation coefficients have limited value in comparing observed and predicted values (Willmott, 1982). Willmott proposes a mean absolute error (MAE) to evaluate model performance defined as P j Hc H o j MAE ¼ ð17þ n where H c is the computed wave height, H o is the observed wave height, and n is the number of observations. For the two flat-bottomed boats the MAE is m and for the two V-hull boats the MAE is m. COMPARISON TO INDEPENDENT DATA Equation 16 was compared to the data of Sorensen (1967) and Zabawa and Ostrom (1980) using the coefficient for the V-hull boats. In the Sorensen data, all parameters are known for the 7 m cabin cruiser. Water depth at the sailing line of the boat was about 9 m. MAXWAV in the Sorensen tests with the 7 m cabin cruiser occurred at average F L of 0.64, F r of 1.3, and C V of 1.1. The comparison of the Sorensen data and the predictions from Equation 16 are shown in Figure 10. The magnitude of the predictions agrees with the data but the trends in the field data with speed variation are not found in the predictions. The Zabawa and Ostrom data do not include details of the two boats used but do state that the boats were provided by the Maryland Department of Natural Resources Marine Police. This agency was contacted and an

14 14 S. T. MAYNORD Figure 10. Observed and computed wave height, 7 m cabin cruiser data from Sorensen (1967) Figure 11. Observed and computed wave height, 5.1 m Boston Whaler, 15 and 46 m from wave gauge, from Zabawa and Ostrom (1980) individual was found (H. Dorsey) who was knowledgeable about the two boats used in the study. He stated that the 4.9 m long Boston Whaler mentioned in the report was actually 5.1 m long, which agreed with an old Boston Whaler catalogue that did not have a 4.9 m boat. The Boston Whaler catalogue showed a boat weight of 454 kg and a maximum load capacity of 643 kg. To estimate total boat weight, the load was assumed to be 75% of the maximum load or 482 kg, resulting in a total boat weight of 454 þ 482 ¼ 936 kg. Concerning the Uniflite Cruiser, Mr Dorsey estimated the total boat weight to be 5909 kg (Mr Dorsey s words- 6.5 tons ). Depths where the boats were operating in the Zabawa and Ostrom tests were 4 m, 3.7 m, 3.0 m and 0.9 m, at distances from the wave gauges of 61 m, 46 m, 30 m and 15 m, respectively. Displacement Froude number for the Boston Whaler and Uniflite Cruiser varied from 1.1 to 5.0 and 1.1 to 3.5, respectively. The comparisons of the Boston Whaler and the Uniflite Cruiser are shown in Figures at distances ranging from 15 m to 61 m. MAXWAV in the Zabawa and Ostrom (1980) tests with the 8.5 m Uniflite Cruiser occurred at average F L of 0.52, F r of 1.05, and C V of MAXWAV

15 WAVE HEIGHT FROM SMALL BOATS 15 Figure 12. Observed and computed wave height, 5.1 m Boston Whaler, 30 and 61 m from wave gauge, from Zabawa and Ostrom (1980) Figure 13. Observed and computed wave height, 8.5 m Uniflite Cruiser, from Zabawa and Ostrom (1980) in the Zabawa and Ostrom (1980) tests with the 5.1 m Boston Whaler occurred at average F L of 0.65, F r of 1.37, and C V of The limited amount of data and the assumptions and estimates regarding weight limit the confidence gained in these comparisons, but results in Figures show the predictive equation to provide the correct trends. Complete wave height data sets like the Johnson Lake data used to develop the predictive equation could not be found and are needed. The data that depart most from the predictive curve are those for the Boston Whaler at 15 m. At this boat position, depth was 0.91 m and depth/boat length was 0.18 which is far less than depth ¼ 0.4 boat length for

16 16 S. T. MAYNORD limited depth effects on power or speed. The Uniflite Cruiser had depth at the boat/boat length as low as 0.35 and exhibited good agreement of measured wave height with the predictive equation. Until more data are available to define shallow water effects on wave height, depth at the boat/boat length should be equal to or greater than 0.35 for application of the predictive equation. SUMMARY AND CONCLUSIONS The increasing number of small boats has raised concerns about their effects on the environment, particularly their waves. Bank erosion is one of the foremost concerns of boat waves but disruption of habitat, resuspension of bottom sediments, and damage to aquatic plants are other areas of concern. A large programme of field measurement of boat waves was conducted on Johnson Lake in Alaska to evaluate boats typically used on the Kenai River. The boat wave study compared wave characteristics of four boats under a variety of loadings, speeds, distances, and motor powers. Over 400 tests were run on Johnson Lake with each test providing wave measurement at four locations. Two wave types were measured. MAXPOW was the wave height at the maximum power of the motor. MAXWAV was the maximum wave height produced by the boat that required runs at a range of speeds to determine the MAXWAV. While the MAXWAV data herein have considerable scatter in magnitude, the conditions at which MAXWAVoccurs are consistent from boat to boat. To prevent generation of maximum wave heights, small boats should operate as far as possible either above or below length Froude number of 0.6, displacement Froude number of 1.3, or beam Froude number of 1.0. A general boat wave height equation is developed based on boat speed, volume displaced by the boat, and distance from the boat, and is applicable to semi-planing and planing boats. The coefficient in the boat wave height equation is determined for the V-hull and flat-bottomed boats used on the Kenai River. The predictive equation was compared to independent data not used in the development and was found to provide the correct trends. Depth/boat length should be equal to or greater than 0.35 for application of the predictive equation. acknowledgements The author thanks the participants in the study. Dr Robert Sanders participated from the USACE Alaska District. From the State of Alaska, participants were Mr Lance Trasky, Mr Dean Hughes, Ms Chris Degernes, and Mr Doug Hill. Volunteers to the State of Alaska participating in the study were Mr Charles Carr, Ms Lee Carr, and Mr Matt Leone. The ERDC field team was Messrs. Terry Waller, Sam Varnell and Wallace Guy. REFERENCES Bhowmik NG Boat-generated waves in lakes. ASCE Journal of the Hydraulics Division 101(HY11): Bhowmik NG, Soong TW, Reichelt WF, Seddik NML Waves generated by recreational traffic on the upper Mississippi River System. Research Report 117. Illinois State Water Survey: Blount DL Reflections on planing hull technology. In A Century of Progress in Power Boats. The Society of Naval Architects and Marine Engineers, Southeast Section, University of Miami: F1 F11. Blount DL, Hankley DW Full-scale trials and analysis of high-performance planing craft data. Transactions- The Society of Naval Architects and Marine Engineers 84: Calkins DE An interactive computer-aided design synthesis program for recreational powerboats. The Society of Naval Architects and Marine Engineers, Transactions 91: Crawford RE, Stolpe NE, Moore MJ The environmental impacts of boating. Proceedings of a workshop at Woods Hole Oceanographic Institution. Technical Report WHOI Dorava JM, Moore GW Effects of Boatwakes on streambank erosion, Kenai River, Alaska. Water Resources Investigations Report US Geological Survey: Anchorage. Maynord ST Boat waves on Johnson Lake and Kenai River, Alaska. ERDC/CHL TR US Army Corps of Engineers Research and Development Center. Savitsky D Planing Craft. Naval Engineers Journal February: Savitsky D, Brown PW Procedures for hydrodynamic evaluation of planing hulls in smooth and rough water. Marine Technology 13(4):

17 WAVE HEIGHT FROM SMALL BOATS 17 Savitsky D, Gore JL Re-evaluation of the planing hull form. Journal of Hydronautics 14(2): Sorensen RM Investigation of ship-generated waves. ASCE Journal of Waterways and Harbours Division 93(WW1): Sorensen RM, Weggel JR Development of ship wave design information. Proceedings of the 19th Coastal Engineering Conference. ASCE: Houston, Texas; Stumbo S, Fox K, Dvorak F, Elliot L The prediction, measurement, and analysis of wake wash from marine vessels. Marine Technology 36(4): Sudol skii AS Formation of waves and boiling of bottom sediments by small motorboats. Meteorologiya I Gidrologiya 3: Willmott CJ Some comments on the evaluation of model performance. Bulletin of the American Meteorological Society 63(11): York D Recreational-boating disturbances of natural communities and wildlife: an annotated bibliography. Biological Report 22. National Biological Survey, US Department of the Interior. Zabawa C, Ostrom C The role of boat wakes in shore erosion in Anne Arundel County, Maryland. Prepared for Maryland Department of Natural Resources.