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Wave types of landslide generated impulse waves
Ocean Engineering 38 (2011) 630–640
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Wave types of landslide generated impulse waves V. Heller a,n,1, W.H. Hager b,2 a b
School of Civil Engineering and the Environment, University of Southampton, Highfield, Southampton, SO17 1BJ, UK VAW, ETH Zurich, CH-8092 Z¨ urich, Switzerland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 July 2009 Accepted 7 December 2010 Editor-in-Chief: A.I. Incecik Available online 17 January 2011
Subaerial landslide generated impulse waves were investigated in a prismatic wave channel. Seven governing parameters, namely the still water depth, slide impact velocity, slide thickness, bulk slide volume, bulk slide density, slide impact angle, and grain diameter, were systematically varied. The generated impulse waves are nonlinear, intermediate- to shallow-water waves involving a small to considerable fluid mass transport. The Stokes wave, cnoidal wave, solitary wave, and bore theories were applied to describe the observed maximum waves. The theoretical and observed features of these four wave types are highlighted. A diagram allows to predict the wave type directly as a function of the slide parameters, the slide impact angle, and the still water depth. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Impulse wave Landslide Nonlinear wave theory Physical model test Wave generation Wave type classification
1. Introduction 1.1. Overview Impulse waves are generated by landslides, rock falls, shore instabilities, snow avalanches, glacier calvings, or asteroids impacting into an ocean, bay, lake, or reservoir. They may reach a considerable height such as in the 1958 Lituya Bay case, Alaska, resulting in a run-up height of 524 m on the opposite shore (Miller, 1960; Fritz et al., 2001) or the 1963 Vaiont case in the Italian Alps (Schnitter, 1964). In the latter case the impulse wave overtopped a dam by about 70 m and destroyed the Longarone village where about 2000 people perished. To prevent such catastrophes mainly passive methods such as evacuation, water level draw-down, freeboard control in artificial reservoirs, or blasting of possible slides are considered. These methods require detailed knowledge on the impulse wave features including type, height, and decay. 1.2. Studies on impulse wave generation and propagation Studies on subaerial landslide generated impulse waves apply five methods (Heller et al., 2009): (i) specific prototype studies (e.g. WCHL, 1970, for Mica Reservoir), (ii) numerical simulations (e.g. Quecedo et al., 2004), (iii) predictions based on field data (e.g. n
Corresponding author. Tel.: + 44 2380592883; fax: + 44 2380677519. E-mail addresses: [email protected] (V. Heller), [email protected] (W.H. Hager). 1 ¨ Formerly, VAW, ETH Zurich, CH-8092 Zurich, Switzerland. 2 Tel.: +41 446324149; fax: + 41 446321192. 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.010
Ataie-Ashtiani and Malek Mohammadi, 2007), (iv) analytical calculations (e.g. Di Risio and Sammarco, 2008), and (v) general model studies. Method (v) is applied herein allowing for the formulation of general statements. Studies with method (v) based on a block model slide were conducted e.g. by Noda (1970), Wiegel et al. (1970), Kamphuis and Bowering (1972), Walder et al. (2003), or Panizzo (2004) in a wave channel (two-dimensional 2D) and by Panizzo et al. (2005) in a wave basin (three-dimensional 3D). Investigations based on granular slide models were accomplished e.g. by Fritz (2002) or Fritz et al. (2004), respectively, Zweifel (2004) or Zweifel et al. (2006), respectively, or Heller and Hager (2010) in 2D and by Huber (1980) and Huber and Hager (1997) in both 2D and 3D. Detailed literature reviews for method (v) are presented in Heller (2007, 2008). Subaerial landslide generated impulse waves involve generally nonlinear and intermediate- to shallow-water waves of small to considerable fluid mass transport (Heller, 2007). They are dispersive (Kamphuis and Bowering, 1972) and cover a wide wave type spectrum. Depending on the wave type the wave profile, amount of fluid mass transport, run-up height, or wave force on a structure differ. The knowledge of the wave type is therefore a prerequisite to reliably predict the effects of impulse waves on the shore line or a dam. 1.3. Wave type classification methods Fig. 1 shows an impulse wave in the (x, z) plane with the basic wave parameters, namely the wave amplitude a, wave height H, wave length L, wave period T, and still water depth h. These parameters are also defined in Fig. 11 for a sinusoidal wave profile.
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Wave type classifications of landslide generated impulse waves mainly exist for wave channel experiments. Four approaches are available to classify impulse waves (i) (ii) (iii) (iv)
optical wave profile inspection; nonlinearity a/H; Ursell parameter U¼(H/L)/(h/L)3 ¼HL2/h3; and wavelet transform analysis.
A linear wave according to method (ii) requires a/H¼ 0.5 and a solitary wave a/H¼ 1. The Ursell parameter U in method (iii) is herein defined with the wave height H in analogy to Le Me´haute´ (1976), Huber (1980), or Panizzo et al. (2005), and not in the classical way with the wave amplitude a. For a linear wave it tends to U-0. According to Ursell (1953) his parameter is a more appropriate criterion to identify linear waves than simply a/H¼0.5 since a linear wave requires not only identical crest and trough amplitudes (Fig. 11(a)) but also small ratios of H/h and H/L. A literature review on wave type classifications is presented in Appendix A. It indicates that all authors observed cnoidal (also referred to as complex solitary, nonlinear transient, transient) waves and solitary (or solitary-like) waves. Further also oscillatory (sinusoidal, nonlinear oscillatory, linear) waves and bores (transient bores) were observed, depending on the applied governing parameter ranges. Existing wave type classifications (Appendix A) have disadvantages if applied as a prediction method: they do not include all governing parameters (e.g. Fritz et al., 2004, and Zweifel et al., 2006, worked with a constant grain diameter and hill slope angle), are based on block and not on granular slides (Noda, 1970; Wiegel et al., 1970; Panizzo et al., 2002, 2005), resulting in design diagram regions where several wave types occurred (Zweifel et al., 2006), or presented no design diagram at all to predict the governing wave type (Huber, 1980; Panizzo et al., 2002). 1.4. Present study This study aims to present a classification for the maximum impulse wave types in the near field caused by granular slides
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based on optical wave profile inspection and the criteria defined in Table 1. The aim is to predict the wave types directly as a function of the governing parameters, such that the wave parameters addressed in Heller and Hager (2010) are not required. It concludes further the 2D impulse wave research initiated by Fritz (2002) and continued by Zweifel (2004), including the effects of grain diameter and hill slope angle and an extension of most dimensionless governing parameter ranges (Heller and Hager, 2010). This study is organized as follows: the physical model and the governing parameters are described in Section 2. The wave type classification is presented in Section 3 together with a discussion of the four observed wave types. The new findings are summarized in Section 4.
2. Physical model 2.1. Experimental set-up The experiments were conducted in 11 m long, b¼0.500 m wide, and 1 m high prismatic and horizontal water wave channel previously employed by Fritz (2002) and Zweifel (2004). The front sidewall was made of glass, whereas the back sidewall and the bottom consisted of continuous steel plates, and glass, respectively. The granular slide material was filled in a slide box and accelerated with the pneumatic landslide generator (Fritz and Moser, 2003) down a 3 m long hill slope ramp. Typically 1.3 m above the still water surface, in hill slope ramp direction, the box front flap opened and the slide moved free on the ramp before it impacted the water body to generate impulse waves. Several measurement systems recorded the slide and wave properties along the channel axis. Two Laser Distance Sensors (LDS) scanned the slide profiles prior to impact the water body, seven Capacitance Wave Gauges (CWG) recorded the wave profiles, and Particle Image Velocimetry (PIV) was applied to determine the velocity vector fields in the slide impact zone. The wave types were determined from the wave profiles and controlled for selected tests with video recordings. Details on the physical model are given by Heller (2007). 2.2. Governing parameters
Fig. 1. Definition sketch of governing parameters and wave features in (x, z) plane.
Fig. 1 shows the relevant parameters for impulse wave generation. The pneumatic landslide generator allowed a systematic and nearly independent variation of seven governing parameters in the test ranges specified in the upper part of Table 2: still water depth 0.150 mrhr0.675 m, slide impact velocity 2:06 m=s rVs r 8:77 m=s, slide thickness 0.050 m rsr0.249 m, bulk slide volume 0.017 m3 rVs r0.067 m3, bulk slide density 590 kg/m3 r rs r 1720 kg/m3, slide impact angle 301r a r901, and grain diameter 2.0 mmrdg r8.0 mm. Note that the bulk slide volume Vs was identical to the inner slide box volume and the corresponding bulk slide density rs ¼ms/Vs was also calculated with Vs and the slide mass ms. The governing parameters Vs and s were related
Table 1 Criteria for the wave type classification. Stokes-like wave
Cnoidal-like wave
Solitary-like wave
Bore-like wave
Wave profile symmetric (to both axes)
Wave profile symmetric (to vertical axis) Trough longer than crest Multiple crests (at least two)
Wave profile symmetric (to vertical axis)
Irregular wave profile (to both axes)
Trough identical long as crest Multiple equivalent crests (at least two) a Eat No air transport aM oh/2 (Eintermediate-water)
Small air transport aM o h ( Eintermediate-water)
One dominant crest Trough nearly absent (at oa/4) small or no air transport Intermediate-water
Steep wave front, flat wave tail One dominant crest Large air transport Intermediate- to shallow-water
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to the slide impact location. For the wave propagation zone the streamwise distance 0 rxr8.90 m and times 0rt r10 s are relevant.
2.3. Governing dimensionless parameters The dimensional analysis applied in Heller et al. (2008) resulted in the dimensionless governing parameters with the test ranges specified in the lower part of Table 2. These parameters include the slide Froude number F ¼Vs/(gh)1/2 (with the gravitational acceleration g) varied in the range 0.86rF r6.83, relative slide thickness S ¼s/h (0.09rS r1.64), relative slide mass M ¼ms/(rwbh2) with rw as the water density (0.11rM r10.02), relative grain diameter Dg ¼dg/h (0.003 rDg r0.040), slide impact angle a (301 r a r901), relative streamwise distance X¼x/h (0.0r Xr59.2), and the relative time Tr ¼t(g/h)1/2 (0 rTr r81). Negligible scale effects with regard to the relative maximum wave amplitude, due to surface tension or kinematic viscosity, for impulse waves involving a granular slide require by rule of thumb hZ0.200 m (Heller et al., 2008). Some tests were conducted with h¼0.150 m (Table 2), and scale effects in this range affect the wave height or wave decay, but hardly change the wave type. Table 2 Test ranges of governing (upper part) and governing dimensionless (lower part) parameters. Symbol
Dimension
Description
Test range
h Vs s Vs
(m) (m/s) (m) (m3) (kg/m3) (1) (mm) (m) (s) (–) (–) (–) (–) (1) (–) (–)
Still water depth Slide impact velocity Slide thickness Bulk slide volume Bulk slide density Slide impact angle Grain diameter Streamwise distance Time Slide Froude number Relative slide thickness Relative slide mass Relative grain diameter Slide impact angle Relative streamwise distance Relative time
0.150–0.675 2.06–8.77 0.050–0.249 0.017–0.067 590–1720 30–90 2.0–8.0 0–8.90 0–10 0.86–6.83 0.09–1.64 0.11–10.02 0.003–0.040 30–90 0–59.2 0–81
rs a dg x t F¼ Vs/(gh)1/2 S ¼s/h M ¼ ms/(rwbh2) Dg ¼ dg/h
a X ¼ x/h Tr ¼ t(g/h)1/2
3. Experimental results and discussion 3.1. Wave type classification The wave length L was calculated with the measured wave celerity c and wave period T as L¼cT. The investigated impulse waves at CWG1 were in the range 5.1rL/hr82.7 and intermediate- to shallow-water waves, therefore. The primary wave was in 95% equal to the maximum wave. About 90% of the generated maximum waves were in the intermediate-water wave range. Further, the Ursell parameter at CWG1 was in the range 2.7rUr10340. Linear waves with U-0 were not observed. The herein investigated impulse waves may therefore be characterized as nonlinear, intermediate- to shallow-water waves with small to considerable fluid mass transport. The maximum waves were allocated to the four nonlinear types: Stokes wave, cnoidal wave, solitary wave, and bore (Appendix B). The hereafter presented wave type classification includes not only impulse waves with a ‘perfect’ but also with a transient profile because of their relevance in practice. The criteria for the classification defined in Table 1 are individually discussed hereafter for each wave type. Impulse waves may change their type even over a short wave channel reach. Bore-like wave profiles in the slide impact zone for instance transform to a solitary- or cnoidal-like wave profile farther away from the impact zone due to energy dissipation and air detrainment. For the present wave classification, the most characteristic wave type of the maximum wave for the spacing between the first and the last CWG (2.7rXr59.2) was selected. Heller et al. (2009) identified the dimensionless wave type product T¼ S1/3M cos[(6/7)a] as the relevant number characterizing a landslide impacting a water body. Fig. 2 relates T versus the slide Froude number F for the 211 experiments of the present study (Fig. 2(a)) and for the present study including the data of Fritz (2002) with 137 runs and Zweifel (2004) with 86 runs (Fig. 2(b)). The symbols refer to Stokes-like waves (J), cnoidal-like waves (&), solitary-like waves (B), and bore-like waves (X). Data described with (*) were conducted in the range of possible scale effects with h¼0.150 m. These runs resulted all in bore-like waves. Open data in Fig. 2(a) refer to runs with rs E590 kg/m3 and dark data to runs with rs E1720 kg/m3; dark data in Fig. 2(b) refer to the present study, open data to runs of Fritz (2002), and light shaded data to runs of Zweifel (2004). All four wave types occurred at all considered slide impact angles a. However, solitary-like wave types are particularly well
Fig. 2. Wave type classification based on the criteria in Table 1: wave type product T¼ S1/3M cos[(6/7)a] and (–) Eq. (1) (¼ instead of o) and (–) Eq. (3) (¼ instead of 4) for (a) present study and (b) data from previous studies (Fritz, 2002; Zweifel, 2004).
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developed for a ¼901 originating from Russell’s (1837) experiments where solitary waves were generated with a vertically dropped block. Three wave type zones may be identified in Fig. 2, namely To
4 7=5 F 5
Stokes-like waves,
4 7=5 F rT r11F5=2 5
cnoidal- or solitary-like waves,
ð1Þ
ð2Þ
and T 4 11F5=2
bore-like waves:
ð3Þ
The cnoidal- and solitary-like waves were not separable in Fig. 2. The reason may be the theoretical similarity between these two in which the solitary wave is a special case of the cnoidal wave (Appendix B). The number of outliers in Fig. 2(a) is 8 (4% of 211) and in Fig. 2(b) 37 (9% of 434), respectively. The relative grain diameter Dg had a negligible effect on the wave type classification. The slide Froude number F and the relative slide mass M are the dominant parameters determining the impulse wave type. Runs with large values of F, S, M, and small a tend to generate bore-like waves because of the large streamwise slide momentum flux component represented by the impulse product parameter P¼FS1/2 M1/4{cos[(6/7)a]}1/2 introduced by Heller and Hager (2010). In contrast, small F, S, M, and large a values tend to generate Stokeslike waves in accordance with a small P. A wave type classification based on P was not successful even though it is the relevant parameter for most wave features in Heller and Hager (2010). In the following, the observed four wave types are discussed individually for the wave propagation zone, whereas their effects on a dam or shore line may be determined after Heller et al. (2009). 3.2. Stokes-like waves Fig. 3(a)–(c) shows a photo sequence of Stokes-like impulse waves generation involving rs ¼608 kg/m3, a ¼601, and small
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relative values of F ¼1.36, S ¼0.23, and M¼0.11 to satisfy Eq. (1). The water splash generated during impact in Fig. 3(a) is relatively small, entraining limited air that affects mainly the primary wave surface. The slide to wave energy conversion is energetically efficient since the slide with rs o rw is totally damped by the surrounding water body and no energy is ‘lost’ during impact on the channel bottom (Heller, 2007). The maximum of the primary impulse wave reaches CWG1 in Fig. 3(c). Several similar pure water waves reach the wave propagation zone at relative distances X¼6.1–10.7 in Fig. 3(d)–(f). The primary wave is not necessarily the maximum wave for Stokes-like waves, in contrast to the other three wave types. Typical wave profiles of Stokes-like impulse waves are shown in Fig. 4. The relative wave profiles Z/h versus relative time Tr are shown from top to bottom at CWG1 in Fig. 4(a) to CWG7 in Fig. 4(g). The gray zone is affected by wave reflection at the wave absorber located at the channel end. These wave profiles allow for a classification according to Table 1. Stokes-like waves feature a nearly symmetrical wave profile both to the vertical and the horizontal axes where multiple crests occur. Due to small governing dimensionless parameters and small wave amplitudes, i.e. much smaller than at wave breaking, negligible air transport is observed in the wave propagation zone. The maximum wave amplitude aM oh/2 was small to assure the intermediate-water range to which the Stokes wave theory applies (Le Me´haute´, 1976). As compared with the theoretical Stokes wave profile (Fig. 11(b)) the observed profiles are not monochromatic, decay, and have no front trough. Huber (1980) suggested to apply the linear wave theory to describe impulse waves such as in Fig. 4 (Appendix A). The following studies may be relevant to determine the effects of Stokes-like waves: Miche (1951) for the run-up of oscillatory waves, Sainflou (1928), Minikin (1950), Wiegel (1964), Novak et al. (2001), or Dean and Dalrymple (2004) for computing the forces of a standing wave (clapotis) on a vertical wall, or Tanimoto et al. (1984) for predicting the force of a shallow-water sinusoidal wave.
Fig. 3. Stokes-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.600 m, rs ¼ 608 kg/m3, a ¼601, F¼ 1.36, S¼ 0.23, M ¼0.11, Dg ¼ 0.008, X ¼6.1–10.7, and Tr E(a) 0.8, (b) 2.4, (c) 4.0, (d) 18.5, (e) 19.8, and (f) 21.0.
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Fig. 4. Stokes-like wave profiles for h ¼ 0.600 m, rs ¼608 kg/m3, a ¼451, F¼ 1.18, S ¼0.17, M ¼ 0.29, Dg ¼0.003 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7; gray area is influenced by wave reflection.
Fig. 5. Cnoidal-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.300 m, rs ¼610 kg/m3, a ¼301, F¼ 2.27, S ¼ 0.40, M¼ 0.45, Dg ¼0.017, X ¼ 10.3–17.6, and Tr E(a) 1.1, (b) 3.4, (c) 5.7, (d) 11.0, (e) 13.1, and (f) 15.1.
3.3. Cnoidal-like wave Fig. 5(a)–(c) shows a photo sequence of a cnoidal-like impulse wave generation involving rs ¼ 610 kg/m3, a ¼301, and medium to
large relative values of F ¼2.27, S ¼0.40, and M ¼0.45 to satisfy Eq. (2). The granular slide material impacts the water body in Fig. 5(a) to generate a considerable water splash in Fig. 5(b). The envelope of the amplitude and the splash profile are visible on the
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wet dark zone on the channel back wall. The splash impacts the water surface between Fig. 5(b) and (c) and a large amount of air is entrained. The impulse wave propagates to the wave propagation zone at relative distances X¼10.3–17.6 in Fig. 5(d)–(f). In Fig. 5(d) the cnoidal-like impulse wave transports still lots of air from the slide impact process. This air escapes in Fig. 5(e) and the wave consists almost only of water in Fig. 5(f). Typical wave profiles of cnoidal-like impulse waves are shown in Fig. 6. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 6(a) to CWG7 in Fig. 6(g). Cnoidal-like waves were selected according to the criteria of Table 1: cnoidal-like waves feature a symmetrical wave profile to the vertical axis where multiple crests occur. The wave troughs are longer than the wave crests and the wave crest amplitudes are larger than the wave trough amplitudes. The maximum amplitude aM is larger than for the Stokes-like impulse waves yet limited to aM oh. Cnoidal-like wave types may transport air as shown in Fig. 5 due to the impact crater formation. As compared with the theoretical cnoidal wave profile described in Appendix B the observed profiles are not shallow-water, but intermediate-water waves. Further they are not periodic and the primary wave has no front trough. The effects of cnoidal-like waves may be determined with the ¨ results of Muller (1995) who tested the run-up and dam overtopping for a wide wave spectrum. Also relevant is the literature review in Section 3.4 given the similar wave profile of cnoidal- and solitary-like waves if e.g. the maximum wave of Fig. 6(d) is compared with that of Fig. 8(c). Further an identical method has to be applied in practice for both cnoidal- and solitary-like waves because they were not separable in Fig. 2.
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dropping box of Russell (1837). The generated water splash is relatively small resulting in relatively small air entrainment in Fig. 7(b) and detrainment in Fig. 7(c). In the wave propagation zone in Fig. 7(d)–(f) at relative distances X¼15.3–24.3 the impulse wave consists only of the water phase. The solitary-like wave is well developed in Fig. 7(e). Its large decay may be observed on the wet dark zone on the channel back wall. The wave profiles of Fig. 7 are shown in Fig. 8. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 8(a) to CWG7 in Fig. 8(g). Solitary-like waves were selected with the criteria shown in Table 1. These waves feature a symmetrical wave profile to the vertical axis with only a singular or solitary dominant crest. The wave trough is nearly absent following the criterion at oa/4. The maximum relative wave amplitudes were 0.49raM/hr0.94, i.e. close to the maximum stable wave height H¼0.78h (McCowan, 1894). Solitary-like waves transport in the wave propagation zone almost no air. These waves were in the intermediate-water range, in contrast to the theory which applies to the shallow-water regime (Appendix B). The wave length and period, respectively, are not infinite and the wave decays (Fig. 7(f)) due to amplitude dispersion, frequency dispersion, and both turbulence in the slide impact zone and in the boundary layers. The observed maximum waves are followed by a small wave trough, in contrast to theory. The effects of solitary waves were object of extensive research: Hall and Watts (1953), Synolakis (1987), Liu et al. (1991), Zelt (1991), Teng et al. (2000), Li and Raichlen (2001), Gedik et al. (2005), and Hughes (2004), among others, investigated the solitary ¨ wave run-up, Muller (1995) focused on both the run-up and dam overtopping, and Ramsden (1996) and Cooker et al. (1997) investigated the solitary wave force on vertical walls.
3.4. Solitary-like wave Fig. 7(a)–(c) shows a photo sequence of a solitary-like impulse wave generation. This wave type was generated using rs ¼609 kg/m3, a ¼901, and large relative values of F ¼3.77, S¼ 0.81, and M¼0.90 satisfying Eq. (2). The vertically impacting granular slide in Fig. 7(a) generates a solitary-like wave in analogy to the vertically
3.5. Bore-like wave Fig. 9(a)–(c) shows a photo sequence of a bore-like impulse wave generation. This wave type was produced using rs ¼1664 kg/m3, a ¼601, and very large relative values of F ¼4.22, S¼ 0.61, and
Fig. 6. Cnoidal-like wave profiles for h ¼ 0.300 m, rs ¼ 610 kg/m3, a ¼ 451, F¼ 2.70, S¼ 0.34, M ¼1.11, Dg ¼ 0.027 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
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Fig. 7. Solitary-like wave generation (a)–(c) and propagation (d)–(f) for h ¼0.300 m, rs ¼ 609 kg/m3, a ¼ 901, F¼ 3.77, S¼ 0.81, M ¼0.90, Dg ¼ 0.017, X ¼15.3–24.3, and Tr E(a) 2.1, (b) 4.1, (c) 6.1, (d) 15.5, (e) 17.5, and (f) 19.5.
Fig. 8. Solitary-like wave profiles from Fig. 7 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
M¼2.47 to satisfy Eq. (3). The impacting slide mass in Fig. 9(a) generates a considerable air cavity. A large amount of the slide energy is ‘lost’ during impact on the channel bottom and the slide to wave energy conversion is energetically inefficient (Heller, 2007). Nevertheless, a relatively large impulse wave is generated in Fig. 9(b) and a large amount of air is entrained in Fig. 9(c). A bore-like wave reaches the wave propagation zone at relative distances X¼14.9–24.3. The bore-like wave is characterized by large energy dissipation and both air entrainment at the wave front and detrainment at the wave tail. The entire water column of this wave type moves horizontally as noted from Fig. 9(f) on the rising air bubbles accumulated prior to the experiment on the channel bottom.
The wave profiles of Fig. 9 are shown in Fig. 10. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 10(a) to CWG7 in Fig. 10(g). The criteria in Table 1 are again applied to classify these wave profiles. Bore-like waves with their typical dominant crest feature an unsymmetrical wave profile both relative to the vertical and the horizontal axes. The steep wave front, the flat wave tail, and the large air transport are typical for these waves. The maximum wave amplitude aM prior to bore formation, consisting rather of a water sheet (Fig. 9(a)) than a well-developed wave profile, was up to 2.5 times the still water depth h. The observed waves were in the intermediate- to shallow-water range, in contrast to the theoretical solitary wave profile which applies to shallow-water
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Fig. 9. Bore-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.300 m, rs ¼ 1664 kg/m3, a ¼601, F¼4.22, S¼ 0.61, M ¼2.47, Dg ¼0.013, X ¼ 14.9–24.3, and Tr E(a) 2.3, (b) 4.6, (c) 6.7, (d) 10.6, (e) 12.9, and (f) 15.0.
Fig. 10. Bore-like wave profiles of Fig. 9 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
waves (Appendix B). The fluid flow is further highly turbulent and rotational. The run-up of bores is discussed by Shen and Meyer (1963), Miller (1968), Yeh (1991), and Hughes (2004). Ikeno et al. (2001) investigated the bore wave force on a vertical wall.
4. Conclusions The wave types of subaerial landslide generated impulse waves were investigated based on the Froude similitude and granular
slide material with 211 new and 223 tests from two previous studies. Seven governing parameters, namely the still water depth h, slide impact velocity Vs, slide thickness s, bulk slide volume Vs, bulk slide density rs, slide impact angle a, and grain diameter dg, were systematically considered. The wave types were determined by optical wave profile inspection and the criteria of Table 1, and they were controlled for selected runs with video recordings. The main results may be summarized as follows: (1) The wave type classification in Fig. 2 is based on the wave type product T ¼S1/3M cos[(6/7)a] and the slide Froude number F.
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(2)
(3)
(4)
(5)
(6)
(7)
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A data inclusion of the two previous studies corroborates this concept of wave type classification. The dominant parameters for the wave type classification are the slide Froude number F¼ Vs/(gh)1/2, the slide thickness S¼ s/h, and the relative slide mass M ¼ms/(rwbh2). The slide impact angle a affects the wave type with cos[(6/7)a] whereas the effect of the relative grain diameter Dg ¼dg/h is negligibly small. Four wave types and transient types were observed which are described with four nonlinear wave theories, namely the Stokes wave, cnoidal wave, solitary wave, and bore theory. Stokes-like waves result generally from small dimensionless parameters F, S, and M. They consist of several waves with symmetrical wave profiles, are intermediate-water waves, and are characterized by small fluid mass transport. Cnoidal-like waves involve generally medium to large dimensionless parameters F, S, and M. They consist of a longer wave trough than the wave crest, are intermediate-water waves, and are characterized by fluid mass transport. Solitary-like waves follow generally from large dimensionless parameters F, S, and M. They consist of one dominant wave crest, the wave trough is nearly absent, and their fluid mass transport is large. Bore-like waves result generally from large dimensionless parameters F, S, and M. They consist of one dominant wave with a large amount of air at the wave front. They transport a large amount of fluid mass.
wave theory for U-N. However, Huber (1980) observed also a number of transient waves based on method (i). Panizzo et al. (2005) applied both methods (iii) and (iv) for wave basin experiments for the waves propagating on the slide length axis (basin symmetry axis with a wave propagation angle of 01). Based on preliminary wave classifications for 2D in Panizzo et al. (2002, 2005) distinguished between linear, transient, and solitary wave characters whereas their impulse waves were rather linear than solitary.
Appendix B. Nonlinear wave theories Appendix B.1. Stokes wave theory The Stokes wave profile is shown in Fig. 11(b) with at as the distance from the still water surface to the wave trough and a as the wave amplitude. As compared with the sinusoidal wave shown in Fig. 11(a), the Stokes wave profile has steeper crests and flatter troughs whereas it is still symmetrical about a vertical plane (Wiegel, 1964). This theory was described by Stokes (1847) and is based on potential (irrotational) flow with a non-hydrostatic pressure distribution. The analytical description of the wave profile involves a power series based on (H/L), commonly to the fifth order
The wave type classification in Fig. 2 may be simply applied in practice since it depends directly on basic parameters such as the slide parameters or the still water depth which can be estimated prior to an event.
Acknowledgements This research was supported by the Swiss National Science Foundation (grant 200020-103480/1) and the Swiss Federal Office of Energy SFOE (contract 152777). Thanks to Dr. H.M. Fritz, Georgia Institute of Technology, and Dr. A. Zweifel, University of Zurich, for providing their experimental data.
Appendix A. Literature review on wave type classifications A literature review on previous wave type classifications based on methods (i)–(iv) in Section 1.3 is presented hereafter. Wiegel et al. (1970), Noda (1970), Huber (1980), and Fritz (2002) or Fritz et al. (2004), respectively, classified the wave types with method (i). Wiegel et al. (1970) described the impulse waves generated by a vertically falling box as solitary waves, complex solitary wave or a series of cnoidal waves, and bores. Noda (1970) presented graphically a wave type classification containing oscillatory waves, nonlinear transient waves, solitary waves, and bores based on own tests and the experiments of Wiegel et al. (1970). Huber (1980) observed sinusoidal waves, cnoidal waves, solitary waves, and transient waves. Fritz et al. (2004) distinguished between weakly nonlinear oscillatory waves, nonlinear transient waves, solitarylike waves, and dissipative transient bores. Zweifel (2004) or Zweifel et al. (2006), respectively, applied method (ii) to describe strongly nonlinear (0.9oa/Hr1.0), moderately nonlinear (0.6 oa/Hr0.9), and weakly nonlinear (0.4o a/Hr0.6) waves and classified the impulse waves in oscillatory waves, transient waves, solitary waves, and bores. Based on method (iii), Huber (1980) suggested the linear wave theory for Uo15, cnoidal wave theory for 15oUoN, and solitary
Fig. 11. Wave profiles with basic parameters for (a) sinusoidal wave, (b) Stokes waves, (c) cnoidal wave, (d) solitary wave, and (e) bore.
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(Fenton, 1985). The water particles describe open orbits; the waves are oscillatory with an intermediate character and a small fluid mass transport, therefore. The range of the Stokes waves is defined according to Le Me´haute´ (1976) as 2 rL/hr20. However, Keulegan (1950) suggested for Stokes waves of fifth order the limitation L/ho10, corresponding to an Ursell parameter of roughly Uo10.
Appendix B.2. Cnoidal wave theory The cnoidal wave theory was developed by Korteweg and de Vries (1895) and its wave profile is shown in Fig. 11(c). The wave crest amplitude is larger than the wave trough and the wave trough portion is longer than both the wave crest and the sinusoidal wave trough portions (Fig. 11(a)). The name cnoidal is derived from the Jacobian elliptic function ‘cn’ to describe its profile (Wiegel, 1960). This wave type is derived by assuming a hydrostatic pressure distribution for the first order (Isobe, 1985) and non-hydrostatic pressure distribution for the second order approximation in combination with irrotational flow (Le Me´haute´, 1976). This theory allows periodic waves to exist in shallow-water (Dean and Dalrymple, 2004). The profile has mainly oscillatory character, although fluid mass transport occurs. This wave type is especially appropriate for U425 and according to Keulegan (1950) valid in the range of L/hZ10. The cnoidal wave theory is spanning the range between the solitary wave theory on the one hand (for T-N) and the linear wave theory on the other hand (Wiegel, 1960; Dean and Dalrymple, 2004).
Appendix B.3. Solitary wave theory The solitary wave type consists only of one water surface elevation and no wave trough such that the wave length L is theoretically infinite (Fig. 11(d)). Russell (1837) investigated solitary waves in a hydraulic model and derived some basic features. The theory of the solitary wave was independently derived by Boussinesq (1871) and Rayleigh (1876) by assuming non-hydrostatic pressure distribution and rotational flow. The solitary wave is translative involving a considerable fluid mass transport (Le Me´haute´, 1976). For this wave type the nonlinearity, tending to steepen the wave front, balances dispersion, which tends to spread the wave front. Therefore, its profile is constant without damping along the travel distance in a horizontal channel of constant width b, if boundary layer friction is neglected. According to Laitone (1960), the solitary wave celerity c is c ¼ ½gðhþ aÞ1=2 :
ðB:1Þ
Eq. (B.1) is widely accepted to describe the celerity of subaerial landslide generated impulse waves (e.g. Kamphuis and Bowering, 1972; Huber, 1980; Fritz, 2002; Heller, 2007).
Appendix B.4. Bore theory The tidal bore and roller wave theory is e.g. applied to describe broken waves in the shoaling region (Wiegel, 1964). A bore wave profile shown in Fig. 11(e) typically consists of a steep front and a flat portion behind the wave crest. The water particles move in the bore propagation direction. The system of equations is based on hydrostatic pressure distribution and irrotational flow. A bore corresponds to a very shallow-water wave with fluid mass transport. The wave profile is evaluated with the momentum, the continuity, and the characteristic equations (Le Me´haute´, 1976).
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Wave types of landslide generated impulse waves V. Heller a,n,1, W.H. Hager b,2 a b
School of Civil Engineering and the Environment, University of Southampton, Highfield, Southampton, SO17 1BJ, UK VAW, ETH Zurich, CH-8092 Z¨ urich, Switzerland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 July 2009 Accepted 7 December 2010 Editor-in-Chief: A.I. Incecik Available online 17 January 2011
Subaerial landslide generated impulse waves were investigated in a prismatic wave channel. Seven governing parameters, namely the still water depth, slide impact velocity, slide thickness, bulk slide volume, bulk slide density, slide impact angle, and grain diameter, were systematically varied. The generated impulse waves are nonlinear, intermediate- to shallow-water waves involving a small to considerable fluid mass transport. The Stokes wave, cnoidal wave, solitary wave, and bore theories were applied to describe the observed maximum waves. The theoretical and observed features of these four wave types are highlighted. A diagram allows to predict the wave type directly as a function of the slide parameters, the slide impact angle, and the still water depth. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Impulse wave Landslide Nonlinear wave theory Physical model test Wave generation Wave type classification
1. Introduction 1.1. Overview Impulse waves are generated by landslides, rock falls, shore instabilities, snow avalanches, glacier calvings, or asteroids impacting into an ocean, bay, lake, or reservoir. They may reach a considerable height such as in the 1958 Lituya Bay case, Alaska, resulting in a run-up height of 524 m on the opposite shore (Miller, 1960; Fritz et al., 2001) or the 1963 Vaiont case in the Italian Alps (Schnitter, 1964). In the latter case the impulse wave overtopped a dam by about 70 m and destroyed the Longarone village where about 2000 people perished. To prevent such catastrophes mainly passive methods such as evacuation, water level draw-down, freeboard control in artificial reservoirs, or blasting of possible slides are considered. These methods require detailed knowledge on the impulse wave features including type, height, and decay. 1.2. Studies on impulse wave generation and propagation Studies on subaerial landslide generated impulse waves apply five methods (Heller et al., 2009): (i) specific prototype studies (e.g. WCHL, 1970, for Mica Reservoir), (ii) numerical simulations (e.g. Quecedo et al., 2004), (iii) predictions based on field data (e.g. n
Corresponding author. Tel.: + 44 2380592883; fax: + 44 2380677519. E-mail addresses: [email protected] (V. Heller), [email protected] (W.H. Hager). 1 ¨ Formerly, VAW, ETH Zurich, CH-8092 Zurich, Switzerland. 2 Tel.: +41 446324149; fax: + 41 446321192. 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.010
Ataie-Ashtiani and Malek Mohammadi, 2007), (iv) analytical calculations (e.g. Di Risio and Sammarco, 2008), and (v) general model studies. Method (v) is applied herein allowing for the formulation of general statements. Studies with method (v) based on a block model slide were conducted e.g. by Noda (1970), Wiegel et al. (1970), Kamphuis and Bowering (1972), Walder et al. (2003), or Panizzo (2004) in a wave channel (two-dimensional 2D) and by Panizzo et al. (2005) in a wave basin (three-dimensional 3D). Investigations based on granular slide models were accomplished e.g. by Fritz (2002) or Fritz et al. (2004), respectively, Zweifel (2004) or Zweifel et al. (2006), respectively, or Heller and Hager (2010) in 2D and by Huber (1980) and Huber and Hager (1997) in both 2D and 3D. Detailed literature reviews for method (v) are presented in Heller (2007, 2008). Subaerial landslide generated impulse waves involve generally nonlinear and intermediate- to shallow-water waves of small to considerable fluid mass transport (Heller, 2007). They are dispersive (Kamphuis and Bowering, 1972) and cover a wide wave type spectrum. Depending on the wave type the wave profile, amount of fluid mass transport, run-up height, or wave force on a structure differ. The knowledge of the wave type is therefore a prerequisite to reliably predict the effects of impulse waves on the shore line or a dam. 1.3. Wave type classification methods Fig. 1 shows an impulse wave in the (x, z) plane with the basic wave parameters, namely the wave amplitude a, wave height H, wave length L, wave period T, and still water depth h. These parameters are also defined in Fig. 11 for a sinusoidal wave profile.
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Wave type classifications of landslide generated impulse waves mainly exist for wave channel experiments. Four approaches are available to classify impulse waves (i) (ii) (iii) (iv)
optical wave profile inspection; nonlinearity a/H; Ursell parameter U¼(H/L)/(h/L)3 ¼HL2/h3; and wavelet transform analysis.
A linear wave according to method (ii) requires a/H¼ 0.5 and a solitary wave a/H¼ 1. The Ursell parameter U in method (iii) is herein defined with the wave height H in analogy to Le Me´haute´ (1976), Huber (1980), or Panizzo et al. (2005), and not in the classical way with the wave amplitude a. For a linear wave it tends to U-0. According to Ursell (1953) his parameter is a more appropriate criterion to identify linear waves than simply a/H¼0.5 since a linear wave requires not only identical crest and trough amplitudes (Fig. 11(a)) but also small ratios of H/h and H/L. A literature review on wave type classifications is presented in Appendix A. It indicates that all authors observed cnoidal (also referred to as complex solitary, nonlinear transient, transient) waves and solitary (or solitary-like) waves. Further also oscillatory (sinusoidal, nonlinear oscillatory, linear) waves and bores (transient bores) were observed, depending on the applied governing parameter ranges. Existing wave type classifications (Appendix A) have disadvantages if applied as a prediction method: they do not include all governing parameters (e.g. Fritz et al., 2004, and Zweifel et al., 2006, worked with a constant grain diameter and hill slope angle), are based on block and not on granular slides (Noda, 1970; Wiegel et al., 1970; Panizzo et al., 2002, 2005), resulting in design diagram regions where several wave types occurred (Zweifel et al., 2006), or presented no design diagram at all to predict the governing wave type (Huber, 1980; Panizzo et al., 2002). 1.4. Present study This study aims to present a classification for the maximum impulse wave types in the near field caused by granular slides
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based on optical wave profile inspection and the criteria defined in Table 1. The aim is to predict the wave types directly as a function of the governing parameters, such that the wave parameters addressed in Heller and Hager (2010) are not required. It concludes further the 2D impulse wave research initiated by Fritz (2002) and continued by Zweifel (2004), including the effects of grain diameter and hill slope angle and an extension of most dimensionless governing parameter ranges (Heller and Hager, 2010). This study is organized as follows: the physical model and the governing parameters are described in Section 2. The wave type classification is presented in Section 3 together with a discussion of the four observed wave types. The new findings are summarized in Section 4.
2. Physical model 2.1. Experimental set-up The experiments were conducted in 11 m long, b¼0.500 m wide, and 1 m high prismatic and horizontal water wave channel previously employed by Fritz (2002) and Zweifel (2004). The front sidewall was made of glass, whereas the back sidewall and the bottom consisted of continuous steel plates, and glass, respectively. The granular slide material was filled in a slide box and accelerated with the pneumatic landslide generator (Fritz and Moser, 2003) down a 3 m long hill slope ramp. Typically 1.3 m above the still water surface, in hill slope ramp direction, the box front flap opened and the slide moved free on the ramp before it impacted the water body to generate impulse waves. Several measurement systems recorded the slide and wave properties along the channel axis. Two Laser Distance Sensors (LDS) scanned the slide profiles prior to impact the water body, seven Capacitance Wave Gauges (CWG) recorded the wave profiles, and Particle Image Velocimetry (PIV) was applied to determine the velocity vector fields in the slide impact zone. The wave types were determined from the wave profiles and controlled for selected tests with video recordings. Details on the physical model are given by Heller (2007). 2.2. Governing parameters
Fig. 1. Definition sketch of governing parameters and wave features in (x, z) plane.
Fig. 1 shows the relevant parameters for impulse wave generation. The pneumatic landslide generator allowed a systematic and nearly independent variation of seven governing parameters in the test ranges specified in the upper part of Table 2: still water depth 0.150 mrhr0.675 m, slide impact velocity 2:06 m=s rVs r 8:77 m=s, slide thickness 0.050 m rsr0.249 m, bulk slide volume 0.017 m3 rVs r0.067 m3, bulk slide density 590 kg/m3 r rs r 1720 kg/m3, slide impact angle 301r a r901, and grain diameter 2.0 mmrdg r8.0 mm. Note that the bulk slide volume Vs was identical to the inner slide box volume and the corresponding bulk slide density rs ¼ms/Vs was also calculated with Vs and the slide mass ms. The governing parameters Vs and s were related
Table 1 Criteria for the wave type classification. Stokes-like wave
Cnoidal-like wave
Solitary-like wave
Bore-like wave
Wave profile symmetric (to both axes)
Wave profile symmetric (to vertical axis) Trough longer than crest Multiple crests (at least two)
Wave profile symmetric (to vertical axis)
Irregular wave profile (to both axes)
Trough identical long as crest Multiple equivalent crests (at least two) a Eat No air transport aM oh/2 (Eintermediate-water)
Small air transport aM o h ( Eintermediate-water)
One dominant crest Trough nearly absent (at oa/4) small or no air transport Intermediate-water
Steep wave front, flat wave tail One dominant crest Large air transport Intermediate- to shallow-water
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to the slide impact location. For the wave propagation zone the streamwise distance 0 rxr8.90 m and times 0rt r10 s are relevant.
2.3. Governing dimensionless parameters The dimensional analysis applied in Heller et al. (2008) resulted in the dimensionless governing parameters with the test ranges specified in the lower part of Table 2. These parameters include the slide Froude number F ¼Vs/(gh)1/2 (with the gravitational acceleration g) varied in the range 0.86rF r6.83, relative slide thickness S ¼s/h (0.09rS r1.64), relative slide mass M ¼ms/(rwbh2) with rw as the water density (0.11rM r10.02), relative grain diameter Dg ¼dg/h (0.003 rDg r0.040), slide impact angle a (301 r a r901), relative streamwise distance X¼x/h (0.0r Xr59.2), and the relative time Tr ¼t(g/h)1/2 (0 rTr r81). Negligible scale effects with regard to the relative maximum wave amplitude, due to surface tension or kinematic viscosity, for impulse waves involving a granular slide require by rule of thumb hZ0.200 m (Heller et al., 2008). Some tests were conducted with h¼0.150 m (Table 2), and scale effects in this range affect the wave height or wave decay, but hardly change the wave type. Table 2 Test ranges of governing (upper part) and governing dimensionless (lower part) parameters. Symbol
Dimension
Description
Test range
h Vs s Vs
(m) (m/s) (m) (m3) (kg/m3) (1) (mm) (m) (s) (–) (–) (–) (–) (1) (–) (–)
Still water depth Slide impact velocity Slide thickness Bulk slide volume Bulk slide density Slide impact angle Grain diameter Streamwise distance Time Slide Froude number Relative slide thickness Relative slide mass Relative grain diameter Slide impact angle Relative streamwise distance Relative time
0.150–0.675 2.06–8.77 0.050–0.249 0.017–0.067 590–1720 30–90 2.0–8.0 0–8.90 0–10 0.86–6.83 0.09–1.64 0.11–10.02 0.003–0.040 30–90 0–59.2 0–81
rs a dg x t F¼ Vs/(gh)1/2 S ¼s/h M ¼ ms/(rwbh2) Dg ¼ dg/h
a X ¼ x/h Tr ¼ t(g/h)1/2
3. Experimental results and discussion 3.1. Wave type classification The wave length L was calculated with the measured wave celerity c and wave period T as L¼cT. The investigated impulse waves at CWG1 were in the range 5.1rL/hr82.7 and intermediate- to shallow-water waves, therefore. The primary wave was in 95% equal to the maximum wave. About 90% of the generated maximum waves were in the intermediate-water wave range. Further, the Ursell parameter at CWG1 was in the range 2.7rUr10340. Linear waves with U-0 were not observed. The herein investigated impulse waves may therefore be characterized as nonlinear, intermediate- to shallow-water waves with small to considerable fluid mass transport. The maximum waves were allocated to the four nonlinear types: Stokes wave, cnoidal wave, solitary wave, and bore (Appendix B). The hereafter presented wave type classification includes not only impulse waves with a ‘perfect’ but also with a transient profile because of their relevance in practice. The criteria for the classification defined in Table 1 are individually discussed hereafter for each wave type. Impulse waves may change their type even over a short wave channel reach. Bore-like wave profiles in the slide impact zone for instance transform to a solitary- or cnoidal-like wave profile farther away from the impact zone due to energy dissipation and air detrainment. For the present wave classification, the most characteristic wave type of the maximum wave for the spacing between the first and the last CWG (2.7rXr59.2) was selected. Heller et al. (2009) identified the dimensionless wave type product T¼ S1/3M cos[(6/7)a] as the relevant number characterizing a landslide impacting a water body. Fig. 2 relates T versus the slide Froude number F for the 211 experiments of the present study (Fig. 2(a)) and for the present study including the data of Fritz (2002) with 137 runs and Zweifel (2004) with 86 runs (Fig. 2(b)). The symbols refer to Stokes-like waves (J), cnoidal-like waves (&), solitary-like waves (B), and bore-like waves (X). Data described with (*) were conducted in the range of possible scale effects with h¼0.150 m. These runs resulted all in bore-like waves. Open data in Fig. 2(a) refer to runs with rs E590 kg/m3 and dark data to runs with rs E1720 kg/m3; dark data in Fig. 2(b) refer to the present study, open data to runs of Fritz (2002), and light shaded data to runs of Zweifel (2004). All four wave types occurred at all considered slide impact angles a. However, solitary-like wave types are particularly well
Fig. 2. Wave type classification based on the criteria in Table 1: wave type product T¼ S1/3M cos[(6/7)a] and (–) Eq. (1) (¼ instead of o) and (–) Eq. (3) (¼ instead of 4) for (a) present study and (b) data from previous studies (Fritz, 2002; Zweifel, 2004).
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developed for a ¼901 originating from Russell’s (1837) experiments where solitary waves were generated with a vertically dropped block. Three wave type zones may be identified in Fig. 2, namely To
4 7=5 F 5
Stokes-like waves,
4 7=5 F rT r11F5=2 5
cnoidal- or solitary-like waves,
ð1Þ
ð2Þ
and T 4 11F5=2
bore-like waves:
ð3Þ
The cnoidal- and solitary-like waves were not separable in Fig. 2. The reason may be the theoretical similarity between these two in which the solitary wave is a special case of the cnoidal wave (Appendix B). The number of outliers in Fig. 2(a) is 8 (4% of 211) and in Fig. 2(b) 37 (9% of 434), respectively. The relative grain diameter Dg had a negligible effect on the wave type classification. The slide Froude number F and the relative slide mass M are the dominant parameters determining the impulse wave type. Runs with large values of F, S, M, and small a tend to generate bore-like waves because of the large streamwise slide momentum flux component represented by the impulse product parameter P¼FS1/2 M1/4{cos[(6/7)a]}1/2 introduced by Heller and Hager (2010). In contrast, small F, S, M, and large a values tend to generate Stokeslike waves in accordance with a small P. A wave type classification based on P was not successful even though it is the relevant parameter for most wave features in Heller and Hager (2010). In the following, the observed four wave types are discussed individually for the wave propagation zone, whereas their effects on a dam or shore line may be determined after Heller et al. (2009). 3.2. Stokes-like waves Fig. 3(a)–(c) shows a photo sequence of Stokes-like impulse waves generation involving rs ¼608 kg/m3, a ¼601, and small
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relative values of F ¼1.36, S ¼0.23, and M¼0.11 to satisfy Eq. (1). The water splash generated during impact in Fig. 3(a) is relatively small, entraining limited air that affects mainly the primary wave surface. The slide to wave energy conversion is energetically efficient since the slide with rs o rw is totally damped by the surrounding water body and no energy is ‘lost’ during impact on the channel bottom (Heller, 2007). The maximum of the primary impulse wave reaches CWG1 in Fig. 3(c). Several similar pure water waves reach the wave propagation zone at relative distances X¼6.1–10.7 in Fig. 3(d)–(f). The primary wave is not necessarily the maximum wave for Stokes-like waves, in contrast to the other three wave types. Typical wave profiles of Stokes-like impulse waves are shown in Fig. 4. The relative wave profiles Z/h versus relative time Tr are shown from top to bottom at CWG1 in Fig. 4(a) to CWG7 in Fig. 4(g). The gray zone is affected by wave reflection at the wave absorber located at the channel end. These wave profiles allow for a classification according to Table 1. Stokes-like waves feature a nearly symmetrical wave profile both to the vertical and the horizontal axes where multiple crests occur. Due to small governing dimensionless parameters and small wave amplitudes, i.e. much smaller than at wave breaking, negligible air transport is observed in the wave propagation zone. The maximum wave amplitude aM oh/2 was small to assure the intermediate-water range to which the Stokes wave theory applies (Le Me´haute´, 1976). As compared with the theoretical Stokes wave profile (Fig. 11(b)) the observed profiles are not monochromatic, decay, and have no front trough. Huber (1980) suggested to apply the linear wave theory to describe impulse waves such as in Fig. 4 (Appendix A). The following studies may be relevant to determine the effects of Stokes-like waves: Miche (1951) for the run-up of oscillatory waves, Sainflou (1928), Minikin (1950), Wiegel (1964), Novak et al. (2001), or Dean and Dalrymple (2004) for computing the forces of a standing wave (clapotis) on a vertical wall, or Tanimoto et al. (1984) for predicting the force of a shallow-water sinusoidal wave.
Fig. 3. Stokes-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.600 m, rs ¼ 608 kg/m3, a ¼601, F¼ 1.36, S¼ 0.23, M ¼0.11, Dg ¼ 0.008, X ¼6.1–10.7, and Tr E(a) 0.8, (b) 2.4, (c) 4.0, (d) 18.5, (e) 19.8, and (f) 21.0.
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Fig. 4. Stokes-like wave profiles for h ¼ 0.600 m, rs ¼608 kg/m3, a ¼451, F¼ 1.18, S ¼0.17, M ¼ 0.29, Dg ¼0.003 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7; gray area is influenced by wave reflection.
Fig. 5. Cnoidal-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.300 m, rs ¼610 kg/m3, a ¼301, F¼ 2.27, S ¼ 0.40, M¼ 0.45, Dg ¼0.017, X ¼ 10.3–17.6, and Tr E(a) 1.1, (b) 3.4, (c) 5.7, (d) 11.0, (e) 13.1, and (f) 15.1.
3.3. Cnoidal-like wave Fig. 5(a)–(c) shows a photo sequence of a cnoidal-like impulse wave generation involving rs ¼ 610 kg/m3, a ¼301, and medium to
large relative values of F ¼2.27, S ¼0.40, and M ¼0.45 to satisfy Eq. (2). The granular slide material impacts the water body in Fig. 5(a) to generate a considerable water splash in Fig. 5(b). The envelope of the amplitude and the splash profile are visible on the
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wet dark zone on the channel back wall. The splash impacts the water surface between Fig. 5(b) and (c) and a large amount of air is entrained. The impulse wave propagates to the wave propagation zone at relative distances X¼10.3–17.6 in Fig. 5(d)–(f). In Fig. 5(d) the cnoidal-like impulse wave transports still lots of air from the slide impact process. This air escapes in Fig. 5(e) and the wave consists almost only of water in Fig. 5(f). Typical wave profiles of cnoidal-like impulse waves are shown in Fig. 6. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 6(a) to CWG7 in Fig. 6(g). Cnoidal-like waves were selected according to the criteria of Table 1: cnoidal-like waves feature a symmetrical wave profile to the vertical axis where multiple crests occur. The wave troughs are longer than the wave crests and the wave crest amplitudes are larger than the wave trough amplitudes. The maximum amplitude aM is larger than for the Stokes-like impulse waves yet limited to aM oh. Cnoidal-like wave types may transport air as shown in Fig. 5 due to the impact crater formation. As compared with the theoretical cnoidal wave profile described in Appendix B the observed profiles are not shallow-water, but intermediate-water waves. Further they are not periodic and the primary wave has no front trough. The effects of cnoidal-like waves may be determined with the ¨ results of Muller (1995) who tested the run-up and dam overtopping for a wide wave spectrum. Also relevant is the literature review in Section 3.4 given the similar wave profile of cnoidal- and solitary-like waves if e.g. the maximum wave of Fig. 6(d) is compared with that of Fig. 8(c). Further an identical method has to be applied in practice for both cnoidal- and solitary-like waves because they were not separable in Fig. 2.
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dropping box of Russell (1837). The generated water splash is relatively small resulting in relatively small air entrainment in Fig. 7(b) and detrainment in Fig. 7(c). In the wave propagation zone in Fig. 7(d)–(f) at relative distances X¼15.3–24.3 the impulse wave consists only of the water phase. The solitary-like wave is well developed in Fig. 7(e). Its large decay may be observed on the wet dark zone on the channel back wall. The wave profiles of Fig. 7 are shown in Fig. 8. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 8(a) to CWG7 in Fig. 8(g). Solitary-like waves were selected with the criteria shown in Table 1. These waves feature a symmetrical wave profile to the vertical axis with only a singular or solitary dominant crest. The wave trough is nearly absent following the criterion at oa/4. The maximum relative wave amplitudes were 0.49raM/hr0.94, i.e. close to the maximum stable wave height H¼0.78h (McCowan, 1894). Solitary-like waves transport in the wave propagation zone almost no air. These waves were in the intermediate-water range, in contrast to the theory which applies to the shallow-water regime (Appendix B). The wave length and period, respectively, are not infinite and the wave decays (Fig. 7(f)) due to amplitude dispersion, frequency dispersion, and both turbulence in the slide impact zone and in the boundary layers. The observed maximum waves are followed by a small wave trough, in contrast to theory. The effects of solitary waves were object of extensive research: Hall and Watts (1953), Synolakis (1987), Liu et al. (1991), Zelt (1991), Teng et al. (2000), Li and Raichlen (2001), Gedik et al. (2005), and Hughes (2004), among others, investigated the solitary ¨ wave run-up, Muller (1995) focused on both the run-up and dam overtopping, and Ramsden (1996) and Cooker et al. (1997) investigated the solitary wave force on vertical walls.
3.4. Solitary-like wave Fig. 7(a)–(c) shows a photo sequence of a solitary-like impulse wave generation. This wave type was generated using rs ¼609 kg/m3, a ¼901, and large relative values of F ¼3.77, S¼ 0.81, and M¼0.90 satisfying Eq. (2). The vertically impacting granular slide in Fig. 7(a) generates a solitary-like wave in analogy to the vertically
3.5. Bore-like wave Fig. 9(a)–(c) shows a photo sequence of a bore-like impulse wave generation. This wave type was produced using rs ¼1664 kg/m3, a ¼601, and very large relative values of F ¼4.22, S¼ 0.61, and
Fig. 6. Cnoidal-like wave profiles for h ¼ 0.300 m, rs ¼ 610 kg/m3, a ¼ 451, F¼ 2.70, S¼ 0.34, M ¼1.11, Dg ¼ 0.027 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
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Fig. 7. Solitary-like wave generation (a)–(c) and propagation (d)–(f) for h ¼0.300 m, rs ¼ 609 kg/m3, a ¼ 901, F¼ 3.77, S¼ 0.81, M ¼0.90, Dg ¼ 0.017, X ¼15.3–24.3, and Tr E(a) 2.1, (b) 4.1, (c) 6.1, (d) 15.5, (e) 17.5, and (f) 19.5.
Fig. 8. Solitary-like wave profiles from Fig. 7 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
M¼2.47 to satisfy Eq. (3). The impacting slide mass in Fig. 9(a) generates a considerable air cavity. A large amount of the slide energy is ‘lost’ during impact on the channel bottom and the slide to wave energy conversion is energetically inefficient (Heller, 2007). Nevertheless, a relatively large impulse wave is generated in Fig. 9(b) and a large amount of air is entrained in Fig. 9(c). A bore-like wave reaches the wave propagation zone at relative distances X¼14.9–24.3. The bore-like wave is characterized by large energy dissipation and both air entrainment at the wave front and detrainment at the wave tail. The entire water column of this wave type moves horizontally as noted from Fig. 9(f) on the rising air bubbles accumulated prior to the experiment on the channel bottom.
The wave profiles of Fig. 9 are shown in Fig. 10. The relative wave profiles Z/h versus relative time Tr are shown from the top to the bottom at CWG1 in Fig. 10(a) to CWG7 in Fig. 10(g). The criteria in Table 1 are again applied to classify these wave profiles. Bore-like waves with their typical dominant crest feature an unsymmetrical wave profile both relative to the vertical and the horizontal axes. The steep wave front, the flat wave tail, and the large air transport are typical for these waves. The maximum wave amplitude aM prior to bore formation, consisting rather of a water sheet (Fig. 9(a)) than a well-developed wave profile, was up to 2.5 times the still water depth h. The observed waves were in the intermediate- to shallow-water range, in contrast to the theoretical solitary wave profile which applies to shallow-water
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Fig. 9. Bore-like wave generation (a)–(c) and propagation (d)–(f) for h ¼ 0.300 m, rs ¼ 1664 kg/m3, a ¼601, F¼4.22, S¼ 0.61, M ¼2.47, Dg ¼0.013, X ¼ 14.9–24.3, and Tr E(a) 2.3, (b) 4.6, (c) 6.7, (d) 10.6, (e) 12.9, and (f) 15.0.
Fig. 10. Bore-like wave profiles of Fig. 9 at (a) CWG1, (b) CWG2, (c) CWG3, (d) CWG4, (e) CWG5, (f) CWG6, and (g) CWG7.
waves (Appendix B). The fluid flow is further highly turbulent and rotational. The run-up of bores is discussed by Shen and Meyer (1963), Miller (1968), Yeh (1991), and Hughes (2004). Ikeno et al. (2001) investigated the bore wave force on a vertical wall.
4. Conclusions The wave types of subaerial landslide generated impulse waves were investigated based on the Froude similitude and granular
slide material with 211 new and 223 tests from two previous studies. Seven governing parameters, namely the still water depth h, slide impact velocity Vs, slide thickness s, bulk slide volume Vs, bulk slide density rs, slide impact angle a, and grain diameter dg, were systematically considered. The wave types were determined by optical wave profile inspection and the criteria of Table 1, and they were controlled for selected runs with video recordings. The main results may be summarized as follows: (1) The wave type classification in Fig. 2 is based on the wave type product T ¼S1/3M cos[(6/7)a] and the slide Froude number F.
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(2)
(3)
(4)
(5)
(6)
(7)
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A data inclusion of the two previous studies corroborates this concept of wave type classification. The dominant parameters for the wave type classification are the slide Froude number F¼ Vs/(gh)1/2, the slide thickness S¼ s/h, and the relative slide mass M ¼ms/(rwbh2). The slide impact angle a affects the wave type with cos[(6/7)a] whereas the effect of the relative grain diameter Dg ¼dg/h is negligibly small. Four wave types and transient types were observed which are described with four nonlinear wave theories, namely the Stokes wave, cnoidal wave, solitary wave, and bore theory. Stokes-like waves result generally from small dimensionless parameters F, S, and M. They consist of several waves with symmetrical wave profiles, are intermediate-water waves, and are characterized by small fluid mass transport. Cnoidal-like waves involve generally medium to large dimensionless parameters F, S, and M. They consist of a longer wave trough than the wave crest, are intermediate-water waves, and are characterized by fluid mass transport. Solitary-like waves follow generally from large dimensionless parameters F, S, and M. They consist of one dominant wave crest, the wave trough is nearly absent, and their fluid mass transport is large. Bore-like waves result generally from large dimensionless parameters F, S, and M. They consist of one dominant wave with a large amount of air at the wave front. They transport a large amount of fluid mass.
wave theory for U-N. However, Huber (1980) observed also a number of transient waves based on method (i). Panizzo et al. (2005) applied both methods (iii) and (iv) for wave basin experiments for the waves propagating on the slide length axis (basin symmetry axis with a wave propagation angle of 01). Based on preliminary wave classifications for 2D in Panizzo et al. (2002, 2005) distinguished between linear, transient, and solitary wave characters whereas their impulse waves were rather linear than solitary.
Appendix B. Nonlinear wave theories Appendix B.1. Stokes wave theory The Stokes wave profile is shown in Fig. 11(b) with at as the distance from the still water surface to the wave trough and a as the wave amplitude. As compared with the sinusoidal wave shown in Fig. 11(a), the Stokes wave profile has steeper crests and flatter troughs whereas it is still symmetrical about a vertical plane (Wiegel, 1964). This theory was described by Stokes (1847) and is based on potential (irrotational) flow with a non-hydrostatic pressure distribution. The analytical description of the wave profile involves a power series based on (H/L), commonly to the fifth order
The wave type classification in Fig. 2 may be simply applied in practice since it depends directly on basic parameters such as the slide parameters or the still water depth which can be estimated prior to an event.
Acknowledgements This research was supported by the Swiss National Science Foundation (grant 200020-103480/1) and the Swiss Federal Office of Energy SFOE (contract 152777). Thanks to Dr. H.M. Fritz, Georgia Institute of Technology, and Dr. A. Zweifel, University of Zurich, for providing their experimental data.
Appendix A. Literature review on wave type classifications A literature review on previous wave type classifications based on methods (i)–(iv) in Section 1.3 is presented hereafter. Wiegel et al. (1970), Noda (1970), Huber (1980), and Fritz (2002) or Fritz et al. (2004), respectively, classified the wave types with method (i). Wiegel et al. (1970) described the impulse waves generated by a vertically falling box as solitary waves, complex solitary wave or a series of cnoidal waves, and bores. Noda (1970) presented graphically a wave type classification containing oscillatory waves, nonlinear transient waves, solitary waves, and bores based on own tests and the experiments of Wiegel et al. (1970). Huber (1980) observed sinusoidal waves, cnoidal waves, solitary waves, and transient waves. Fritz et al. (2004) distinguished between weakly nonlinear oscillatory waves, nonlinear transient waves, solitarylike waves, and dissipative transient bores. Zweifel (2004) or Zweifel et al. (2006), respectively, applied method (ii) to describe strongly nonlinear (0.9oa/Hr1.0), moderately nonlinear (0.6 oa/Hr0.9), and weakly nonlinear (0.4o a/Hr0.6) waves and classified the impulse waves in oscillatory waves, transient waves, solitary waves, and bores. Based on method (iii), Huber (1980) suggested the linear wave theory for Uo15, cnoidal wave theory for 15oUoN, and solitary
Fig. 11. Wave profiles with basic parameters for (a) sinusoidal wave, (b) Stokes waves, (c) cnoidal wave, (d) solitary wave, and (e) bore.
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(Fenton, 1985). The water particles describe open orbits; the waves are oscillatory with an intermediate character and a small fluid mass transport, therefore. The range of the Stokes waves is defined according to Le Me´haute´ (1976) as 2 rL/hr20. However, Keulegan (1950) suggested for Stokes waves of fifth order the limitation L/ho10, corresponding to an Ursell parameter of roughly Uo10.
Appendix B.2. Cnoidal wave theory The cnoidal wave theory was developed by Korteweg and de Vries (1895) and its wave profile is shown in Fig. 11(c). The wave crest amplitude is larger than the wave trough and the wave trough portion is longer than both the wave crest and the sinusoidal wave trough portions (Fig. 11(a)). The name cnoidal is derived from the Jacobian elliptic function ‘cn’ to describe its profile (Wiegel, 1960). This wave type is derived by assuming a hydrostatic pressure distribution for the first order (Isobe, 1985) and non-hydrostatic pressure distribution for the second order approximation in combination with irrotational flow (Le Me´haute´, 1976). This theory allows periodic waves to exist in shallow-water (Dean and Dalrymple, 2004). The profile has mainly oscillatory character, although fluid mass transport occurs. This wave type is especially appropriate for U425 and according to Keulegan (1950) valid in the range of L/hZ10. The cnoidal wave theory is spanning the range between the solitary wave theory on the one hand (for T-N) and the linear wave theory on the other hand (Wiegel, 1960; Dean and Dalrymple, 2004).
Appendix B.3. Solitary wave theory The solitary wave type consists only of one water surface elevation and no wave trough such that the wave length L is theoretically infinite (Fig. 11(d)). Russell (1837) investigated solitary waves in a hydraulic model and derived some basic features. The theory of the solitary wave was independently derived by Boussinesq (1871) and Rayleigh (1876) by assuming non-hydrostatic pressure distribution and rotational flow. The solitary wave is translative involving a considerable fluid mass transport (Le Me´haute´, 1976). For this wave type the nonlinearity, tending to steepen the wave front, balances dispersion, which tends to spread the wave front. Therefore, its profile is constant without damping along the travel distance in a horizontal channel of constant width b, if boundary layer friction is neglected. According to Laitone (1960), the solitary wave celerity c is c ¼ ½gðhþ aÞ1=2 :
ðB:1Þ
Eq. (B.1) is widely accepted to describe the celerity of subaerial landslide generated impulse waves (e.g. Kamphuis and Bowering, 1972; Huber, 1980; Fritz, 2002; Heller, 2007).
Appendix B.4. Bore theory The tidal bore and roller wave theory is e.g. applied to describe broken waves in the shoaling region (Wiegel, 1964). A bore wave profile shown in Fig. 11(e) typically consists of a steep front and a flat portion behind the wave crest. The water particles move in the bore propagation direction. The system of equations is based on hydrostatic pressure distribution and irrotational flow. A bore corresponds to a very shallow-water wave with fluid mass transport. The wave profile is evaluated with the momentum, the continuity, and the characteristic equations (Le Me´haute´, 1976).
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