In this study, both laboratory and numerical experiments are conducted to investigate stem waves propagating along a vertical wall developed by the incidence of monochromatic waves. The results show the following features: for small-amplitude waves, the wave heights along the wall show a slowly varying undulation. Normalized wave heights perpendicular to the wall show a standing wave pattern. The overall wave pattern in the case of small-amplitude waves shows a typical diffraction pattern around a semi-infinite thin breakwater. As the amplitude of incident waves increases, both the undulation intensity and the asymptotic normalized wave height decrease along the wall. For larger-amplitude waves with smaller angle of incidence, the measured data clearly show stem waves. Numerical simulation results are in good agreement with the results of laboratory experiments. The results of present experiments favorably support the existence and the properties of stem waves found by other researchers using numerical simulations. The characteristics of the stem waves generated by the incidence of monochromatic Stokes waves are compared with those of the Mach stem of solitary waves.

Coastal structures have been increasingly constructed in deep water regions as the size of ships becomes larger. In such deep water regions, a vertical-type structure is preferred to save construction costs. In the case of a vertical structure, stem waves occur when waves propagate obliquely against the structure. Thus, there is a need for careful consideration to secure appropriate free board and stability of caisson blocks.

Based on laboratory experiments of the reflection of a solitary wave
propagating obliquely against a vertical wall, Perroud (1957) reported the
existence of three types of waves when the angle between incident wave ray
and a vertical wall is below 45

While the stem waves generated by the sinusoidal waves have drawn less
attention in recent years, the Mach stem induced by the interaction between
the line solitons in the shallow-waters has continuously attracted the
attention of the researchers. Since the pioneering work of Miles (1977a, b)
on the obliquely interacting solitary waves, the soliton interactions have
been extensively studied. Miles (1977b) developed an analytical solution to
predict the amplification of the stem wave along the wall as a function of
the interaction parameter,

Even though the existence and the properties of stem waves for sinusoidal waves are well known theoretically via numerical simulations (e.g., Yue and Mei, 1980; Yoon and Liu, 1989), they are not yet fully supported by physical experiments. Berger and Kohlhase (1976) conducted hydraulic experiments to show the existence of stem waves for the cases of sinusoidal waves. Their experimental data, however, failed to produce clear stem waves, possibly due to partial reflection from the beach, diffraction from the ends of vertical wall, or insufficient space in the wave basin. Lee et al. (2003), Lee and Yoon (2006) and Lee and Kim (2007) performed laboratory experiments to investigate stem waves for sinusoidal waves and compared the measured waves with the numerical results obtained using a nonlinear parabolic approximation equation model. Their hydraulic experiments demonstrated stem waves for some cases with a relatively large incident wave. However, the stem waves were not clearly developed because of both the narrowness of the wave basin and the reflected waves from the beach. Only four cases of incident wave conditions were tested in their experiment. Thus, the experimental data were not sufficient to investigate the properties of stem waves. Moreover, the numerical results for the cases of large angle of incidence were not highly accurate because of the small-angle parabolic model employed for their numerical simulations. Thus, there is still need to perform a precisely controlled experiment to investigate the existence and the properties of stem waves.

In this study, precisely controlled laboratory experiments are conducted to investigate the characteristics of stem waves developed by the incidence of monochromatic waves. The measured data are compared with numerical simulations and analytical solutions. In the following section, the numerical simulation method and the analytical solution employed in this study are summarized. In Sect. 3, the experimental setup and procedure are briefly presented. In Sect. 4, the measured wave heights are compared with numerically simulated results and analytical solutions. In Sect. 5, the characteristics of the stem waves generated by the incidence of monochromatic Stokes waves are compared with those of the Mach stem of solitary waves. In the final section, the major findings from this study are summarized.

Definition sketch of wave field around a vertical wedge.

Coordinate system for numerical simulations:

In this study, the stem waves that have developed along a vertical wall over
a constant water depth are investigated for the cases of monochromatic waves.
Figure 1 shows a sketch of the wave field around a vertical wedge. The
monochromatic waves are symmetrically incident towards the tip of the wedge.
The

To compare with our experiments, the latest version of REF/DIF, a wide-angle
nonlinear parabolic approximation equation model developed by Kirby et al. (2002), is employed to simulate stem waves. The REF/DIF model can deal with
the refraction–diffraction of Stokes waves of third-order nonlinearity over a
slowly varying depth and current. Due to the use of parabolic formulation the
reflection in the main direction of propagation is forbidden, but not in the
transverse direction. In this study, the water depth is uniform, and no
ambient current is present. With no current and energy dissipation on a
constant water depth and by selecting a (1, 1) Padé approximant in the
model, the governing equation of the REF/DIF model is simplified
as

The third term of Eq. (1) is the correction term obtained by selecting the
(1, 1) Padé approximant for the wide-angle parabolic approximation.
According to Fig. 2 of Kirby (1986) the accuracy of the waves propagating
obliquely to the main direction of propagation, i.e.,

The conventional parabolic approximation equation, i.e., the nonlinear
Schrödinger equation of Yue and Mei (1980), is obtained if this term is
neglected. The last term in Eq. (1) describes the nonlinear
self-interaction of waves. Figure 2 shows the coordinate system for the
present numerical simulation in comparison with that of Yue and Mei (1980).
In the present simulation the incident waves are prescribed obliquely along
the

If the side boundary opposite to the vertical wall is located far from the
wall, the no-flux boundary condition, Eq. (6), can also be used. However, to
save the computational resources the obliquely incident plane wave condition
is prescribed along the side boundary at

Along the outer side no boundary condition is necessary, because Eq. (1) is
a parabolic-type differential equation. The grid size,

For the latter we use the nonlinear parameter,

Chen (1987) developed an analytical solution for the Helmholtz equation in
polar coordinates to solve the combined reflection and diffraction of
monochromatic waves due to a vertical wedge. The analytical solution is given
in a polar coordinate as shown in Fig. 1 as

Experimental facility and wave gauge array.

Experimental wave conditions (

Measuring points in hydraulic experiments.

Hydraulic experiments are carried out in the multidirectional irregular wave
generation basin of the Korea Institute of Construction Technology (see
Fig. 3). The basin used in the laboratory experiments is 42 m long, 36 m
wide, and 1.05 m high. A snake-type wave generator consisting of 60 wave
boards, each with dimensions of 0.5 m in width and 1.1 m in height and
driven by an electronic servo piston, is installed along the 36 m long
bottom wall of the wave basin. Free surface displacements are measured using
0.6 m long capacitance-type wave gauges with a measuring range of

Definition sketch of the experimental setup.

The present experiment and wave conditions of the real-world cases
(after Le Méhauté, 1976). The solid red triangles represent the
incident waves tested in this study and empty blue circles represent the
swell wave conditions. The

Figure 4 shows the configuration of the experimental setup and model
installation. A 30 m long vertical wall is installed along the left lateral
side of the basin in four different orientations. A dissipating gravel beach
with a

The incident wave conditions are summarized in Table 1. The title of each test case is composed of three alphabet characters and a numeric digit. The first letter, M, stands for “monochromatic” waves. The letters S or L represents “shorter” or “longer” waves in terms of period, respectively. The letters S, M, or L represents “small”, “medium”, or “large” waves in terms of wave height, respectively. Finally, the numeric digit represents the angle of incidence.

The wave periods of

In the real world, we can assume the situation where the swell is incident on
a breakwater. Swell waves are the regular longer period waves created by
storms far away from the beach. Swell waves tend to have longer periods than
wind waves. The wave period of swell lies between 10 to 15 s. Breakwaters
are generally constructed at a depth of about 10 to 20 m. If the wave height
is 1 to 3 m, the swell wave conditions can be within the range of Stokes
wave, as shown in Fig. 5. In the figure the empty blue circles represent the
swell wave conditions and the red triangles represent the incident waves
tested in this study. It can be seen that the incident waves tested in this
study belong to the Stokes range. The dispersion effect of the Stokes waves
is much stronger than that of the solitary waves. Thus, the characteristics
of stem waves in this study should be very different from those of the
solitary waves. In Fig. 5, the

In the experiments, wave heights are measured along both the vertical wall
(

Wave pattern in front of a vertical wall
(

Prior to the main experiments the performance of the wave generator was
tested. For this test no vertical wall was placed in the wave basin. After
the initiation of wave generation the time histories of free surface
displacement were recorded at three incident-wave-measuring points, as shown
in Fig. 4. The first part of data with a sufficiently long time is discarded
in evaluating the wave height to avoid the start-up transients, and the wave
height and period are obtained using the zero-upcrossing method. The tests
show that the target waves are well generated, and that the bottom friction
is negligible within the test area of the wave basin. In particular, three
wave gauges aligned in a wave propagation direction with a specified distance
are placed at the incident-wave-measuring point located near the gravel beach
with a

In this study, experiments on the formation of stem waves near a vertical wall are conducted and the measured wave heights are compared with results calculated using both the wide-angle parabolic approximation equation numerical model, REF/DIF, and the analytical solution of Chen (1987). All the figures for the experimental and calculated data are presented in the Supplement to avoid an excess of figures.

Prior to presenting the experimental and numerical results, the definitions of the stem angle and the stem width are discussed. The definition of stem width is rather controversial. Yue and Mei (1980) defined the stem width as the distance from the wall to the edge of the uniform wave amplitude region. However, it is not an easy task to locate the edge of the flat region. Berger and Kohlhase (1976) defined the stem width for the periodic waves as the distance along the stem crest lines from the wall to the first node line of the standing wave pattern, which is easier to identify from the measured data. However, Soomere (2004) obtained the analytical stem length using the KP equation for the obliquely interacting two solitary waves. As pointed out by Li et al. (2011) the crest lines of the stem wave, the incident, and the reflected solitons measured in their experiment are not straight, and they do not meet at a point. In reality, the analytical solutions of the KP equation deviate slightly from the pattern observed in the experiment. Thus, Li et al. (2011) proposed the edge of the Mach stem as the intersection of the linear extensions of the stem and the incident-wave crest lines.

Definition sketch for the stem angle and the stem boundary.

Three-dimensional plots of normalized wave height for

Three-dimensional plots of normalized free surface displacements

For the periodic waves the wave pattern is more complicated because many wave
components are superposed. Thus, the definitions of the stem boundary and the
stem angle should be different from the case of solitary waves. As shown in
Figs. 2a and 7, when the stem waves are fully developed, the stem
boundary is nearly parallel to the first node line. Thus, as suggested by
Berger and Kohlhase (1976), the experimental stem angle

Contour plots of the instantaneous normalized free surface for

Comparison of calculated and measured normalized wave heights along
the wall as a function of nonlinear parameter

Comparison of calculated normalized wave heights along the wall for
various nonlinear parameter values of

This

In this study the stem angle,

The stem width

Figure S1 shows the
comparisons between the measured, numerically simulated,
and analytically calculated wave heights,

Figures S2 and S3 show the comparisons of wave heights

Figure S4 shows normalized wave heights along the vertical wall for the cases
of MSM series (i.e.,

In the results shown in Figs. S5 and S6 the stem waves of uniform wave height
are found under the conditions of

The results from laboratory experiments are in good agreement with those of the results of REF/DIF model. However, the analytical solutions of Chen (1987) do not agree well with the measured data, probably because of nonlinear interactions between incident and reflected waves. The discrepancy between the analytical solution of Chen (1987) and the measured data decreases as the angle of incidence increases. This can be attributed to the decrease in the intensity of nonlinear interactions between incident and reflected waves as the angle of incidence increases.

Figures S7, S8, and S9 show the comparisons of the measured, numerically
simulated, and analytically calculated results for the cases of MSL series
(

Figures S10, S11, and S12 show comparisons between the measured, numerically
simulated, and analytically calculated wave heights

Figure S13 shows normalized wave heights along the vertical wall for the cases
of MLM series (

Figures S16, S17, and S18 show comparisons of the measured, numerically
simulated, and analytically calculated results of MLL series (

Figures 8a and b show the comparison of the three-dimensional plots of
normalized wave height for MLS1 and MLL1 cases, respectively, based on the
numerical results of REF/DIF. For the nonlinear case, the overall amplitudes
are much smaller and the stem waves are developed along the wall, as shown in
Fig. 8b. The stem wave height is nearly constant and the width of the stem
waves tended to increase along the wall. Figure 9a and b present
the comparison of the three-dimensional plots of normalized free surface
displacements,

In conclusion, the results of the laboratory experiments are in good agreement with those of the numerical simulations. However, the analytical solution cannot reproduce the stem waves. In addition, given the same incident angle condition, the stem waves in the cases of MLL series show the largest stem width. Moreover, the widths of the stem waves tend to increase as the nonlinear property of the incident waves increases. This further demonstrates the effect of nonlinearity of incident waves on the development of stem waves as suggested by Yue and Mei (1980) and Yoon and Liu (1989).

Yue and Mei (1980) proposed that a single parameter,

Comparison of calculated and measured stem angle

It is well known that the stem waves are generated by the nonlinear interactions between the incident and the reflected waves. When the angle between the incident and the reflected waves is small and the amplitude of two waves is small but finite, two waves attract each other and form a new wave with a single crest: the so-called stem wave. The amplitude of the stem wave is larger than the incident wave, and that of the reflected wave is smaller. Three waves meet at a point due to both the continuous growth of the crest length of the stem wave and the phase shift of the reflected wave. All the mechanisms observed in the formation of a Mach stem wave for the solitary waves also apply for the monochromatic Stokes waves, but the intensity of nonlinear interaction is weaker than that of solitary waves.

Comparison of amplification ratios,

Yue and Mei (1980) proposed the slope ratio

The characteristics of stem waves developed by monochromatic Stokes waves investigated in this study are compared with those of the solitary waves.

For comparison purposes the amplification ratio,

The interaction parameter

In this study, precisely controlled experiments are conducted to investigate
the existence and the properties of stem waves developed along a vertical
wedge for the cases of monochromatic Stokes waves. Numerical and analytical
solutions are also obtained and compared with the measured data. The results
obtained from this study are summarized as follows.

For small-amplitude waves, the wave height along the wall shows slowly
varying undulations with the average value of

As the amplitude of incident waves increases, the undulation intensity
decreases along the wall. For larger-amplitude waves with smaller angle of
incidence, i.e., larger

Stem waves can develop when the nonlinear parameter

The resonant interactions between the incident and reflected waves predicted for solitary waves are not observed for the periodic Stokes waves. The amplification ratios along the wall do not exceed 2 for the case of Stokes waves, while those can reach 4-fold for the solitary waves.

The existence and the properties of stem waves for sinusoidal waves found theoretically via numerical simulations are favorably supported by the physical experiments conducted in this study. Experimental data obtained in this study can be used as a useful tool to verify nonlinear dispersive wave numerical models.

All figures for the experimental and calculated data are presented in this supplement.

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

This study was performed by a project of “Investigation of large swell waves and rip currents and development of the disaster response system (No. 20140057)” sponsored by the Ministry of Oceans and Fisheries of Korea. Edited by: Victor Shrira Reviewed by: Tarmo Soomere and four anonymous referees