UFL/COEL2002/011
WAVE TRANSFORMATION BY TWODIMENSIONAL
BATHYMETRIC ANOMALIES WITH SLOPED TRANSITIONS
by
Christopher J. Bender
and
Robert G. Dean
Partial funding by:
Bureau of Beaches and Wetland Resources
Florida Department of Environmental Protection
Tallahassee, Florida
2002
Wave Transformation by TwoDimensional
Bathymetric Anomalies with Sloped Transitions
by
Christopher J. Bender
and
Robert G. Dean
Partial funding by:
Bureau of Beaches and Wetland Resources
Florida Department of Environmental Protection
Tallahassee, Florida
Executive Summary
The reflection and transmission of normally incident waves by twodimensional
trenches and shoals of finite width with sloped transitions between the depth changes are
studied. Three methods are developed using linearized potential theory. The step
method is valid in arbitrary water depth and is an extension of the solution of Kirby and
Dalrymple (1983) for asymmetric trenches that allows for sloped transitions to be
approximated by a series of steps. The slope method is an extension of Dean (1964) that
allows trenches and shoals with a linear transition between changes in depth to be
modeled in the shallow water limit. A numerical method is also developed using a
backward space stepping routine commencing from the downwave side of the trench or
shoal to model the wave field for an arbitrary bathymetry in the shallow water limit.
The reflection and transmission coefficients are compared for both symmetric and
asymmetric trenches and shoals with abrupt transitions and sloped transitions. The
sloped transitions cause a reduction in the reflection coefficient as compared to trenches
with abrupt transitions of the same crosssectional area; a reduction that increases as the
waves progress from long waves to shorter period waves. For symmetric bathymetric
anomalies complete transmission is found for certain dimensionless wavelengths, a result
not found for asymmetric trenches. The wave transformation by domains with Gaussian
forms is also investigated with results indicating minimal reflection for waves beyond the
shallow water limit. Comparison between the three methods indicates good agreement in
the shallow water limit for the cases studied.
Several new results were established during the study. The wave field
modification is shown to be independent of the incident wave direction for asymmetric
changes in depth, a result shown by Kreisel (1949) for a single step. For asymmetrical
bathymetric anomalies with the same depth upwave and downwave of the anomaly a zero
reflection coefficient occurs only at klhm = 0. For asymmetrical bathymetric anomalies
with unequal depths upwave and downwave of the anomaly the only k1hl value at which
the reflection coefficient equals zero is that approached asymptotically at deepwater
conditions.
Table of Contents
Executive Summary ...................................................................................................
List of Figures ................................................................................................................iii
1. Introduction......................................................................................................... 1
2. Step M ethod: Formulation and Solution ............................................. ................ 3
2.1. Abrupt Transition ................................................................ .................. 6
2.2. Gradual Transition .................................................................... .................... 8
3. Slope M ethod: Formulation and Solution...............................................................10
3.1. Single Transition ............................................................... ........................... 11
3.2. Trench or Shoal .......................................................... .................................. 13
4. Numerical M ethod: Formulation and Solution .................................... ....... 15
5. Results and Comparisons ......................... . .. ..................... 17
5.1. Arbitrary W ater Depth ..................................................... .......................... 17
5.1.1. Comparison to Previous Results ....................................... 18
5.1.2. Symmetric Trenches and Shoals ....................................... ....... .... 19
5.1.3. Asymmetric Trenches ............. ........ ............................................. ........... 28
5.2. Long W aves............ ......................................... ............................................. 32
5.2.1. Symmetric Trenches and Shoals .................................................... 33
5.2.2. Asymmetric Trenches ...................................................... ..................... 37
6. Summary and Conclusions................................... .............................................. 39
7. Acknowledgements ............................................................. ............................... 40
8. References ............................................................... ............................................ 40
ii
List of Figures
Figure 1: Definition sketch for trench with vertical transitions........................................ 6
Figure 2: Definition sketch for trench with stepped transitions........................................ 8
Figure 3: Definition sketch for linear transition....................................... ............ .. 12
Figure 4: Definition sketch for trench with sloped transitions........................................ 14
Figure 5: Comparison of reflection coefficients from step method and Kirby and
Dalrymple (1983 Table 1) for symmetric trench with abrupt transitions
and normal wave incidence: h3= hi, h2/hl = 3, W/hi = 10............................ 18
Figure 6: Comparison of transmission coefficients from step method and Kirby and
Dalrymple (1983 Table 1) for symmetric trench with abrupt transition and
normal wave incidence: h3 = hi, h2/hl = 3, W/h1 = 10............................. ........ 19
Figure 7: Setup for symmetric trenches with same depth and different bottom widths
and transition slopes. .................................................. ............................ 20
Figure 8: Reflection coefficients versus k1hi for trenches with same depth and
different bottom widths and transition slopes. Only onehalf of the
symmetric trench crosssection is shown.................................... ............. 21
Figure 9: Transmission coefficients versus k1hl for trenches with same depth and
different bottom widths and transition slopes. Only onehalf of the
symmetric trench crosssection is shown.................................... ............. 21
Figure 10: Reflection coefficient versus the number of evanescent modes used for
trenches with same depth and transition slopes of 5000, 1, and 0.1................. 22
Figure 11: Reflection coefficient versus the number of steps for trenches with same
depth and transition slopes of 5000, 1, and 0.1.............................................. 23
Figure 12: Reflection coefficients versus klhl for trenches with same bottom width
and different depths and transition slopes. Only onehalf of the symmetric
trench crosssection is shown................................................. ................ 24
Figure 13: Reflection coefficients versus k1hi for trenches with same top width and
different depths and transition slopes. Only onehalf of the symmetric
trench crosssection is shown................................................. ................ 25
Figure 14: Reflection coefficients versus klhl for trenches with same depth and
bottom width and different transition slopes. Only onehalf of the
symmetric trench crosssection is shown.................................................. 26
Figure 15: Reflection coefficients versus klhl for shoals with same depth and
different top widths and transition slopes. Only onehalf of the symmetric
shoal crosssection is shown.......................................... ......................... 26
Figure 16: Reflection coefficients versus k1ih for Gaussian trench with C1 = 2 m and
C2 = 12 m and ho = 2 m. Only onehalf of the symmetric trench cross
section is show n................................................................................................ 27
Figure 17: Reflection coefficients versus k1hi for Gaussian shoal with C1 = 1 m and
C2 = 8 m and ho = 2 m. Only onehalf of the symmetric shoal crosssection
is show n............................................................................................................ 28
Figure 18: Reflection coefficients versus k1hi for a symmetric abrupt transition
trench, an asymmetric trench with different s, and S2 values and a mirror
image of the asymmetric trench. ................... .. .......................... 29
Figure 19: Reflection coefficients versus k1hi for an asymmetric abrupt transition
trench, an asymmetric trench with different hi and h5 values and sl equal
to S2 and a mirror image of the asymmetric trench with s1 = s2........................30
Figure 20: Reflection coefficients versus klhl for an asymmetric abrupt transition
trench, an asymmetric trench with different hi and h5 values and s, not
equal to s2 and a mirror image of the asymmetric trench with Sl = S2.............. 31
Figure 21: Reflection coefficient versus the space step, dx, for trenches with same
depth and different bottom width and transition slopes. Only onehalf of
the symmetric trench crosssection is shown ............................................. 32
Figure 22: Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with transition slope equal to 5000. Only onehalf of the
symmetric trench crosssection is shown ....................................................... 33
Figure 23: Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with transition slope equal to 1. Only onehalf of the
symmetric trench crosssection is shown......................................... 34
Figure 24: Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with transition slope equal to 0.1. Only onehalf of the
symmetric trench crosssection is shown ................................................ 35
Figure 25: Conservation of energy parameter versus k3h3 for three solution methods
for same depth trench case with transition slope equal to 1. Only onehalf
of the symmetric trench crosssection is shown............................................. 35
Figure 26: Reflection coefficients versus k1hi for step and numerical methods for
Gaussian shoal (ho = 2 m, C1 = 1 m, C2 = 8 m). Only onehalf of the
symmetric shoal crosssection is shown.................................... ............ 36
Figure 27: Reflection coefficients versus k3h3 for step and numerical methods for
Gaussian trench in shallow water (ho = 0.25 m, C1 = 0.2 m, C2 = 3 m).
Only onehalf of the symmetric trench crosssection is shown...................... 37
Figure 28: Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with symmetric abrupt transition trench and asymmetric
trench with unequal transition slopes equal to 1 and 0.1.............................. 38
Figure 29: Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with asymmetric abrupt transition trench and
asymmetric trench with unequal transition slopes equal to 1 and 0.2.............. 39
1. Introduction
There is a long history of the application of analytic methods to investigate wave
field modifications by bathymetric changes. Early efforts centered on the effects on
normally incident long waves of an infinite step, trench, or sill of uniform depth in an
otherwise uniform depth domain. The fluid domain in many of the solutions was divided
into two regions: one region comprising the fluid within the change in bathymetry and
the other region consisting of the remainder of the fluid domain. This technique has been
employed in many studies with a progression from normally incident long waves over a
step to solutions that account for oblique incidence and currents along a trench.
Lamb (1932) was one of the first to develop a long wave approximation for the
reflection and transmission of a normally incident wave at a finite step using the
matching technique. Bartholomeauz (1958) performed a more thorough analysis of the
finite depth step problem and found that the Lamb solution gave correct results for the
reflection and transmission coefficients for lowest order (kh) where k is the wave number
and h is the water depth upwave of the step.
To solve the case of normal wave incidence and arbitrary relative depth over a sill
or a fixed obstacle at the surface, Takano (1960) employed an eigenfunction expansion of
the velocity potentials in each constant depth region and matched them at the region
boundaries. The set of linear integral equations was solved for a truncated series.
Dean (1964) investigated long wave modification by linear transitions in channel
depth and/or width. A solution was defined with planewaves of unknown amplitude and
phase for the incident and reflected waves with the transmitted wave specified. In the
region of linear variation in depth and/or width, both transmitted and reflected wave
forms, were represented by Bessel functions. The solutions were shown to converge to
those of Lamb (1932) for the case of an abrupt transition.
Newman (1965a) studied wave transformation due to normally incident waves on
a single step between regions of finite and infinite water depth with an integralequation
approach. Newman (1965b) examined the propagation of water waves past long
obstacles by constructing a domain with two steps placed "back to back" and applying
the solutions of Newman (1965a). Complete transmission was found for certain water
depth and pit length combinations.
The variational approach was used by Mei and Black (1969) to determine the
scattering of surface waves by submerged and floating bodies. Black et al. (1971)
applied the variational approach to determine the scattering caused by oscillating or fixed
bodies that may be submerged or floating. In Black et al. (1971) both infinitely long
horizontal cylinders of rectangular cross section and vertical cylinders of circular cross
section were studied; rendering the problem two and threedimensional, respectively.
Lassiter (1972) employed the variational approach to study waves normally incident on a
trench where the depths before and after the trench may be different (the asymmetric
case).
The symmetric 2D trench problem was studied by Lee and Ayer (1981), who
utilized a transform method. In this study the fluid domain is divided into two regions
with one uniform depth domain of infinite length overlying a finite rectangular region
representing the trench component below the uniform seabed level. The problem was
solved by matching the normal derivative of the potential function in each region along
the common boundary. Numerical results were compared to those from a boundary
integral method with good agreement and to results from a laboratory experiment, which
support the general trends seen in the numerical results.
Using a boundary integral method, Lee et al. (1981) studied the interaction of
water waves with trenches of irregular shape. The method was compared with good
agreement to results from Lee and Ayer (1981) for a rectangular trench. A case with
"irregular" bathymetry was demonstrated in a plot of the transmission coefficient for a
trapezoidal trench; however the complete dimensions of the trench were not specified.
Miles (1982) solved for the diffraction by a 2D trench for obliquely incident long
waves using a procedure developed by Kreisel (1949) that conformally mapped a domain
containing certain obstacles of finite dimensions into a rectangular strip. To add the
capability of solving for obliquely incident waves, Miles used the variational formulation
of Mei and Black (1969).
The problem of obliquely incident waves over an asymmetric trench was solved
by Kirby and Dalrymple (1983) using a modified form of Takano's method (1960).
Differences in the results of Kirby and Dalrymple and those of Lassiter (1972) were
found in a direct comparison with a boundary integral method solution verifying the
results of Kirby and Dalrymple. Lee and Ayer (1981, see their Figure 2) also
demonstrated differences in their results and those of Lassiter (1972). The Kirby and
Dalrymple (1983) solution was later extended to include the effects of currents flowing
along the trench in Kirby et al. (1987).
The previous studies have all investigated the interaction of water waves with
changes in bathymetry. With the exception of Dean (1964) and Lee et al (1980) the study
domains have featured an abrupt transition. A more realistic representation of natural
trenches and shoals should allow for gradual transitions (sloped sidewalls).
The focus of the present study is the propagation of water waves over a 2D
trench or shoal of more realistic geometry. This will extend the study of Dean (1964)
that investigated wave modification by a sloped step and Lee et al. (1980), which did not
directly address the effect of the transition slope on the reflection and transmission
coefficients. Three solution methods are developed for linear water waves: (1) the step
method, (2) the slope method and (3) a numerical method. The step method is valid in
arbitrary water depth while the slope method and the numerical method are valid only for
shallow water conditions. The step method is an extension of the Takano (1960) solution
as modified by Kirby and Dalrymple (1983) that allows for a trench or shoal with
"stepped" transitions that approximate a specific slope or shape. The slope method is an
extension of the Dean (1964) solution that allows for linear transitions between the
changes in bathymetry for a trench or shoal creating regular or irregular trapezoids. The
numerical method employs a backward spacestepping procedure for arbitrary (but
shallow water) bathymetry with the transmitted wave specified.
2. Step Method: Formulation and Solution
The twodimensional motion of monochromatic, smallamplitude water waves in
an inviscid and irrotational fluid of arbitrary depth is investigated. The waves are
normally incident and propagate in an infinitely long channel containing a two
dimensional obstacle (trench or shoal) of finite width. Details of the fluid domain and the
formulation of the solution vary depending on the case studied: abrupt transition or
gradual transition with the slope approximated by the step method.
The step method is an extension of the Takano (1960) formulation for the
propagation of waves over a rectangular sill. The method of Takano (1960) was
extended in Kirby and Dalrymple (1983) to allow for oblique wave incidence and again
in 1987 by Kirby et al. to include the effects of currents along the trench. In the present
formulation the method of Takano, as formulated in Kirby and Dalrymple (1983), was
generally followed for normal wave incidence.
The solution starts with the definition of a velocity potential:
S(x,z, t)= j (x, z)e' (j =1 J) (2.1)
where j indicates the region, J is the total number of regions (3 for the case of a trench or
sill with an abrupt transition), and a is the angular frequency. The velocity potential
must satisfy the Laplace Equation:
+ 2 z)+= 0 (2.2)
Zx 2 )0XZ)=
the freesurface boundary condition:
a9(x,z)0 (2
+az =0 (2.3)
8z g
and the condition of no flow normal to any solid boundary:
0(x)= 0 (2.4)
On
The velocity potential must also satisfy radiation conditions at large Ixl.
The boundary value problem defined by Eq. (2.2), the boundary conditions of
Eqs. (2.3) and (2.4), and the radiation condition can be solved with a solution in each
region of the form:
S(x,z)= A; cosh[k (h+ + z)] e ik,(x) +s[, C (h + z)]ei(xxi)
n=1
(j=1J), (n=10 o) (2.5)
In the previous equation A1l is the incident wave amplitude coefficient, A1 is the
reflected wave amplitude coefficient and Aj+ is the transmitted wave amplitude
coefficient. The coefficient B is an amplitude function for the evanescent modes,
(n = 1 oo), at the boundaries, which are standing waves that exponentially decay with
distance from the boundary. The values of the wave number for the propagating modes,
kj, are determined from the dispersion relation:
a2 = gk, tanh(kh) (j = 1 J) (2.6)
and the wave numbers for the evanescent modes, Kj,n are found from:
2 = gKt,, tan(K~.,hj) (j =1> J), (n = 1 > co) (2.7)
In each region a complete set of orthogonal equations over the depth is formed by Eqs.
(2.5) to (2.7).
To gain the full solution, matching conditions are applied at each boundary
between adjacent regions. The matching conditions ensure continuity of pressure:
#. = O,+ (x = xJ ) ( = 1  J 1) (2.8)
and continuity of horizontal velocity normal to the vertical boundaries
0A= ( = j= 1 > J 1) (2.9)
ax ax
The matching conditions are applied over the vertical plane between the two regions:
(h, < z 0) if h, < h, or ( h., z <0 ) if h, > h+.,.
In order to form a solution, one wave form in the domain must be specified,
usually the incident or the transmitted wave. Knowing the value of the incident, reflected
and transmitted wave amplitudes, the reflection and transmission coefficients can be
calculated from:
K, = a (2.10)
a,
a, cosh(k,h, )
KT= a cosh(kjh) (2.11)
a, cosh(k k)
where the cosh terms account for the change in depth at the upwave and downwave ends
of the trench/shoal for the asymmetric case. A convenient check of the solution is to
apply conservation of energy considerations:
K + a, nJ kI = 1 (2.12)
a, nIk,j
where nj is the ratio of the group velocity to the wave celerity,
n = 1+ s hj (2.13)
j 2 sinh(2k h.)
2.1. Abrupt Transition
The solution of Takano (1960) for an elevated sill and that of Kirby and
Dalrymple (1983) for a trench are valid for abrupt transitions (vertical walls) between the
regions of different depth. For these cases the domain is divided into three regions (J = 3)
and the matching conditions are applied over the two boundaries between the regions.
The definition sketch for the case of a trench with vertical transitions is shown in Figure 1
where W is the width of the trench.
rZ l 
1R xr Op
X XI X2
hi h2; h3
Region 1 iRegion 2 Region 3
.. W L,
Figure 1: Definition sketch for trench with vertical transitions.
Takano constructed a solution to the elevated sill problem by applying the
matching conditions [Eqs. (2.8) and (2.9)] for a truncated series (n = 1> N) of
eigenfunction expansions of the form in Eq. 2.5. Applying the matching conditions
results in a truncated set of independent integral equations each of which is multiplied by
the appropriate eigenfunction; cosh[kj(hj+z)] or cos[Kj,n(hj+z)]. The proper eigenfunction
to use depends on whether the boundary results in a "step down" or a "step up"; thereby
making the form of the solution for an elevated sill different than that of a trench. With
one wave form specified, the orthogonal properties of the eigenfunctions result in 4N+4
unknown coefficients and a closed problem.
By applying the matching conditions at the boundary between Regions 1 and 2
(x = xl), 2N+2 integral equations are constructed. For the case of a trench with vertical
transitions (Figure 1) the resulting equations are of the form:
0 0
J (x,z)cosh[kl(h, +z)]dz = Jq2(x,,z)cosh[kli(hi +z)dz (2.14)
A, hi
0 0
fj (xi, z) cos[A,,. (h, + z)iz = 2 (x,, z) cos[ (h, + z)lz (n = N) (2.15)
hi hl
So ,z)cosh[(h2 +z)]z = (x,)cosh[k(h2 + z)]
ix iax
hl h (2.16)
= (x,,z)cosh[k2(h.+z)]dz
0 ax
S ,(x z)cos[ (h2 + z)z = J (x, z) cos[t2, (h2 + z)
hi hx (n= 1 N) (2.17)
ax
The limits of integration for the right hand side in Eqs. (2.16) and (2.17) are shifted from
hi to h2 as there is no contribution to the horizontal velocity for ( h2 < z < h,) at x =
xt and ( h2 < z < h3) at x = x2, for this case. In Eqs. (2.14) and (2.15) the limits of
integration for the pressure considerations are (h, 5 z <0) at x = xl and ( h3 z <0) at
X =X2.
At the boundary between Regions 2 and 3 the remaining 2N+2 equations are
developed. For the case of a trench the downwave boundary is a "step up", which
requires different eigenfunctions to be used and changes the limits of integration from
the case of the "step down" at the upwave boundary [Eqs. (2.14) to (2.17)].
0 0
2(x2, z)cosh[k3(h3 + z)]dz = J03(xz)cosh[k3 (h3 + z)]dz (2.18)
hA h3
0 0
j2(x2,z)cos[K,(h3+z)dz= 3(x2,z)cos[CK3,n(h3+z)}iz (n=1N) (2.19)
h3 /3
S (x2,z)cosh[k,(h2 + z)z = (xz)cosh[(h +z) (2.20)
ax ax
h2 ax h3 ax
S(x, z) cos[12(h2 + z)z = (x,,z)cos[K,(h2 + z)}z (n = 1 N) (2.21)
h h3
At each boundary the appropriate evanescent mode contributions from the other
boundary must be included in the matching conditions. The resulting set of simultaneous
equations may be solved as a linear matrix equation. The value of N (number of non
propagating modes) must be large enough to ensure convergence of the solution. Kirby
and Dalrymple (1983) found that N = 16 provided adequate convergence for most values
of klhl.
2.2. Gradual Transition
The step method is an extension of the work by Takano (1960) and Kirby and
Dalrymple (1983) that allows for a domain with a trench or sill with gradual transitions
(sloped sidewalls) between regions. Instead of having a "step down" and then "step up"
as in the Kirby and Dalrymple solution for a trench, or the reverse for Takano's solution
for an elevated sill, in the step method a series of steps either up or down are connected
by a constant depth region followed by a series of steps in the other direction. A sketch
of a domain with a stepped trench is shown in Figure 2. In this method, as in the case of
a trench or a sill, a domain with J regions will contain J1 steps and boundaries.
z l T1 
X Xi X2 X3 X4 X,
hi h2 h3 h4 I h h
I !
I4 r
I Region 1 (R1) R 2 j R I Region 4 R5 .gion 6
Figure 2: Definition sketch for trench with stepped transitions.
Each region will have a specified depth and each boundary between regions will have a
specified x location where the matching conditions must be applied.
At each boundary the matching conditions are applied and depend on whether the
boundary is a "step up" or a "step down." With the incident wave specified, a set of
equations with 2(J1)N+2(J1) unknown coefficients is formed.
The resulting integral equations are of the form: for (j = 1 > J 1)
if ( h, > h.+ ) at x = xj then the boundary is a "step down";
0 0
J (xi, z) cosh [k (h + z)z= fJ+1 (x, z) cosh [k (hj + z)}z (2.22)
hj h1
0 0
fJ,(x,,z)cos[r,,,(h,+ z)z= j+(x,,z)cos[,,(h ,+ z)i (n=l>N) (2.23)
hi hi
J (xy,z)cosh[k j+(h+ + z) = Yi(xJ,z)cosh[k+,(hj, + z)lz (2.24)
hi hj+l
J ,(x, z)cos[.,, (h+ +z)}z= i j (xJZ)cos[KJ+,+(h+, +z)J z
hi hJ+I
(n =1 N) (2.25)
if ( hj < h+, ) at x = xj then the boundary is a "step up";
0 0
o 0
SO(x., z)cos yh[Ki,(h,+, + z)z= fJi +(x,z)cos [+,(h+l( + z)}z (n 1> N)(2.27)
hijl h +1
hJI ha+l
0J i^,z)cosh[k(h z)}z J a+1(x,,z)cosh[kj (hj+z)}z (2.28)
Sj 9 j+1
S(xj,,z)cos[,,(h +z)}hz= J ^0l(xj, Z)cos[,,K(hj +z)z (n=1+N)(2.29)
hi hj+l
At each boundary (xj) the appropriate evanescent mode contributions from the adjacent
boundaries (xj1, xj+i) must be included in the matching conditions. The resulting set of
simultaneous equations is solved as a linear matrix equation with the value of N large
enough to ensure convergence of the solution.
3. Slope Method: Formulation and Solution
The slope method is an extension of the analytic solution by Dean (1964) for long
wave modification by linear transitions. Linear transitions in the channel width, depth,
and both width and depth were studied. The solution of Dean (1964) is valid for one
linear transition in depth and/or width, which in the case of a change in depth allowed for
an infinite step, either up or down, to be studied. In the slope method a domain with two
linear transitions allows the study of obstacles of finite width with sloped transitions.
The long wave formulation of Dean (1964) for a linear transition in depth was
followed. By combining the equations of continuity and motion the governing equation
of the water surface for long wave motion in a channel of variable crosssection can be
developed. The continuity equation is a conservation of mass statement requiring that the
net influx of fluid into a region during a time, At, must be equal to a related rise in the
water surface, ri. For a channel of uniform width, b, this can be expressed as:
[Q(x) Q(x + Ax)]At = bAx[r(t + At) 7(t)] (3.1)
where Q(x) and Q(x+Ax) are the volume rates of flow into and out of the control volume,
respectively. The volume flow rate for the uniform channel can be expressed as the
product of the cross sectional width, A, and the horizontal velocity, u, in the channel:
Q= Au (3.2)
By substituting Eq. (3.2) into Eq. (3.1) and expanding the appropriate terms in their
Taylor series while neglecting higher order terms, Eq (3.1) can be rewritten as:
(Au) =bb (3.3)
ax at
The hydrostatic pressure equation is combined with the linearized form of Euler's
equation of motion to develop the equation of motion for small amplitude, long waves.
The pressure field, p(x,y,t), for the hydrostatic conditions under long waves is:
p(x,z,t)= pg[r(x,t) z] (3.4)
Euler's equation of motion in the x direction for no body forces and linearized motion is:
S= (3.5)
p ax at
The equation of motion for small amplitude, long waves follows from combining Eqs.
(3.4) and (3.5):
g (3.6)
ax at
The governing equation is developed by differentiating the continuity equation [Eq. (3.3)]
with respect to t:
aa 8 al. a aulb
[(Au)=b L  A =ba2 (3.7)
at 9x at 9x at t
and inserting the equation of motion [Eq. (3.6)] into the resulting equation, Eq. (3.7)
yields the result:
g All = ba (3.8)
ax L ax dat
Eq. (3.8) is valid for any small amplitude, long wave form and expresses rj as a function
of distance and time. Eq. (3.8) can be further simplified under the assumption of simple
harmonic motion:
q(x, t) = qr (x)ei(a+a) (3.9)
where a is the phase angle. Eq. (3.8) can now be written as:
g [bh a + U] 2 0 (3.10)
b ax ax x
where the subscript ri(x) has been dropped and the substitution, A = bh, was made.
3.1. Single Transition
The case of a channel of uniform width with an infinitely long step either up or
down was a specific case solved in Dean (1964). The definition sketch for a "step down"
is shown in Figure 3. The three regions in Figure 3 have the following depths:
Region 1, x < x; h= h (3.11)
x
Region 2, x1
x2
Region 3, x > x2; h=h3 (3.13)
Region 1 Region 2 Region 3
L:  7 
hi 2, hi hI
Figure 3: Definition sketch for linear transition.
For the regions of uniform depth, Eq. (3.10) simplifies to:
gh + a'2r= 0 (3.14)
8x
a2r2
which has the solution for ir of cos(kx) and sin(kx) where k = and X is the wave
length. The most general solution of r(x,t) from Eq. (3.14) is:
rq(x,t)= B cos(kx t+a )+ Bcos(kx + t+a2) (3.15)
The wave form of Eq. (3.15) consists of two progressive waves of unknown amplitude
and phase: an incident wave traveling in the positive x direction and a reflected wave
traveling in the negative x direction.
For the region of linearly varying depth, Eq. (3.12) is inserted in Eq. (3.10)
resulting in a Bessel equation of zero order:
x a + L P = 0 (3.16)
ax ax
where
P = 2 (3.17)
ghA
The solutions of rl(x) for Eq. (3.16) are:
i(x)= Jo(2pl12x112) and Yo(2/12x"2) (3.18)
where Jo and Yo are zeroorder Bessel functions of the first and second kind, respectively.
From Eq. (3.18) the solutions for rl(x,t) in Region 2 follow:
q(x, t) = B3Jo (2lli2xl/2) cos(at + a3) + B3Y (2fl'/2x112) sin(at + a))
(3.19)
+ B4Jo (2pfl/2x2)cos(at + a4) B4Y0(2fl'/2x'/2) sin(at + a4)
The wave system of Eq (3.19) consists of two waves of unknown amplitude and phase;
one wave propagating in the positive x direction (B3) and the other in the negative x
direction (B4).
The problem described by Figure 3 and Eqs. (3.15) and (3.19) contains eight
unknowns: B14 and al,4. Solution to the problem is obtained by applying matching
conditions at the two boundaries between the three regions. The conditions match the
water surface and the gradient of the water surface:
77j = ljl atx = xj (j = 1,2) (3.20)
= atx=xj (j = 1,2) (3.21)
ox ax
Eqs. (3.20) and (3.21) result in eight equations (four complex equations), four from
setting ot = 0 and four from setting t = , which can be solved for the eight unknowns
as a linear matrix equation.
3.2. Trench or Shoal
The slope method is an extension of the Dean (1964) solution that allows for a
domain with a trench or a sill with sloped transitions. Two linear transitions are
connected by a constant depth region by placing two solutions from Dean (1964) "back to
back." A trench/sill with sloped side walls can be formed by placing a "step down"
upwave/downwave of a "step up." The definition sketch for the case of a trench is shown
in Figure 4.
In the slope method the depths are defined as follows:
Region 1, x < x,; h = h (3.22)
Region2, x,
Region 3, x2 < x < x3; h= h (3.24)
Region 4, x3 < x < x4; h = h3 s2(x3) (3.25)
Region 5, x > x4; h = h (3.26)
where hi, h3, hs, si, s2, and W are specified. With the new definition for the depth in
02
regions 2 and 4, the definition of the coefficient P in Eq. (3.16) changes to / = and
gsi
2
S= in regions 2 and 4, respectively.
gs2
. W 
Figure 4: Definition sketch for trench with sloped transitions.
The matching conditions of Eqs. (3.20) and (3.21) are applied at the four boundaries
between the regions. With the transmitted wave specified and by setting at = 0 and
at = for each matching condition a set of 16 independent equations is developed.
2
Using standard matrix techniques the eight unknown amplitudes and eight unknown
phases can then be determined.
The reflection and transmission coefficients can be determined from
KR =aR and K, = a (3.27), (3.28)
a, a,
Conservation of energy arguments in the shallow water region require:
K + K 2F 1 (3.29)
This method can be extended to the representation of long wave interaction with any
depth transition form represented by a series of line segments.
4. Numerical Method: Formulation and Solution
A numerical method was developed to determine the long wave transformation
caused by a trench or shoal of arbitrary, but shallow water bathymetry. A transmitted
wave form in a region of constant depth downwave of the depth anomaly is the specified
input to the problem. Numerical methods are used to space step the wave form
backwards over the trench or shoal and then into a region of constant depth upwave of the
depth anomaly where two wave forms exist; an incident wave and a reflected wave.
As in the long wave solution of Section 3, the continuity equation and the
equation of motion are employed to develop the governing equation for the problem.
The continuity equation and the equation of motion in the x direction are written in a
slightly different form than in Eqs. (3.3) and (3.6) of Section 3:
a = aq (4.1)
at ax
gh7 = x (4.2)
ax at
Taking the derivative of Eq. (4.1) with respect to t and the derivative of Eq. (4.2) with
respect to x results in the governing equation for this method:
82r I 2rg dh ar=
gh2 g O (4.3)
8t2 a2 dx 8x
where the depth, h, is a function of x and 1 may be written as a function of x and t:
r = r(x)ei' (4.4)
Inserting the form of rI in Eq. (4.4) into the governing equation of Eq. (4.3) casts the
equation in a different form:
gh ) a21(x) g dh l(x)0 (45)
ax2 a2 dx ax
Central differences are used to perform the backward space stepping of the
numerical method.
SF(x + Ax) 2F(x)+ F(x Ax) (4.6)
Ax2
F(x) = F(x+Ax) F(xAx) (4.7)
2Ax
Inserting the forms of the central differences into Eq. (4.5) for 71 results in:
(x) +gh [rl(x+Ax)2rl(x) + (xr Ax) g dh [7(x+Ax) (xAx)]= (4.8)
a(x) + =AX2 I 2(8
For the backward space stepping calculation, Eq. (4.8) can be rearranged:
7(xA g dhA) + + (x) g1 2gh1
gh g dh 1
o 2'Ax2 o2 dx 2Ax
To initiate the calculation, values of rl(x) and r(x+Ax) must be specified in the constant
depth region downwave from the depth anomaly. If the starting point of the calculation is
taken as x = 0 then the initial values may be written as:
q7(0) = H (4.10)
2
qr(Ax) = [cos(kAx) i sin(kAx)] (4.11)
2
The solution upwave of the depth anomaly comprises of an incident and reflected
wave. The form of the incident and reflected waves are specified as:
t71 = cos(kx ot s,) (4.12)
2cos(kx+
r7R =HR cos(kx + ot cR) (4.13)
2
where the s's are arbitrary phases. At each location upwave of the depth anomaly the
total water surface elevation will be the sum of the two individual components:
Hr
q, = r, + r = H cos(kx e,) cos(ot) + sin(kx e,) sin(ot)
2
HR
+ R cos(kx eR) cos(ot) sin(kx 6R) sin(ot)
2
=cos(ot) H cos(kx ,)+ Hcos(kx ER
2 2R
+ sin(o) sin(kx e) ' sin(kx 6R
2)1H
2 2
= I2 +2cos Ot *) e'=taf(n (4.14)
Using several trigonometric identities, Eq. (4.14) can be reduced further to the form:
= T H + H+2 HHIHR cos(2kx ESR c)COS(ot e) (4.15)
which is found to have maximum and minimum values of:
rTax = (H, +H,) (4.16)
7Tr. = H(H HR) (4.17)
Eqs. (4.16) and (4.17) are used to determine the values of HI and HR upwave of the
trench/shoal, and allow calculation of the reflection and transmission coefficients.
5. Results and Comparisons
5.1. Arbitrary Water Depth
The step method was used to study wave transformation by 2D trenches and
shoals in arbitrary water depth. The focus of the study was trenches or shoals with sloped
transitions, which for the step method was performed by approximating a uniform slope
as a series of steps of equal size. For shallow water conditions the results of the step
method were compared to the results of the slope method and the numerical method,
which are longwave models.
5.1.1. Comparison to Previous Results
A comparison to results from Kirby and Dalrymple (1983) was made to verify the
step method (Figures 5 and 6). In these figures the magnitudes (i.e. the phases are not
reported) of the reflection coefficient, KR, and magnitudes of the transmission coefficient,
KT, are plotted versus the dimensionless wave number, klhl, using the step method for
the case of a symmetric trench with vertical transitions and normal wave incidence. Also
plotted are the values (shown as "*") of Kr and Kt for three different klhj, values taken
from Table 1 in Kirby and Dalrymple (1983) with good agreement between the results.
0.5 ..ii
0.45 (*) Kirby and Dalrymple (1983) Table 1
0.4
0.35
0.3
v 0.25
0.2
0.15
0.1
0.05
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
kh,
Figure 5: Comparison of reflection coefficients from step method and Kirby
and Dalrymple (1983 Table 1) for symmetric trench with abrupt transitions
and normal wave incidence: h3 = hi, h2/hi = 3, W/h1 = 10.
The results in the figure include 16 evanescent modes, an amount which was found by
Kirby and Dalrymple (1983) to provide adequate convergence for most values of klhl.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
kh,
Figure 6: Comparison of transmission coefficients from step method and
Kirby and Dalrymple (1983 Table 1) for symmetric trench with abrupt
transition and normal wave incidence: h3 = hi, h2/hl = 3, W/h1 = 10.
A comparison to the results of Lee et al. (1981) was also carried out. Good
agreement was found between the results of the step method and those from Lee et al. for
several cases of a trench with abrupt transitions. A direct comparison between the results
of the step method and data from the one case in Lee et al. (1981) for a trench with
gradual transitions was not made as the complete dimensions of the trench were not
specified in the article and were not able to be obtained at present.
5.1.2. Symmetric Trenches and Shoals
To study the effects of sloped transitions on the wave transformation, a number of
trench and shoal shapes were examined. For the first component of the analysis, the
crosssectional area was kept constant for several different symmetric trench
configurations. The objective was to investigate wave transformation for various slopes
with a fixed crosssectional trench area in a twodimensional domain.
The first set of trench shapes has the same crosssectional area and depth with
different bottom widths and transition slopes as shown in Figure 7. The transition slopes
range from an extreme value of 5000 that represents an abrupt transition, to a gradual
transition with a slope of 0.1. In this figure of the bathymetry, and those that follow, the
still water level is at elevation = 0 m. The trench with the abrupt transitions contains one
step while ten equally sized steps approximate the slopes in the other three trenches.
The reflection and transmission coefficients versus the dimensionless wave number, klhl,
are shown in Figures 8 and 9, respectively for the trenches with the same depth. An inset
figure of the trench through the centerline is included in each figure with four different
line types indicating the corresponding trench configurations. The reflection coefficients
oscillate with decreasing peaks as klhl increases and with klhl values of complete
transmission (KR = 0). Decreasing the transition slope is seen to reduce the reflection
caused by the trench, especially at larger values of klhl. As the slope is reduced, the
location of the maximum value of KR is shifted and the number of instances where
complete transmission occurs is also seen to increase greatly. The plot of the
transmission coefficients shows the same features as the reflection coefficients with the
effect on the wave field reduced as the slope is decreased. For all four trench
configurations and all values of klhl, conservation of energy requirements were satisfied.
1.5 I
2
(4) (3) (2)' (1)
S3
3.5
i t tIo r" Trench Slope
.. (1) 5000
S(2) 1
4. L,i  (3) 0.2
4.5 1
X Distance (m)
Figure 7: Setup for symmetric trenches with same depth and different
bottom widths and transition slopes.
0 0.5 1 1.5
kh,
Figure 8: Reflection coefficients versus k1hi for trenches with same depth
and different bottom widths and transition slopes. Only onehalf of the
symmetric trench crosssection is shown.
Figure 9: Transmission coefficients versus klhl for trenches with same
depth and different bottom widths and transition slopes. Only onehalf of
the symmetric trench crosssection is shown.
The results shown in Figures 8 and 9 are for 16 evanescent modes taken in the
summation and trenches with 10 steps approximating the sloped transitions. The
influence of the number of evanescent modes taken in the summation on the reflection
coefficient is shown in Figure 10 for the same depth trenches with slopes of 5000, 1, and
0.1 at a specified value of klhi.
0.195
09 Trench (1): Slope = 5000
0 k,h, = 0.95
0.185
0. 5 10 15 20 25 30 35 40
0.242
0.241 Trench (2): Slope = 1
Y 0.24 k1h = 0.7
0.239
0.238 '
0 5 10 15 20 25 30 35 40
0.282 1
0.2815 Trench (4): Slope = 0.1
0.2805
00.28 kth = 01
0 28"'5'  i  i   i    35 
S0 5 10 15 20 25 30 35 40
Number of Evanescent Modes in Summation
Figure 10: Reflection coefficient versus the number of evanescent modes
used for trenches with same depth and transition slopes of 5000, 1, and 0.1.
The reflection coefficient is seen to converge to a near constant value with increasing
number of evanescent modes. A value of 16 modes is found to produce adequate results
and was used for all the step method calculations.
The influence of the number of steps used in the slope approximation is shown in
Figure 11 for the same depth trench configurations. The reflection coefficient is seen to
converge to a steady value with increasing number of steps used for the trenches with
slopes of 1 and 0.1; however for the abrupt transition trench (slope = 5000) convergence
with an increasing number of steps was not found. Convergence for the abrupt transition
was found as the step number decreased to 1, which represents an almost vertical wall.
This could be due to the small distance between points that results when dividing the
nearly vertical wall into an increasing number of segments. For this reason the abrupt
transition trenches were configured with 1 step, resulting in an almost vertical wall and
for all other slopes 10 steps were used. This convention was followed for all the step
method calculations.
0.27
0.26 Trench (1): Slope = 5000
0.25 kh, = 0.95
0.24
0.23'
0 5 10 15 20 25 30
0.3
Trench (2): Slope = 1
p0.295 kh = 0.7
0.29
0 5 10 15 20 25 30
0.285
S0.28 Trench (4): Slope = 0.1
kth, = 0.1
0.275
0 5 10 15 20 25 30
Number of Steps
Figure 11: Reflection coefficient versus the number of steps for trenches
with same depth and transition slopes of 5000, 1, and 0.1.
A second way to maintain a constant crosssectional area is to fix the bottom
width of the trench and allow the trench depth and transition slopes to vary. The
reflection coefficients for four trenches developed in this manner with the same bottom
width are shown in Figure 12. An inset figure is included to show the trench dimensions
through the centerline with slopes of 5000, 1, 0.2 and 0.05 being used. For this case, as
the slope decreases the depth of the trench must also decrease to maintain the fixed value
of the crosssectional area. The maximum reflection coefficient is seen to decrease and
shift towards a smaller value of k1hi as the slope decreases, due to the associated decrease
in the depth of the trench. Generally the depth of the trench determines the magnitude of
KR and the trench width determines the location of the maximum value of KR.
0 0.5 1 1.5
k1h1
Figure 12: Reflection coefficients versus k1hi for trenches with same
bottom width and different depths and transition slopes. Only onehalf of
the symmetric trench crosssection is shown.
Keeping the trench crosssectional area and top width fixed while allowing the
transition slopes and depth to change, results in another series of trench configurations.
The reflection coefficients for four trenches with the same crosssectional area and top
width are shown in Figure 13. The slopes of the transitions are 5000, 5, 2, and 1. The
trench with the smallest slope is found to produce the largest value of Kr due to the large
depth associated with that slope; however the abrupt transition results in the largest KR
values after the first maximum. For this case decreasing the slope does not shift the
locations of the maximum values of KR as much as in the same depth and same bottom
width cases, although no slopes less than one were used as they do not satisfy the
constraints of the domain. These three cases have demonstrated that trenches with the
same crosssectional area can have very different reflective properties depending on the
trench configuration.
Figure 13: Reflection coefficients versus k1hi for trenches with same top
width and different depths and transition slopes. Only onehalf of the
symmetric trench crosssection is shown.
The effect of the transition slope can be viewed in another manner when the
crosssectional area of the trench is not fixed. For a fixed depth and bottom width,
decreasing the transition slope results in a trench with a larger crosssectional area. Plots
of the reflection coefficients versus k1hi for four trenches with the same depth and bottom
width, but slopes of 5000, 0.2, 0.1, and 0.05 are shown in Figure 14. Decreasing the
transition slope is seen to reduce KR, even though the trench crosssectional area may be
much larger.
The step method is also valid for the case of a submerged shoal with sloped
transitions approximated by a series of steps. Figure 15 shows the reflection coefficients
versus klhl for four different shoal configurations with transition slopes of 5000, 0.5, 0.2
and 0.05. Decreasing the transition slope is found to reduce the value of KR as was the
case for the trench, however the reduction in KR is not as significant for large klhl as in
the trench cases.
o0.15
0 0.5 1 1.5
kh,
Figure 14: Reflection coefficients versus klhl for trenches with same depth
and bottom width and different transition slopes. Only onehalf of the
symmetric trench crosssection is shown.
Figure 15: Reflection coefficients versus klhl for shoals with same depth
and different top widths and transition slopes. Only onehalf of the
symmetric shoal crosssection is shown.
A uniform transition slope, approximated by a series of uniform steps, was used
in the previous cases for the wave transformation caused by symmetric trenches and
shoals. In the following cases the wave transformation by depth anomalies with variable
transition slopes is investigated. The domain was created by inserting a Gaussian form
for the bottom depth into an otherwise uniform depth region. The equation for the
Gaussian shape centered at Xo was:
(xx,)2
h(x)= h+Ce 2C2 (5.1)
where ho is the water depth in the uniform depth region and CI and C2 are shape
parameters with dimensions of length. To implement the step method, steps were placed
at a fixed value for the change in depth to approximate the Gaussian shape. Two extra
steps were placed near ho where the slope is very gradual and one extra step was placed
near the peak of the 'bump' to better simulate the Gaussian form in these regions.
The reflection coefficients versus k1hi for a Gaussian trench with C1 and C2 equal
to 2 m and 12 m, respectively, and a uniform water depth, ho, equal to 2 m is shown in
Figure 16.
0.25 2
2.2
2.4
0.2 2.6
E2.8
0 3
0.15
3.2
3.4
0.1 3.6
3.8
4
40 30 20 10 0
0.05 X Distance (m)
0
0 0.5 1 1.5
kh,
Figure 16: Reflection coefficients versus klhl for Gaussian trench with C1 =
2 m and C2 = 12 m and ho = 2 m. Only onehalf of the symmetric trench
crosssection is shown.
An inset figure is included to show the configuration of the stepped trench and the form
of the Gaussian bump. The Gaussian form is approximated by the step method with a
step at every 0.05 m change in depth. The reflection caused by the Gaussian trench is
seen to differ considerably from that for the trenches with constant transition slopes
shown previously where KR showed significant oscillations that diminished as k1hi
increased, even with gradual transition slopes. The reflection coefficient is seen to have
one peak near klhl equal to 0.1 with minimal reflection for larger values of klhl.
The reflection caused by a Gaussian shoal is shown in Figure 17 for Ci and C2
values of 1 m and 8 m, respectively. The steps approximating the Gaussian form are
placed a 0.05 m depth intervals. As for the case of the Gaussian trench, a single peak in
KR occurs followed by minimal reflection at larger values of klhl.
0.16
0.12
Figure 17: Reflection coefficients versus k1hi for Gaussian shoal with C1 =
1 m and C2 = 8 m and ho = 2 m. Only onehalf of the symmetric shoal cross
section is shown.
5.1.3. Asymmetric Trenches
The trenches considered previously have been symmetric with the upwave and
downwave water depths equal and the same slope for the upwave and downwave
transitions. Figure 18 shows the reflection coefficients versus k1hi for 3 trenches with the
same depth upwave and downwave of the trench: a symmetric trench with an abrupt
transition, an asymmetric trench with a steep (si = 1) upwave transition and gradual (s2 =
0.1) transition and a mirror image of the asymmetric trench with a gradual (sl = 0.1)
upwave transition and steep (s2 = 1) downwave transition. The trench configurations are
shown in the inset diagram with dashed and dotted lines. The order of the transition
slopes (steep slope first versus gradual slope first) is seen to have no effect on the value
of KR; therefore only one dotted line is plotted in the KR versus k1hi plot. Stated another
way, the direction of the incident wave (positive x direction versus negative x direction)
does not matter for this case. For the asymmetric trenches with the different transition
slopes, complete transmission only occurs at k1hi = 0, which is different from the case for
the abrupt transitions and the previous cases where s, equals s2.
0.35
0.25 10 20 30 40 50 '
1 l X Distance (m)
Wave
2
0.1 %2.5
iA
C I
0 3
0.25 3.5
04 
0 0.5 1 1.5
kh.
Figure 18: Reflection coefficients versus k2hl for a symmetric abrupt
transition trench, an asymmetric trench with different s, and s2 values and a
mirror image of the asymmetric trench.
Asymmetric trenches can also be studied with different depths upwave and
downwave of the trench. Figure 19 plots KR versus klhl for domains with different
depths upwave (hj) and downwave (hj) of the trench. The case of an abrupt transition
(solid line) with the downwave depth greater than the upwave depth can be compared to
the case with transition slopes of 0.2 (dashed line) with the same crosssectional area.
Complete transmission does not occur for either the trench with the abrupt transition or
the trench with sloped transitions. The trench with sloped transitions is found to cause
less reflection with an irregular oscillation in KR with increasing k1hi as compared to the
abrupt transition.
0.3 1.5
* Wave
2 "
0.25 92.5 .
C I
0.2 
4
4.5
10 20 30 40 50 60
p0.15 X Distance (m)
0.1 
o / 'I I /
0. i
Oo "i
0 0.5 1 1.5
kh,
Figure 19: Reflection coefficients versus klhl for an asymmetric abrupt
transition trench, an asymmetric trench with different hi and hs values and
sl equal to s2 and a mirror image of the asymmetric trench with SI = Sz.
For this asymmetric case KR does not approach zero as khil approaches zero as was
found when hi was equal to hj, but approaches the long wave value for a single abrupt
step (either up or down) of elevation Ihl hjI. Also plotted (dotted line) are the results for
an asymmetric trench with the depth upwave of the trench greater than the depth
downwave of the trench with transition slopes of 0.2, which is a mirror image of the
trench shown with the dashed line. The reflection coefficients for this case are identical
in magnitude to (but differ in phase from) the mirror image trench if KR is plotted versus
the shallower constant depth, kjhj; therefore only one dashed line for KR is plotted. This
result is another case of the incident wave direction not affecting the magnitude of the
reflection coefficients for an asymmetric trench as shown in Figure 18.
Another asymmetric trench case is shown in Figure 20 where the upwave and
downwave depths and transition slopes are different with the upwave depth being larger.
In this figure two asymmetric trenches with the same crosssectional area are compared: a
trench with an abrupt transition (solid line) and a trench with an upwave transition slope
(si) equal to 1 and a downwave slope (s2) of 0.2 (dashed line). The sloped transitions are
seen to reduce significantly the value of KR, especially as klhl increases.
0.4
2 Wave
0.35 .
3
0.3 t .
5
0.25 / 20 40 60
X Distance (m)
0.15
0.1 v "
V I
% 'It
0 0.5 1 1.5
k1hi
Figure 20: Reflection coefficients versus klhi for an asymmetric abrupt
transition trench, an asymmetric trench with different hi and hs values and
sl not equal to S2 and a mirror image of the asymmetric trench with si = s2.
As shown in the previous figure, KR does not approach zero as klhi approaches zero for
this case where the upwave and downwave depths are not equal. Also plotted is a mirror
image of the asymmetric trench with the sloped transitions (dotted line).Only one dashed
line for KR is plotted, as the reflection coefficients for this case are identical to that for the
mirror image trench if KR is plotted versus the deeper constant depth, kshs. This result is
another example of the independence of KR to the incident wave direction as shown in
Figures 18 and 19.
5.2. Long Waves
Direct comparison of the step method to the slope and numerical methods can
only be made in the long wave region. While the slope method and numerical method
are limited by the long wave restrictions, it is in this region that changes in bathymetry
are most effective at modifying the wave field.
The slope method does not contain any variables that will affect the accuracy of
the results such as the number of evanescent modes or the number of steps as in the last
section. The numerical method however is sensitive to the spacing between points used
in the backward space stepping procedure described in Section 4. Figure 21 shows the
value of KR versus the step spacing, dx, using the numerical method for three trench
configurations at a single value of klhl for the three of the same depth cases (Figure 7)
presented earlier. The reflection coefficient is seen to converge for dx < 1 m and a value
of dx = 0.5 m was found to be adequate for the numerical model.
0.36 1.5 '
2
0.355 
_2.5 kh, =0.142
0.35 3
3.5. ... ..
0.345 4
4.5 1......
0.34 0 10 20 ...... k,h,= 0.1355
X Distance (m) ...............
0.335 ...... .........
aa'2G... ............... ..
0.33
0.325
kh,= 0.1293
0.32 
0.315 I
0 1 2 3 4 5 7 8 9 10
dx (m)
Figure 21: Reflection coefficient versus the space step, dx, for trenches with
same depth and different bottom width and transition slopes. Only onehalf
of the symmetric trench crosssection is shown.
5.2.1. Symmetric Trenches and Shoals
To compare the three solution methods, the same depth trench configurations
shown in Figure 7 were used. In order to maintain long wave conditions throughout the
domain the limit of k3h3 < 9n/10 was taken with the subscript 3 indicating the constant
depth region inside the trench between the transitions. Figure 22 shows KR versus k3h3
for the three solution methods for the same depth trench case with a slope of 5000. The
slope method and the numerical method are found to yield nearly the same value of KR at
each point with the peak value occurring at a slightly smaller k3h3 than for the step
method. Overall, the agreement between the three models is found to be very good for
this case.
0.35
03
0.25
(solid) = step method
(dash) = slope method
0.2 (o) = numerical method
*; 1.5
S25
0.15 
S3.5
4
0.05 4.5
0 10 20
X Distance (m)
0 0.05 0.1 0.15 0.2 0.25 0.3
k3h3
Figure 22: Reflection coefficients versus k3h3 for three solution methods for
same depth trench case with transition slope equal to 5000. Only onehalf
of the symmetric trench crosssection is shown.
Figures 23 and 24 show KR versus k3h3 for a slope of 1 and 0.1, respectively for
the same depth trench case with all three models results plotted. Good agreement is
found for the results of the three models. In Figure 24 the numerical model results match
those of the slope method up to the peak value for KR, but as k3h3 increases after the peak
the numerical results match those of the step method and the slope method results diverge
slightly. The similarity in the results of Figures 22 through 24 provides reasonable
verification for the three solution methods.
While the conservation of energy requirement is satisfied exactly in the step
method and the slope method, for the cases studied, the numerical method is found to
have some variance in the conservation of energy requirement for the space step used in
Figures 22 through 24. Figure 25 shows the conservation of energy parameter versus
k3h3 for the same depth trench with a slope of 1. As noted, the step method and the slope
method have values exactly equal to 1, while the numerical method results are equal to
one for small k3h3 with increasing deviation from one, although very small, as k3h3
increases.
0.35
0.25 
(solid) = step method
S(dash) = slope method
0.2 / (o) = numerical method
1.5
2
0.15
.0I 3
0.1 3.
3.0.05
4.
0.05 4.5
0 10 20
X Distance (m)
0 0.05 0.1 0.15 0.2 0.25 0.3
k3h3
Figure 23: Reflection coefficients versus k3h3 for three solution methods for
same depth trench case with transition slope equal to 1. Only onehalf of
the symmetric trench crosssection is shown.
Figure 24: Reflection coefficients versus k3h3 for three solution methods for
same depth trench case with transition slope equal to 0.1. Only onehalf of
the symmetric trench crosssection is shown.
1.0001
) 1.5
L 0.9999
2
0 92.5
1C
3 (solid) = step method
S3.5 (x)= slope method
o 0.9998 (o) = numerical method
4
4.5
0 10 20
X Distance (m)
0.9997
0 0.05 0.1 0.15 0.2 0.25 0.3
k3h
Figure 25: Conservation of energy parameter versus k3h3 for three solution
methods for same depth trench case with transition slope equal to 1. Only
onehalf of the symmetric trench crosssection is shown.
4O. 1!
Using the step method the reflection coefficients for Gaussian trenches and shoals
were shown to have a single peak followed by minimal reflection as klhl increased.
Figure 26 shows a comparison of the step method and numerical method results for a
shoal with ho = 2 m, C1 = 1 m, and C2 = 8 m. The results for the step method were shown
previously in Figure 17. The slope method can only contain two slopes and therefore
cannot be employed to model a Gaussian form and is not included in the following
comparisons. The numerical results show good agreement with those of the step method
with a slight divergence at higher k1hl values near the shallow water limit.
0.28 .
0.24 / (solid) step method, dh= 0.1 m
(o) numerical method, dx = 0.5 m
0.2
0.16
/ 1 71
0.12.2
'1.2
1.4 o
0.08 1.6
1 1.8 o
0.04 o o
/0 20 10 0 o
X Distance (m)
0 L'iIiLii
0 0.05 0.1 0.15 0.2 0.25 0.3
Figure 26: Reflection coefficients versus klhl for step and numerical
methods for Gaussian shoal (ho = 2 m, C1 = 1 m, C2 = 8 m). Only onehalf
of the symmetric shoal crosssection is shown.
Figure 27 is a comparison of the results of the step method and the numerical
method for a Gaussian trench in shallow water. The step method has a step for every
0.01 m in elevation and the numerical method has a space step of 0.1 m. The domain is
in very shallow water (ho = 0.25 m, CI = 0.2 m, C2 = 3 m) to extend the numerical
method results into the region of minimal reflection following the initial peak in KR.
The numerical method results agree closely to those of the step method and verify the
small KR values of the step method that occur as k3h3 increases after the initial peak in
KR.
0.25
0.25
0.3
0.2 
0 0.35
0.15 0.4
0.45 
10 8 6 4 2 0
0.1 X Distance (m)
(solid) step method
0.05 (o) numerical method
01 0 10
0 0.05 0.1 0.15 0.2 0.25 0.3
k3h3
Figure 27: Reflection coefficients versus k3h3 for step and numerical
methods for Gaussian trench in shallow water (ho = 0.25 m, C1 = 0.2 m, C2
= 3 m). Only onehalf of the symmetric trench crosssection is shown.
5.2.2. Asymmetric Trenches
The case of a trench with unequal upwave and downwave slopes as shown in
Figure 18 is now compared for the three solution methods. In order to maintain shallow
water conditions throughout the domain a limit of k3h3 < dn10 was taken for the slope
and numerical methods. Figure 28 shows the value of KR versus k3h3 for the same depth
trench with an abrupt transition (solid line) and for si and S2 equal to 1 and 0.1,
respectively. The bottom width for all three cases was maintained the same. The slope
and numerical method results show good agreement with the step method solution for
both cases. Due to the location of the shallow water limit, the increase of KR for the step
method at k3h3 equal to 0.4 cannot be verified directly.
y\ / \ %
or
0.1 
(solid) step method, abrupt transition
( ) step method, st not equal to s2
0.05 (+) numerical method
(o) slope method
0!
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
k3h3
Figure 28: Reflection coefficients versus k3h3 for three solution methods for
same depth trench case with symmetric abrupt transition trench and
asymmetric trench with unequal transition slopes equal to 1 and 0.1.
The case of an asymmetric trench with unequal upwave and downwave slopes, and
depths as shown in Figure 20 is now compared for the three solution methods. For the
slope and numerical methods a limit of k3h3 < 7r/10 was taken to maintain shallow water
conditions. Figure 29 shows the value of KR versus k3h3 for an asymmetric trench with
an abrupt transition (solid line) and for st and s2 equal to 1 and 0.2, respectively (dashed
line). The bottom width was changed to maintain a constant crosssectional area. The
slope and numerical method results show good agreement with the step method solution
for both cases with less divergence in the KR values at larger k3h3 for the sloped transition
trench.
sI
0.15 /
2r
0.1
4 / \I
S/ / (solid) step method, abrupt transition
0.05 ,5 1 ( ) step method, s not equal to s2
(+) numerical method
0 20 40 60 (o) slope method
X Distance (m)
0 iti7i
0 0.1 0.2 0.3 0.4 0.5 0.6
k3h3
Figure 29: Reflection coefficients versus k3h3 for three solution methods for
same depth trench case with asymmetric abrupt transition trench and
asymmetric trench with unequal transition slopes equal to 1 and 0.2.
6. Summary and Conclusions
The interaction of linear water waves with twodimensional trenches and shoals
has been demonstrated using three methods. The three methods show good agreement
for the cases presented for various bathymetric changes with abrupt and gradual
transitions between changes in depth. Gradual transitions in the depth for both symmetric
and asymmetric bathymetric changes are seen to reduce the reflection coefficient,
especially for nonshallow water waves. Linear transitions are shown to result in
instances of complete wave transmission for symmetric trenches and shoals, while for
asymmetric trenches and shoals complete transmission does not occur. Changes in depth
that are Gaussian in form are demonstrated to result in a single peak of the reflection
coefficient in the long wave region, followed by minimal reflection for shorter
wavelengths.
Several new results have been found in this study: (1) The wave field
modifications are shown to be independent of the incident wave direction for asymmetric
changes in depth; a result shown by Kreisel (1949) for a single step, (2) For asymmetrical
bathymetric anomalies with hi = hj, a zero reflection coefficient occurs only at klhl = 0,
and (3) For asymmetrical bathymetric anomalies with hi # hj, the only k1hi value at
which KR = 0 is that approached asymptotically at deepwater conditions.
7. Acknowledgements
An Alumni Fellowship granted by the University of Florida sponsored this study
with partial support from the Bureau of Beaches and Wetland Resources of the State of
Florida.
8. References
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