TRANSFORMATION, BREAKING AND RUN-UP OF A LONG WAVE OF FINITE HEIGHT

On studying the transformation, breaking and run-up of a relatively steep wave of a short period, the theory for waves of permanent type has given us many fruitful results. However, the theory gradually loses its applicability as a wave becomes flat, since a considerable deformation of the wave profile is inevitable in its propagation. 
In § 1, a discussion concerning the transformation of a long wave in a channel of variable section is presented based on the non-linear shallow water theory. Approximate solutions obtained by G. B. Whitham's method (1958) are shown. Further, some brief considerations are given to the effects of bottom friction on wave transformation. 
In § 2, breaking of a long wave is discussed. Breakings on a uniformly sloping beach and on a beach of parabolic profile are considered and the effects of beach profile on breaking are clarified. Finally in § 3, experimental results on wave run-up over l/30 slope are described in comparing with the Kaplan's results.


INTRODUCTION
On studying the transformation, breaking and run-up of a relatively steep wave of a short period, the theory for waves of permanent type has given us many fruitful results.However, the theory gradually loses its applicability as a wave becomes flat, since a considerable deformation of the wave profile is inevitable in its propagation.
In § 1, a discussion concerning the transformation of a long wave in a channel of variable section is presented based on the non-linear shallow water theory.Approximate solutions obtained by G. B. Whitham's method (1958) are shown.Further, some brief considerations are given to the effects of bottom friction on wave transformation.
In § 2, breaking of a long wave is discussed.Breakings on a uniformly sloping beach and on a beach of parabolic profile are considered and the effects of beach profile on breaking are clarified.
Finally in § 3, experimental results on wave run-up over l/30 slope are described in comparing with the Kaplan's results.

TRANSFORMATION OF A LONG WAVE OF FINITE
The characteristic equations to be derived from these hyperbolic equations are where Next, a compressive wave propagating shoreward will be considered.It is well known that a compressive wave continues to deform its profile in its propagation and eventually breaks by the curling of the wave front.And from the physical point of view, the wave is considered to form a bore after breaking.In this meaning the theory of a bore in shoaling water given by H. B. Keller, D. A. Lavine, and G. B. Whitham (i960) would give some informations on the transformation of a deformed long wave.
Suppose that particle velocity and propagation velocity of a wave in shoaling water are related to surface elevation by the following expressions: where Since the above relations are precise solutions for the case of a uniform depth, they must also be approximately applied in shoaling water, provided the change of water depth within the distance between the wave front and the crest is not large.It is of interest to compare the transformation of a wave governed by the above relations with that of a bore, since the above relations differ a little from the bore conditions as are shown in the following relations : where V / is the particle velocity just behind the bore and C / is the propagation velocity of a bore.-arf J5

•rl El
Turning to the details of the relation given by Eq.(l-12), it will be seen that M increases monotonically as h decreases when M y o However, this manner of variation does not exist in the height £ .The maximum of <£ occurs when In the above analysis two interesting features are found, one of them being that the rate of amplification of wave height in decreasing depth decreases as the relative wave height increases and the other being the existence of the maximum of wave height.

2. TRANSFORMATION IN A CHANNEL OF VARIABLE WIDTH.
Suppose a channel has a variable width and a uniform depth and let the dimensionless channel width be given by b -t>(x) • Dimensionless variables are defined as follows : The positive characteristic equations of the conservation equations of mass and momentum, in the dimensionless form, are By the combination of the above two equations, one has When h = / is considered, substitution of Kqs.(l-9) and (l-lO) into Eq.(l-2l)yields By using the same dimensionless variables as in 1.1 the positive characteristic equations of conservation equations of mass and momentum on a uniformly sloping beach, f^ss f-x., can be given by where k = k'/S , mnd S = beach slope. .Substitution of Eqs.(l-9) and (1-10) into the equation which is obtained by combining the above two equations leads to Provided the terms smaller than (fiS-/) can be neglected, Eq.(l-29) approxi-where /V -yT+M Lded the terms SD mately integrates to k ££ A 0 e~4 /S («£+{)* i e*f>(-^6) (1-30-1) where In a special case of oj= 0 » lt=4-the above relation can be reduced to kQZ AoE -* /S "e*/*/-2£ ( ~£ + 2 )} (1-30-2) In the case of a uniform depth the attenuation of wave height will be expressed by Where k'-3 /Cg * and the dimensionless variables are similar to those in 1. 2. Substitution of Eqs.(l-9) and (l-lO) into Eq.(l-3l)gives 2 k' ( VH^T ~ I ) 2 which can be integrated as where A/=s /T+T <thd M> is fi/ at X =* Q As an example, attenuation of long wave under the condition k = 0.0f is illustrated in Figure 5.However, the value of roughness factor, k!~0.of is only an example and experimental studies are necessary to discuss further.

BREAKING DP A LONG WAVE IN SHOALING WATER.
A long wave continues to deform due to the difference of its local propagation velocity and eventually breaks by the curling of its front.From the mathematical point of view, breaking points are expressed by an envelope of intersections of characteristic curves.In the following a wave which has a non-zero slope at the wave front and propagates shoreward into quiescent water is considered.
At first, a uniformly sloping beach with a depth of /i*"*=S(4> -X ) will be considered, here x is the beach slope.It takes a dimensionless form of

k= /
(2-1) Considering a characteristic curve eCx/ett» V-t-c t which starts from the origin at time -6=-t , one has the following equation from the relation given by Eq.(l-6) :

%'•}-£*,**&+ &tf+&*?±-(2 -17)
Since the initial condition is x =. o when -6 -T , the solution of Eq.(2-16) which satisfies the initial condition is assumed to have the form where 4n is a constant to be determined from Eq.(2-16).Substitution of Eq.( 2 of bottom friction is neglected, the conservation equations of mass and momentum in the non-linear shallow-*l , h. , and X are defined in Figure 1, V is velocity, ±* is time, and $* is the gravitational acceleration.The asterisks denote dimensional quantities.The following dimensionless variables are introduced for the sake of simplicity : TRANSFORMATION, BREAKING AND RUN-UP OF A LONG WAVE OF FINITE HEIGHT X m Z*/to* , k -hVhf > 1=1 Vho* where Q* = distance from the origin to the shoreline fa * = water depth at the origin Let the depth be given by h.-hex) , the basic equations are as follows when these dimensionless variables are substituted : Eq.(l-5) into Eq.(l-6)leads to ct( V+ ZC) -cth/(V-hC)*= well known relation for waves of small height.Under the boundary condition •? = 1 0 at b m bo » Eq.(i-22) becomes * \-jT7%) (ffTW 0f ) (1 " 25) The relation between (ty%) and (Me) obtained from Eq.(l-25) is illustrated in Figure 4.The wave transformation in a channel of variable width shows different character from the preceding case of variable depth.The rate of amplification of wave height in converging channel continuously increases as the relative height of wave increases.1. 3. EFFECTS OF BOTTOM FRICTION ON WAVE TRANSFORMATION.The conservation equation of momentum which accounts for the effects of bottom friction is given by v ** + v * V l* = ~?% -k'(v* 2 /h*+ ?*) (1 _ 26 ) where k'-^^/C* 2 and C e *= Che'zy'S toujhness factor

Fig. 6 *
Fig. 6* Breaking time tfc as a function of initial slope m for a uniformly sloping beach.