(V cos 9) 

 F = - ... (3) 



(gd T ) 1/2 



where the Q axis represents values of dimensionless frequency 



(2ir) /d T \ 1/2 

 = -^ (4) 



T \g 

 Each curve is a fixed value of wavelength factor, Rt, where 



_ / V \ _ wavelength with current , . 



L XL/ wavelength without current 



Negative values of F correspond to waves with a component of phase velocity 

 in the direction opposite to that of the current. The phase velocity is the 

 velocity that a point on the crest or trough is moving. The curve R, = 1.0 is 

 the case where a current is absent; i.e., where L = L., L. fulfilling the 

 usual dispersion relationship (eq. 1). 



The curve F = FM in Figure 2 represents the limiting effective Froude num- 

 bers for which waves can propagate. If F is less than FM for a particular 

 fi, then waves cannot propagate against the current for the particular Q and 

 F values. In other words, the current acts as a filter under certain condi- 

 tions and prevents waves from ever reaching a given -point. The limiting value 

 of "effective current velocity," V cos 9, for which waves of a given period, 

 T, in a mean depth of water, d-p, can propagate is the "stopping velocity," 

 VST, given by 



VST = (FM) (gd^) 1 ' 2 (6) 



VST is negative (since FM is negative) and depends on dimensionless fre- 

 quency, Q (since _ FM, as shown in Fig. 2, depends on Q) , and mean depth, 

 drp (through the (gd-r.) 1 ' 2 factor). If a wave is propagating against the cur- 

 rent (i.e. , cos 9 is less than zero) , then the wave cannot reach any area 

 where the current speed V is greater than the local value of VST. Such an 

 unreachable area in physical space or on a plot like Figure 2 is called the 

 "forbidden region." Waves traveling with the current (i.e., with a cos 9 

 greater than or equal to zero) have no forbidden region. 



The Table gives the minimum effective Froude number, FM, and the ratio 



A _ wavelength without current 

 - average (in time) depth 



T 



for various values of dimensionless frequency, il. For a particular SI 



value, this table (used in conjunction with Fig. 2 and linear interpolation) 

 allows the following to be found: (a) whether a wave can propagate against 

 the current (i.e., is F less than FM) , and (b) if a wave can propagate, the 

 value of its current modified wavelength, L y . Using the value of (L^/drj,) 

 from the table and R, from Figure 2, Ly is computed from 



12 



