Hidrológiai Közlöny 1968 (48. évfolyam)

8. szám - Varrók Endre: A nyomásvonal alakulása töltésezett folyók mellett

348 Hidrológiai Közlöny 1968. 8. sz. Varrók E.: A nyomásvonal alakulása 2 TC°k =0 (-1)* (2£+l)

2 sin[(2/fc+l)ö)(í-T)] (6) The seeond factor on the right-hand side of (5) is the exponential function: it is, as a matter of fact, the exponent C of this function that we are now looking for. Let us rearrange (5) to obtain C in explicit form: '"[-WH Making use of the recorded data h(x; t) we could obtain from this equation the value of C, but we do not yet know the value of r to be substituted into (6) to obtain h

o(0; t—x). The delay x can be obtained by comparing the peaks of the undamped wave, and of the damped waves observed at the various outlets, respectively. The delay r[s] is related to the distance x [cm] and the cycle frequency a> [°/sec] as follows: ' 4,7( OW This is identical in structure with the function 2a

2a> (8) (9) Fig. 6. Diagram to interpret the notations of Eq. (6) 6. ábra. Értelmező ábra a (6) egyenlethez The figure clearly shows that the amplitude of the wave decreases with distance and that the delay against the fundamental wave increases with distance. This delay entails that at a given instant t, the phase angle of the local pressure fluctuation will be different at different points x, and, conversely, phases of a given phase angle occur at different instants at different points x. In nature, this means that at a distance x from the river, the phase at a given instant ti of the river stage will be reproduced with a delay r, i. e. at the instant ti + x, or in other words, the phase of the aquifer pressure at a given instant ti is the consequence of a river stage that occurred earlier, at the instant ti— r. Hence, pressure fluctuation in aquifer records, as it were, the „past" of the river stage. Returning now to our recorded data let us illustrate the above considerations by Fig. 6. where the ordinates AíBí intersect the respective curves at pressures of identical phase angle. Let us now find the function that best fits the recorded data and satisfies the boundary conditions. We have assumed that the amplitude of the pressure fluctuation is damped exponentially, i. e. that the function looked for can be written in the form h(x;t) = h

o(0;t — x)e~° (5) The function h

o(0; t—r) on the right-hand side of this equation is none other than Eq. (4), displaced in time by x: A

0(0;«-T) = which occurs in heat flow problems. The value 4,70 in the denominator of the expression under the radical is a constant depending on the dimensions and nature of the soil prism as well as on the nature of the liquid used. For a flood wave of given period T and for a fixed point x, x is a constant and thus C can be obtained from (7). The values of C computed from the observed values define graphs shown in Fig. 7. Examining these curves in a purely qualitative way for the time being, we may state that C is a periodic function increasing with increasing x. The function has maxima at every instant to + iiT/4 (where t

0 is the instant when the first flood wave, i. e. the experiment as a whole, is started and n is a positive odd integer). The dependence of C on the parameters of the flood wave, e. g. its cycle frequency, is still not clear, however. We have computed C values for various waveforms. The diagrams obtained were invariably similar to Fig. 7, but not so similar as to permit a direct comparison. Owing to the variance of T, the peaks occurred at different instants even when referred to a common time of beginning t„. A different variable had therefore to be introduced. Comparison was effected onthcbasis of the identity of phase angles. We have found that the exponent C as plotted vs. a has maxima at the points with ot=cot = nnl2, where n is again an odd positive integer. Fig. 7. shows that the curves of the individual pressure-gauge outlets coincide: having determined the temporal variation of pressure at a single outiét,.one may construct the whole set of curves by displacing the curve first determined parallel to itself in the direction of the ordinate axis. The amount of displacement is obviously somé function of x. Hence, the function describing the set of Fig. 7. Variation of exponent G vs. time and distance x 7. ábra. A C exponenciális kitevő hely és idő szerinti változása