Hidrológiai Közlöny 1968 (48. évfolyam)

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

Varrók E.: A nyomásvonal alakulása Hidrológiai Közlöny 1968. 8. sz. 349 curves consists of two parts, one of whicli is solely time-dependent and defines the temporal variation of C for any value of x, whereas the second, additive part is solely dependent on x and attributes the variation of C as defined above to a certain point of the z-axis. Let us first discuss the variation of C vs. x. The curves in Fig. 7. reveal that it is most expedient to determine the desired function C\(x) from the ordinates C belonging to the instants t — 0, t = T\2, t = T, as these are the loci of the minima of the other function, C vs. time, unknown for the time being. Rigorously speaking, we could only use the C values belonging to the above-named instants, but the points in question are singularities of the function and Eq. (7) makes no sense at those instants. This difficulty may be circumvented by taking into consideration that the variation of the functions is so slight in the neigbourhoods of these instants that the error committed when data from the close vicinity of these points are alsó used is negligible. Plotting the ordinates marked C

x{x) in Fig. 7. vs. x yields a second-degree parabola C^x) = bx

2 = 0,0000135a;

2 (10) {Fig. 5) where x is in centimetre units. Computing by (10) the C

1 value of every pressure gauge outiét and subtracting the constant obtained from every C value of the outiét in question yields a plot of the desired function C

2(t) which depends purely on time. As indicated previously, the time function in question is periodic and its entire rangé is positive. It would be expedient to express the periodic variation of C

2 in terms of the river stage, but this is rendered impossible by the negatíve portion of the rangé of this latter. This, however, can be remedied by elevating the negatíve values to any power of even exponent. It is further expedient to perform a transformation which renders the function in the denominator dimensionless. This can be achieved W x, [cm] Fig. 8. Observed data referring to the function C

1 (x). 8. ábra. A G

x(x) függvényre vonatkozó mérési adatok by dividing both sides of Eq. (4) (the flood-wave function) by H/2 : 2

h

0(0; t) H XH (-D* 2 Z /óz. , i \

2 sin[(2£+lM] (11) W (2&+1)

2 k =0 This transformation results, moreover, in a bounded flood-wave function: (12) Squaring (11) yields the function in Fig. 9.We liave examined whether the curves in Fig. 7, of identical shape for any value of x, can be reproduced by an adequate constant distension or compression of the ordinates of the curve shown as Fig. 9. The examinaFig. 9. Temporal variation of the square of the dimensionless flood-wave function 9. ábra. A dimenzió nélküli vizállásfüggvény négyzetének időbeli változása tion has proved the assumption to be correct: indeed it is possible to obtain the ordinates of the desired C

2 function by multiplying the ordinates of the curve in Fig. 9 by 0,5: CS) = O5^)

2- d3) Bv (10) and (13), the looked-for exponent of the exponential function is it)' (14

) and the empirical form of Eq. (5) is ~bx

2 -0,5 (V* f h(x;t) = h

0(0;t-T)e •e l 1

1 > (15) This function yields the pressure arising at any distance x and at any instant t as a consequence ofa flood wave rio (0; t). Por practice, it is only the extremes — in the present case, the pressure maxima — that carry any special significance. The maxima in question can be derived by putting h

ö(0]t)mnx=HI2 : h(x-,t)

m^=^e~

bx

2 .e-°'

5 = 0,303^-e6a;

2 (16) Finally it was examined that the conditions required to make the experimentál equation (15) satisfy Eq. (2). For the purpose, we have written (15) into a dimensionless form: h(x; t) -bx í 2/io \ ' — C (17)