530 
= fle 
APPENDIX Cy 
CORRECTION ON PIEZO-ELECTRIC RECORDS DUE TO 
FINITE SIZ. 
S, Butterworth 
It is assumed that the pressure recorded at any instant is the mean pressure over the surface 
of the gauge. 
The gauge is Edge On to the wave and the law for Pressure-Space is taken to be 
pe pet 
Fy P(s—pxe dy? x? - 2 x sp0cc08) (1) 
Let the radius of the gauge be a and consloer the moment when the wave front has reached the position 
AB in Figure 1 (Appendix C).° Let A 8B subtend on angle 2q@ at the centre of the gauge and let a narrow 
strip C D of width 5 x paralle) to A B subtend on angle 26, Then the contribution of C D to the total 
pressure is 
p.cd. 8x = -2a* p sin?O8 0 (2) 
since CD = 2a sin@, x 3 a (cos@-cos¢), 8k =-asinO sO 
Integrating from x = 0,to x=c [a (1~- cos d)) that is from@ = dp to = 0, we have for the integral 
pressure 2a Pp p sin” 6 d@ where p is expressed in terms of @ by replacing 
x by a (cos @ - cos d) in (1) 
Making this substitution in the series in (1), integrating term by term and dividing by 7 a? we find 
for the mean pressure 
Pm = PiA+aB+ par Ct ......) (3) 
where A, B, C are functions of m viz:- 
Ae 4 G- 4d sin 2) 
B = = {es P-t sin 2) -3 sin? o} (4) 
¢ = 4 B cos* db (D- 4 sin 2¢) - 5 cosh sin? p+ d - & sin “@)} 
The formulae give the following values for A, B, C. 
op. A 8 c £ = (1-cosd 
279 0° 0 0 0 0 
30° 0,029 0,001 0.00 0 .134 
60° 0.196 0.040 0,04 0.500 
90° 0.500 0. 212 0.06 1.000 
120° 0.805 0.580 0.24 1.500 
150° 0.971 0.866 0.50 1.866 
180° 1.000 1.000 0.62 2.000 
By means of this table and equation (3), the course of the pressure-t ime curve recorded by the gauge 
up to the moment the wave fromt has traversed the gauge diameter may be obtained for any given values 
of yt and a, for if tig is the time to traverse the gauge diameter and t the time to traverse the distance 
thr. = c/2a, 
° 
ThuS cevece 
