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(4) What should be the dimensions of the crystals? (Diameter might be half that of the tank, e.g. 1 cm. Thickness should be such that the first resonances of the two crystals are two or three megacycles on either side of the carrier, if water-alcohol is used. With mercury the thickness is less critical and may be either as with water-alcohol or may have resonance equal to carrier.
(5) Should the inside of the tank be rough or smooth? (Smooth).
(6) What should be the dimensions of the tank? (Standard tanks to give a delay of 1 ms. should be about 5' long whether water-alcohol or mercury. Diameter ½').
(Keep all the tanks within one degree Fahrenheit in temperature. Correct systematic temperature changes by altering the pulse frequency.)
In order to be able to answer these questions various mathematical problems connected with the delay lines will have to be solved.
(ii) Electromagnetic conversion efficiency. The delay line may best be considered as forming an electrical network of the kind usually (rather misleadingly) described as ‘four-pole’, i.e. a network which has one input current and one input voltage which together determine an output voltage and current. Such a network is described by three complex numbers at each frequency. In the case where there is little coupling between the output and input, which will apply to our problem, we may take these quantities to be the input and output admittances and the ‘transfer admittance’. Strictly speaking we should specify whether the output is open circuit or short circuit when stating the input impedance, but with weak coupling these are effectively the same; similarly for the output impedance. The transfer admittance is the current produced at one end due to unit voltage at the other, and does not depend on which end has the voltage applied to it. In the case of the delay lines the input and output admittances will be effectively the capacities between the crystal electrodes. We need only determine the transfer admittance.
We shall consider the following idealised case. Two crystals of thicknesses d and d' are immersed in a liquid, with their faces perpendicular to the x-axis. The liquid extends to infinity in both the positive and the negative x-directions, and both liquid and crystals extend to infinity in the y and z directions (Fig. 40). The distance between the near side faces of the crystals is ℓ It is assumed that there is considerable attenuation of sound waves over a distance of the order of ℓ but hardly any over a distance of the order of d or d'.
These assumptions are introduced largely with a view to eliminating the possibility of reflections. In practice the reflections would be eliminated by other means. For instance, the infinite liquid on the extreme right and left would be replaced by a short length of liquid in a stub of not very regular shape, so that the reflected waves would not be parallel to the face of the crystal. More likely still, of course, we should have some entirely different medium there.
The physical quantities involved are:
(a) The density ρ. We write ρ for the density of the crystal and ρ1 for that of the liquid. Likewise a suffix 1 will indicate liquid values throughout.
