and therefore the velocity at a is
i.e. the velocity at the inside edge of the crystal is
where .
Assuming that the exciting voltage is longitudinal we may say that
The effect of the medium between the two crystals we will not consider
just yet. Let us simply assume that
We have now to consider the effect of the receiving crystal. Fortunately
we can deal with this by the principle of reciprocity. When applied to
a mixed electrical and mechanical system this states that the velocity
produced at the mechanical end by unit voltage at the electrical end is
equal to the current produced at the electrical end by unit force at the
mechanical end. Hence
.
To these equations we may add that the ratio of force to pressure is the
area A’ of the receiving crystal, and that the ratio of pressure to
velocity is the mechanical characteristic impedance ζ1. Combining
we obtain
.
Let us now assume that the input to the valve from the receiving
crystal consists of a tuned circuit with a fairly low ‘Q’ as in Fig. 41.
Then/