Page:Alan Turing - Proposed Electronic Calculator (1945).pdf/35

 (b) The pressure p. In the case of the crystal this is understood to mean the xx-component of stress.

(c) The displacement ξ in the x-direction.

(d) The velocity v in the x-direction.

(e) The radian frequency w.

(f) The elasticity η. This is the rate of change of pressure per unit decrease of logarithm of volume due to compression.

(g) The velocity of propagation c.

(h) The mechanical characteristic impedance ζ.

(i) The reciprocal radian wave length β.

(j) The piezo-electric constant ε. This gives the induced pressure due to an electric field strength of unity. This field strength should normally be thought of as in the x-direction, but we shall have to consider the case of a field in the y or z direction briefly also.

These quantities are related by the equations

$$c = \sqrt{\eta/\epsilon}, \zeta = \sqrt{\eta/\rho}, \beta = \frac{w}{c}, v = iw\xi, iw\rho v = -\frac{dp}{dx}, p = -\eta\frac{d\xi}{dx} + E\epsilon$$

In what follows we assume that all quantities such as p, v, ξ depend on time according to a factor eiwt, which we omit.

We now consider the ‘transmitting crystal’, which we suppose extends from x = -a to x = a where d = 2a. The solution of the equations will be of form

$$p = E\epsilon + B\cos\beta x$$

within the crystal, i.e. for |x| < a. Since the pressure is continuous we shall have

$$p = (E\epsilon + B\cos\beta a)e^{i\beta_1(a-x)} \mbox{ if } x > a.$$

This gives for the velocity

$$v = \frac{1}{w\rho}\cdot-B\beta \sin\beta x = -B\zeta^{-1} \sin\beta x \mbox{ if }|x| < a$$

$$v = \zeta_1^{-1} (E\epsilon + B \cos \beta a)e^{i\beta_1(a-|x|)} \mbox{ syn } x \mbox{ if }|x| > a$$

Continuity of velocity now gives

$$B\Big(cos\beta a + \frac{i\zeta_1}{\zeta}sin\beta a\Big) = -E\epsilon$$ and/