Model calculations for ultrasonic plate-liquid-plate resonators: Peak frequency shift by liquid density and velocity variations

Ultrasonic resonators-consisting of the liquid sample, enclosed by two planar piezoelectric transducer plates-permit a direct and accurate determination of liquid sound velocities c(L) at kilohertz and megahertz frequencies. Such plate-liquid-plate (PLP) resonators offer a resolution Delta c(L)/c(L) < 10(-3), which is important for analytical work in chemistry, bio- and physico-chemistry and for some technical applications. A simple one-dimensional resonator model is derived which permits calculation of the spectrum of the longitudinal (nonharmonic) eigenfrequencies f(n) (n = 1, 2, 3, 4...) as a function of liquid and resonator parameters. This model obtains the peak f(n) shift by variation of liquid velocity c(L) and density rho(L) as well for plane wave propagation; the effect from variations in rho(L) has been neglected in most ultrasonic studies so far. Caused by a frequency-dependent phase shift for sound reflection at both liquid/transducer interfaces, the differential quotients df(n)/dc(L) and df(n)/d(rho L) of the 'real' resonator deviate from those of an 'ideal' PLP resonator with perfect, 'hard' reflection (reflection factor = 1) and harmonic overtones nf(L) (f(L) is the liquid fundamental frequeny; one half liquid wavelength lambda(L)/2 equals transducer separation x). The figures show typical acoustic impedance and admittance spectra, which are calculated for a model configuration; they illustrate features of the harmonic numbers in the liquid cavity and longitudinal mode counting. Equations are given for the dimensionless, normalized differential quotients df(n)/dc(L).c(L)/f(n) and df(n)/d(rho L).rho(L)/f(n). Plots demonstrate systematic aberrations from 'ideal' resonator behaviour, which can affect high-precision sound velocity measurements.