The following assumptions are made when considering an ideal resonator:

• The quartz transducers are arbitrarily thin, and their radius is large compared with the wavelength, so that no diffraction losses occur: R_transducer ≫ λ.
• The cell enclosure does not disturb the wave field because its acoustic impedance is matched to that of the liquid.
• The transducers reflect completely (acoustically rigid termination); the reflection coefficients satisfy r_s = r_e = 1.
• The transmitting transducer emits a plane wave.
• The total losses are very small.

Under these assumptions, consider the idealized resonator arrangement in Fig. 4.2. Harmonic excitation of the transmitting transducer W_s propagates through the liquid as a plane acoustic wave:

where

When the emitted wave reaches the receiving transducer W_e, it is reflected back towards the transmitter, taking into account the receiver reflection coefficient r_e. A reflection coefficient r_s can likewise be assigned to the transmitting transducer. The pressure associated with the wave is:

Considering the magnitude of the pressure wave at the receiving transducer, P_e, leads mathematically to a series relating P_e to the distance travelled by the sound wave, with transducer spacing l:

This series converges (for r_e·r_s < 1 and αl ≪ 1) to:

Original mathematical notation is preserved in the extracted image assets.