Given these boundary conditions, the piezo will output a voltage which is basically proportional to acoustic pressure input. This technique works pretty well, and indeed results in the desired flat frequency response up to where the piezoelectric begins to have a resonance. In fact, if one truly has a matched backing impedance, and the bond between the piezo and the backing is sufficiently thin, then one can have a reasonably flat response up to about 1.5 f0, where f0 is the half wavelength thickness resonance of the piezo. Output from a piezoelectric layer is inherently zero at 2 f0 owing to the fact that the integral of a full cycle of a sine wave is always zero.
Low frequency performance tends to be limited by the real part of the electrical load impedance, resulting in a simple RC time constant, or sometimes the lossy dielectric properties of the piezo result in a similar effect. A SPRH-S-1000 will have a total capacitance of around 118 pF, including active element (22 pF), 8" RG174 cable, and SMC female connector. A Tek TDS724A oscilloscope, set to have a high input impedance and DC coupling, will present a load impedance equivalent to 10 pF in parallel with 1 MOhm. AC coupling will result in a much higher DC component (> 20 MOhms), however this does not seem to affect the output seen by a hydrophone at low frequency by much. The RC time constant obtained by multiplying 1 MOhm by 118 pF is 0.118 msec corresponding to a low frequency cutoff of 8.5 kHz. If one gently taps the tip of a SPRH-S-1000 terminated as above, one will see a pulse of about 10 mV with a -6 dB pulse width of around 10 ms, though this is highly variable. (Note that touching your hydrophone tip with solids of any sort voids your warranty).
A more repeatable experiment was done with compressed air, where a SPRH-S-1000 was subjected to a pressure of 40 psi, which was suddenly released using a mechanical valve. In this experiment, a negative pulse with a very repeatable size and shape was seen, and the negative pulse amplitude was about -3.32 mV with a -6dB width of 31 ms. While these results were repeatable, they contained numerous interesting artifacts and failed to resemble the desired Heavyside function due to the fact that we are not operating at a point where the hydrophone response curve is flat (and the air pressure is probably not actually a clean drop in pressure due to guided wave propagation in the tubes and pipes of the experiment). The 3.32 mV, divided by 40 psi results in a conductance of 1.204E-8 V/Pa, or -278.4 GdB, and the hydrophone at 1 MHz had a response of -243 GdB. The 35 missing dB are probably due to roll off proportional to f below 8.5 kHz.
Mechanically, the hydrophone must be fixed in a holder to see the really low frequencies, as the hydrophone itself becomes small with respect to an acoustic wavelength and will move about "in the breeze" so to speak if it is not constrained.
The "theory of operation", therefore, is the simple fact that the voltage out of a piezo material, when terminated into a high impedance, is proportional to the stress through it. The real trick to designing hydrophones systems is making sure you get all of the boundary conditions right, and eliminate all of the acoustic resonances that can perturb the frequency response. Given a good design, hydrophones can yield a flat frequency response over several decades of frequency, far greater than we are used to based on experience with resonant devices. Unfortunately, the operation of a device in the above described "subresonant mode" results in voltages that are very small compared to that which can be obtained under resonant conditions. For this reason, hydrophones should not be used where great sensitivity is needed. If the ultimate in sensitivity is needed, then narrower bandwidth, custom tailored transducers need to be considered. If flat frequency behavior below 10 kHz is required, then other technologies, such as micromachined pressure sensors should be considered.