The second term under the integral sign is due to pressure fluctuations which may be caused by flow or structural vibrations. Dimensional analysis yields:

This dependence on the sixth power of the flow is characteristic of dipoles on a compact scale. However, even at the low Mach numbers prevailing in underwater acoustics, sources are not always compact and consequently a different speed dependence will apply.

Generally, the flow is not affected by structural vibrations since the amplitudes are much smaller than the characteristic length scales of the fluid. An exception is the well-known phenomenon of propeller “singing” where blade vibrations tend to increase the coherence of a periodically shed vortex wake which, in turn, feeds more energy into the structure.

At moderate to high speeds, the propulsor is the most important contributor to a vehicle's radiated noise. The major sources are due to the following effects:

1. The response of rotating blade rows to flow disturbances caused by upstream hydrofoils, such as stator vanes, control surfaces and other fixed appendages. The radiation consists of tonals at blade passage frequencies whose amplitudes are randomly modulated, as shown in Fig. (1).

2. The response of stationary or rotating blade rows to ingested turbulence which may be generated on the hull of the vehicle.

3. The response of rotating blade rows to self-generated flow distortions, as shown in Fig. (2) and of stators located downstream.

4. The scattering of boundary layer turbulence from trailing edges.

5. Radiation from structures which are coupled to the propulsor through hydrodynamic, acoustic and structural paths.

An authoritative treatise covering these topics is provided by Blake ⁽⁶⁾ in “Mechanics of Flow-Induced Sound and Vibration”.

In the mid-sixties, the fluctuating thrust of open propellers subjected to a turbulent inflow was of interest. A study of this problem was reported by Sevik⁽⁷⁾ at the Seventh Symposium on Naval Hydrodynamics. Experiments were performed in the Water Tunnel at the Applied Research Laboratory of the Pennsylvania State University. Its test section has a diameter of 1.22 m and a length of 4.27 m. The advantage of this facility is that relatively large propellers can be tested at high Reynolds numbers. Fig. (3) shows the installation of a ten bladed propeller in the test section. It had a diameter of 20.3 cm and its chord length was 2.54 cm. The fluctuating thrust was measured with a high impedance dynamometer which had a linear response over the frequency range of interest.

FIG. 1 ACOUSTIC LEVELS AT BLADE RATE FREQUENCY (FROM STRASBERG)