E.J. Richards, J. L. Willis, and D. J. M. Williams 293 
15.4.4. Noise from Roughness 
Willis' results on the effect of single roughnesses are still in a preliminary 
stage [2] and it would be unwise to use these results other than qualitatively at 
the moment. It is interesting, however, to note (Fig. 15.9) that the rms pressure 
from even a large single roughness apparently does not exceed twice the smooth- 
surface result. Thus, at first sight, the effect of roughness would seem small. 
If, however, we assume that the influence ofeach ridge is independent of the next 
one, that ridges are separated by distances equal to their height, and that the 
areas over which these pressures are correlated are similar to those measured 
in the single-roughness experiments, then a rather different picture emerges. In 
Fig. 15.17, the noise measured at a typical distance from the submarine is 
estimated for varying degrees of roughness. It must be emphasized that the 
effect of roughness will depend critically on the height of the protrusion on the 
boundary layer or its viscous sublayer and that these results are purely illus- 
trative. It is seen, nevertheless, that in an extreme case of roughness, we can 
explain the high observed noise environment, although these figures are un- 
questionably extremes and other possibilities should not be excluded. 
It is difficult to obtain practical observations ofthe effects of roughness. The 
buoyant-body tests [12] of Haddle and Skudrzyk give spectra, but no over-all 
levels. However, ifthe calculations of radiated noise are made on the above basis, 
we obtain an estimated over-all level increase of 11 db for a 0.1-in. step. This 
compares well with the increases of about 10 db given in [12]. 
15.4.5. Reradiated Noise from Body Vibrations 
As with an aircraft fuselage, the possible modes of vibration of a submarine 
hull fall into two categories. Firstly, there are the over-all modes of vibration 
of the whole cylinder; these have been considered by Arnold and Warburton [13] 
ignoring frames or stiffeners and later by Miller and Junger [14, 15] who have 
included stiffeners. Secondly, there are the localized vibrations of the panels, 
the boundaries of which are defined by the stiffener spacing. Some symmetric 
modes of vibration of these have been considered theoretically, but it has been 
found from testing of practical structures that the modes which are excited are 
not necessarily symmetrical and are therefore extremely difficult to define. 
If the stiffeners in the submarine hull are ignored, the frequencies of the 
over-all modes of vibration may be obtained (Fig. 15.18) directly from Arnold 
and Warburton's work. Junger's paper [11] on radiation considers these over-all 
modes of a circular cylinder and shows that there is no resultant radiation if 
the sound wavelength is greater than the axial structural wavelength. This "cut - 
off" line is also shown in Fig. 15.18. It is clear in this case, therefore, that the 
only over-all modes of vibration which can radiate are those which have long 
axial wavelengths and relatively short circumferential wavelengths (e.g., when 
n=12, the circumferential wavelength is 5.24 ft, and the axial wavelength is 
greater than 44 ft). It is felt, without mathematical backing it must be confessed, 
that a boundary layer is unlikely to excite this type of mode. 
Over-all flexural modes of the submarine will not cause any resultant 
radiation because the frequencies will be low, the sound wavelength relatively 
long, and complete cancellation will occur. The only other mode of interest is the 
