328 Lecture 17 
10.0m 
plate ehiekne os 0 wavelengths for longitudinal 
x Re waves in air or in seawater 
40 equiv. beam th. and Xp for bendingwaves 
in steel or aluminium plates 
h(m) d ~_ (thickness h) or beams (equiv. 
20 (e) ~ > th. hL for wood actual aN 
is 10 % smaller 
1.00m ; SS 
SN Php 
0.40 = SS 
Sle 
0.20 
my 
(B) 
010m = 5000 m/s for steel, aluminium 
an = 4000 m/s for wood 
= VI/A (radius of gyration of cross section) 
0.040 
= h/2V3 for plates 
20 = 3.46 iy for beams SS 
0.01m 
1 2 4 8 i) €2 (sS 2S Aso) Soo) | 2 4 8 iS s2 3) 
——_ Hz kHz 
Fig. 17.6. Wavelengths for longitudinal waves in air or in sea water and for bending waves in plates of 
thickness h; some equivalent I-beam DIN-numbers are indicated. The lines for constant A must not be 
extended to the right. 
the "pumping-around" effect. A similar situation exists when the bending waves 
in a plate are of such a form that nearby parts of the plate vibrate in opposite 
directions within a distance of, say, halfthe wavelength associated with the sound 
radiated into the surrounding fluid medium. The frequency f,, for which cg =c, 
is called the critical frequency; here c, is the velocity for longitudinal waves 
in the fluid. The critical frequency is given by 
Cw 
for = 2ncpi, (15) 
as may easily be derived. For frequencies f>f.,, the radiation is very efficient 
and shows pronounced directional effects. The frequency range discussed in 
this paper is such that Ag < 2m, approximately, for the lower limit; and f <f,, 
for the upper limit. 
It is supposed that bending waves are easily excited in typical ship struc- 
tures consisting of plates and beams. These waves are presumed primarily 
responsible for the radiation of sound. Thus it seems worthwhile to consider 
them in more detail, especially with respect to mechanical impedances and 
radiated sound power. 
.9. MECHANICAL IMPEDANCE 
In describing the response of a mechanical structure exposed to alternating 
forces, it is almost impossible to avoid extremely simplified mathematical 
models of the physical reality. Thus, a ship will be considered as a composition 
of flat plates and straight beams in this paper. It may be hoped that conclusions 
drawn from theories for these simple configurations will, in broad outlines, 
also be valid for more complicated configurations. 
