334 Lecture 17 
17.6. RADIATION OF SOUND 
In following the sound path of the present noise control problem we arrive 
at the vibrating shell plates. There are indications that only relatively small 
portions of the shell actually radiate sound into the surrounding water. The 
simplification, "flat plate with bending waves," may therefore represent a not 
too unrealistic approach. 
We borrow at once two formulas for the radiated sound power from a paper 
by Heckl [12]. They were derived for radiation into a rectangular duct. This 
means that the radiating plate is slightly better loaded than a comparable plate 
radiating into an infinite half-space. This effect probably implies a factor of 2 
in power. For the present purpose this order of magnitude may be neglected. 
The formula for a rectangular plate excited by a point force Fes reads 
P, = Fy pas?(oobta/ 95 ) (30) 
2M? c,, 
where S is the area of the plate radiating into water (of density p, and sound 
velocity c,), Ag, is the bending wavelength, and 7 is the loss factor of the plate. 
It is supposed that f<f,,. Moreover, it must be supposed that the mass M of the 
plate is corrected to include the additional water load as mentioned before. 
The same remarks apply to the formula for a plate excited along a line of 
length 4 (the width of the plate); the power is given by 
52 1+Ag5 /mS 
P, = F 2 py 
7 ff P. 4nM2 bt 
(31) 
In both instances the radiated power consists of two additive components, one 
due to the over-all vibrations of the plate—the corresponding term contains 7 — 
the other due to the deformation near the excitation point or line. Nearly always, 
7 Will be so small that the latter contribution to the total power may be neglected. 
It follows that the radiated power may be reduced by increasing the loss factor 
if the exciting force is kept constant. 
Inserting the expressions for the point or line impedances from Eqs. (26) 
and (27), one finds for the radiated power as a function of the velocity veg of the 
excitation point or line that 
Py =O0A5\vicnipmice Ace (32) 
and 
P, = 0.43 vg PyCwbrcr (33) 
where A., equals the wavelength in the plate at the critical frequency fcr (the 
formulas given are derived for f < f.,). For constant-velocity excitation, clearly, 
radiated noise reduction cannot be achieved by increasing the loss factor 7! 
It is now worthwhile to discuss these formulas. The results of some labo- 
ratory measurements (in air) are shown in Figs. 17.10 and 17.11. In Fig. 17.10 
the radiated sound power for half-octave-band noise excitation with constant 
force as measured in a reverberant room is compared with the theory of Eq. 
(30). The agreement is satisfactory. In Fig. 17.11 the agreement of the experi- 
mental data with the theory of Eq. (32) is not so good. For a velocity of the 
excitation point of 1 m/sec, Eq. (32) indicates a power level of 120 db (41 w), 
