Figure 13. High-speed motion pictures of a 1-inch stee! sphere entering the water at a velocity of 

 35 ft/sec. The time interval between frames is 1.6 millisec. (From Ref. 34.) 



to the waterborne sounds produced by the impact, the splash, and the volume pulsa- 

 tions of air trapped below the surface. The only systematic attempt to relate the 

 sounds quantitatively to the size and velocity of the impinging object seems to be that 

 of Franz [36], some of whose results will be presented toward the end of this section. 



In principle, the understanding of the generation of sounds by surface dis- 

 turbances requires no theoretical considerations other than those which enter into 

 the question of the origin of all sounds. Thus those relations which determine solutions 

 of the wave equation in terms of the motions and pressures occurring at the bounding 

 surface of the fluid are valid and, in principle, applicable to the determination of the 

 sound field in water. Here, the "bounding surface" of the fluid, i.e. the water, includes 

 the entire air-water interface as well as the boundary at which the water is in contact 

 with any penetrating object. But the determination of the boundary conditions neces- 

 sary to specify the sound field may, in a given practical situation, present a hydro- 

 dynamical problem whose solution is unattainable. It is therefore expedient to seek 

 a more direct way of expressing the relation between the data characterizing the cause 

 of the disturbance and the sound field which results. 



The dominant boundary condition on the underwater sound field of sources 

 at or near the free surface is, of course, the requirement that fluctuations in pressure 

 be zero at the free surface. For sound sources which disturb the free surface only 

 acoustically, the modification of the field resulting from the presence of the surface 

 is easily described in terms of out-of-phase reflections at the surface. Where a radical 

 disturbance in the surface itself constitutes the sound source, the situation is more 

 complicated. For simplicity, we consider only disturbances which are symmetrical 

 about a vertical axis. The sketch, Figure 14, indicates the vertical entry of an axially 

 symmetric body of some specified density and shape. We can further specify the body 

 and its motion in terms of a characteristic linear dimension, L, and the velocity, U, 

 with which it strikes the surface (at time t = 0). The figure indicates the significance 

 of the coordinates, r and Q. We are interested in the sound pressure, p s , which, at 

 sufficiently great distances from the disturbance, obeys the wave equation, 



c 2 V 2 Ps - d 2 p s /dt 2 = 0. (16) 



The solution can be expressed in terms of spherical harmonics (Lamb [6], § 

 292). For our purpose, however, it is more useful to consider the sound field as that 



259 



