1190 
Sie 
above a "fast". one and investigate the extent to which such a concentration is 
tikely to oceur, We need a numerical estimate of the likely difference of 
velocities. |f we suppose that we aro near a river-mouth, it Is possible 
that a layer of fresh water will Ile above the salt water for a considerable 
distance out to sea, The velocity of sound in fresh water is about 4% less 
than in salt which turns out to be of the right order of magnitude to give 
an observable effect of the type for which we are looking. |\f we attempt 
to attribute such effects to temperature variation in the water alone, we meet 
with a number of difficulti2s so serious as almost to rule out this possibility. 
(a) The temperature difference between the two layers to give a velocity 
difference Of 4% would have to be about 15° C., a large value. 
(db) Even if such temperature differences were possible, the transition 
between the layers would not be as sharp as it can be between fresh 
and salt water, and reflection effects would be correspondingly 
reduced, 
(c) In order that the energy should be concentrated near the surface, It 
would be necessary for the "slew" or cold layer to be above the hot. 
Such a state of affairs does sometimes occur, but it is rare, belng 
essentially unstable, except in the region between 4° C. and zero, 
over which range the change in velocity is very small, 
Statement of the iWathematical Problem. 
To fix our ideas, consider a sca of salt water, with a ten-foot layer 
of fresh water above it. We wish to investigate the propagation of sound 
wavzs of various frequencies, and this being known we can infer the way in 
which a pulse of arbitrary shape will be propagated from a point source, For 
mathematical reasons, it is difficult to treat directly the cas2 of an Infinitely 
deep sea, so we consider the case of a ten-foot layer of fresh water above a 
hundred—foot layer of salt water resting on an acoustically rigid bottom. 
(This is probably not unlike the conditions that do in fact exist near a 
river=-mouth). We first review two avallablo methods for dealing with the 
corresponding problem in a uniform sea, 
(a) The method of images. The effect of the surface and bottom is 
replaced by that of an infinite scries of imagss of the source, and 
the effect at a distant point is computed as the sum of all these, 
The resulting serics is very slowly convergent, but can be transformed 
in various ways, provided that we assume perfect reflection at bottom 
and surface. Application to the present problem, wher= we are 
concerned with a partially reflecting surface, would te difficult. 
(b) The method of normal modes. We use cylindrical co-ordinates, 
the axis of symmetry being the vertécal through the point source, 
Let the free surface be the plane y = O and let y be the distance 
below this surface, Let: 
Yo be the depth of the source; 
h be the depth of the boundary between the two layers; 
| be the thickness of the lower layer, 
If the velocity of sound is uniform, a typical solution of the wave cquation 
satisfying the boundary conditions at the free and rigid surface is:- 
baw steialete 
