128 
PACIFIC SCIENCE, Vol. XVII, January 1963 
of continuous dilution by the addition of bioge- 
nous material. An expression giving the dilution 
rate can be derived under the following assump- 
tions : ( 1 ) sand transported along the shore con- 
sists of a mixture of biogenous and terrigenous 
material, where the fraction of each is B and 1-B 
respectively; (2) this sand is transported along 
the shore at a rate Q which is constant with dis- 
tance x along the coast; ( 3 ) biogenous material 
is supplied to the beach, from offshore, at a 
constant rate F{ per unit of coast length, where 
it is completely mixed with the sand transported 
along the shore; and (4) there is a net deposi- 
tion of this mixed sediment at a rate F 0 per unit 
of coast length, such that Fj = F 0 . 
Conditions 1 through 4 lead to- the following 
general expression for the concentration of bi- 
ogenous material in the beach sand as a function 
of the longshore transport rate Q, the rate of sup- 
ply of biogenous material F 1? and distance along 
the coast x: 
F 
In B-rr-^-x + constant of integration 
where In B is the natural logarithm of the 
carbonate fraction (see Appendix I). This is a 
straight line with slope F t /Q when B is plotted 
on a logarithmic and x on a linear scale (Fig. 
11, bottom). If the terrigenous material is sup- 
plied to the coast in large quantities and at a 
single point, as at the mouth of the Waimea 
River, B will be almost zero near the river and 
will increase exponentially with distance from 
the river — as it is observed to- do. 
The relation derived here is useful in that it 
not only predicts the form of the concentration 
change, but also indicates that, if either the long- 
shore transport of sediment Q, or the deposition 
rate F is known, the other can be computed. The 
equation for the curve of concentration change 
with distance from the Waimea River is: 
In B= 1.4 x — 4.6 
where In B is the natural logarithm of the car- 
bonate fraction, x is in nautical miles, and 
Fj/Qn 1.4 has the units of nautical miles" 1 . 
Neither F; nor Q are known, but very rough 
order-of-magnitude estimations can be made for 
both. The rate of accumulation of calcareous sed- 
iment on the Mana Coastal Plain, divided by the 
length of coast line, gives a measure of F ls while 
the sediment yield from the Waimea River 
drainage basin provides an estimate for the lit- 
toral transport rate Q near the mouth of the 
Waimea River. 
The volume of sediment in the Mana Coastal 
Plain down to a depth of 60 ft below sea level is 
approximately 4 x 10 10 ft 3 . If it is assumed that 
60% of this is of biogenous origin, and that it 
has accumulated along a coastal length of 15 
nautical miles during a time interval of 8,500 
years 8 , then the rate of supply of biogenous ma- 
terial to the coast, F i? is about 7000 cubic yd per 
mile per year. This gives a value for the littoral 
transport rate in the Waimea region of: 
Q = Fi/ 1.4-= 5,000 yd 3 per year. 
The Waimea drainage basin, with an average 
effective precipitation of 53 inches per year 
(Table 1), would be expected to yield about 
420 tons of sediment per square statute mile per 
year (Langbeinand Schumm, 1958: fig. 3). As- 
suming half of this to be sand-size or larger ma- 
terial, and using the conversion from weight to 
volume of 60 lbs per ft 3 (given by Langbein 
and Schumm, 1958), one obtains a total yield of 
sand-size material from the Waimea basin of 
25,000 yd 3 per year. Since the Waimea River 
does not have a prograding delta, the yield from 
the river must nearly equal the littoral transport 
rate Q. 
Beach surveys of 1926 and 1950 by the Corps 
of Engineers (1955: par. 52) indicated a net 
loss of sand from Waimea Beach of 5,000 yd 3 
per year. The littoral transport rate is probably 
greater than the loss, and the Corps of Engineers 
estimate it to be about 20,000 yd 3 per year. Con- 
sidering the nature of the assumptions leading 
to the calculations of littoral transport rate in 
the preceding paragraphs, values of 5,000 and 
25,000 yd 3 per year are remarkably, if not for- 
tuitously, similar. One must recognize that, in 
so far as the actual transport rates at Waimea are 
concerned, these calculations may eventually 
prove to be merely a mental exercise. If this be 
the case, it Is hoped that they have at least served 
to Illustrate a valid principle. 
8 The rate of 60 ft sea level rise in 8,500 years was 
selected because this seems to be the most accurate 
and significant part on the sea level rise curves, pre- 
sented by Shepard (I960: fig. 4), McFaden (1961: 
fig. 9), and Jalgersma and Pannekoek (I960': fig. 3). 
