386
12 Internal Flows in Marine Organisms
neglecting the gradient oA/ox. Differentiating Eq. (12.31) and substituting
into Eq. (12.33) we obtain:
OU 20: op
- + - - = 0 .
ox a ot
(12.34)
Eliminating velocity u from Eqs. (12.32) and (12.34) yields equation for fluid
pressure p(x, t):
(12.35)
in which:
c = fJ;.
(12.36)
Equation (12.35) is the well known wave equation, and the quantity c is the
wave speed. If the tube is thin walled and the material obeys Hooke's law (see
Sect. 10.2), wave speed is given by (Fung, 1997):
(12.37)
in which h is the wall thickness. The value of c is known as the Moens-Korteweg
wave speed. For mammals the size of a dog, the velocity c is of the order of
5-8 m/s (Pedley, 1980).
To solve the wave equation (12.35), the boundary and initial conditions have
to be known. However, even without complete knowledge of these conditions,
it can be shown that the solution is a function that represents a progressive
wave with the speed c:
p=!(x-ct),
(12.38)
where! is an arbitrary function. For illustration, let us assume that at t = 0,
the pressure pulse has a form (Fig. 12.4):
{
7rX
sin - , 0 < x < 1
p(x,O) =
1
- -
0,
x < 0, x > I.
(12.39)
Let the velocity c be 5 m/s. At the time t = 5 s later, the same pressure pulse
is translated to the right. The value of the pressure pulses remains constant as
long as value of (x - ct) is the same. Thus, at t = 5 s, the point A will is at