Page 17 - Essentials of applied mathematics for scientists and engineers

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PARTIAL DIFFERENTIAL EQUATIONS IN ENGINEERING 7
1.3.4 Boundary Conditions
Four types of boundary conditions are common in conduction problems.

a) Heat flux prescribed, in which case k∂u/∂n is given.

b) Heat flux is zero (perhaps just a special case of (a)), in which case ∂u/∂n is zero.
c) Temperature u is prescribed.
d) Convection occurs at the boundary, in which case k∂u/∂n = h(U − u).

Here n is a length in the direction normal to the surface, U is the temperature of the fluid
next to the surface that is heating or cooling the surface, and h is the coefficient of convective
heat transfer. Condition (d) is sometimes called Newton’s law of cooling.

1.4 THE VIBRATING STRING
Next we consider a tightly stretched string on some interval of the x-axis. The string is vibrating
about its equilibrium position so that its departure from equilibrium is y(t, x). The string is
assumed to be perfectly flexible with mass per unit length ρ.
Fig. 1.4 shows a portion of such a string that has been displaced upward. We assume
that the tension in the string is constant. However the direction of the tension vector along the
string varies. The tangent of the angle α(t, x) that the string makes with the horizontal is given
by the slope of the wire, ∂y/∂x,

V (x)/H = tan α(t, x) = ∂y/∂x (1.16)

If we assume that the angle α is small then the horizontal tension force is nearly equal to
the magnitude of the tension vector itself. In this case the tangent of the slope of the wire

FIGURE 1.4: An element of a vibrating string