Page 18 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

8 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
at x + x is

V (x + x)/H = tan α(x + x) = ∂y/∂x(x + x). (1.17)

The vertical force V is then given by H∂y/∂x.The net vertical force is the difference between
the vertical forces at x and x + x, and must be equal to the mass times the acceleration of
2
2
that portion of the string. The mass is ρ x and the acceleration is ∂ y/∂t .Thus
2
2
ρ x∂ /∂t = H[∂y/∂x(x + x) − ∂y/∂x(x)] (1.18)
Expanding ∂y/∂x(x + x)inaTaylor series about x = 0 and neglecting terms of order
2
( x) and smaller, we find that

ρy tt = Hy xx (1.19)
which is the wave equation. Usually it is presented as

2
y tt = a y xx (1.20)
2
where a = H/ρ is a wave speed term.
Had we included the weight of the string there would have been an extra term on the
right-hand side of this equation, the acceleration of gravity (downward). Had we included a
damping force proportional to the velocity of the string, another negative term would result:

ρy tt = Hy xx − by t − g (1.21)

1.4.1 Boundary Conditions
The partial differential equation is linear and if the gravity term is included it is nonhomo-
geneous. It is second order in both t and x, and requires two boundary conditions (initial
conditions) on t and two boundary conditions on x. The two conditions on t are normally
specifying the initial velocity and acceleration. The conditions on x are normally specifying the
conditions at the ends of the string, i.e., at x = 0and x = L.

1.5 VIBRATING MEMBRANE
The partial differential equation describing the motion of a vibrating membrane is simply an
extension of the right-hand side of the equation of the vibrating string to two dimensions.
Thus,
2
ρy tt + by t =−g +∇ y (1.22)

2
In this equation, ρ is the density per unit area and ∇ y is the Laplacian operator in either
rectangular or cylindrical coordinates.