Chapter 3 The Wave Equation 217
When these expressions are substituted into Eq. (2), we have
−Ttan(
φ(x,t))
+Ttan(
φ(x+ x,t))
−ρ xg=ρ x∂(^2) u
∂t^2
. (3)
Recall from elementary calculus that tan(φ(x,t))is the slope of the string at
(x,t)and hence may be expressed in terms of the (partial) derivative with
respect tox:
tan(
φ(x,t))
=∂u
∂x(x,t), tan(
φ(x+ x,t))
=∂u
∂x(x+ x,t).Substituting these into Eq. (3), we have
T(
∂u
∂x(x+^ x,t)−∂u
∂x(x,t))
=ρ x(
∂^2 u
∂t^2 +g)
.
On dividing through by x, we see a difference quotient on the left:T
x(
∂u
∂x(x+ x,t)−∂u
∂x(x,t))
=ρ(
∂^2 u
∂t^2+g)
.
In the limit as x→0, the difference quotient becomes a partial derivative
with respect tox, leaving Newton’s second law in the form
T∂^2 u
∂x^2 =ρ∂^2 u
∂t^2 +ρg, (4)
or
∂^2 u
∂x^2
=^1
c^2∂^2 u
∂t^2+^1
c^2g, (5)wherec^2 =T/ρ.Ifc^2 is very large (usually on the order of 10^5 m^2 /s^2 ), we
neglect the last term, giving the equation of the vibrating string
∂^2 u
∂x^2=^1
c^2∂^2 u
∂t^2, 0 <x<a, 0 <t. (6)This equation is called thewave equationin one dimension. Two- and three-
dimensional versions will be treated in Chapter 5.
In describing the motion of an object, one must specify not only the equa-
tion of motion, but also both the initial position and velocity of the object.
The initial conditions for the string, then, must state the initial displacement
of every particle — that is,u(x, 0 )— and the initial velocity of every particle,
∂u/∂t(x, 0 ).
For the vibrating string as we have described it, the boundary conditions are
zero displacement at the ends, so the boundary value–initial value problem for