218 Chapter 3 The Wave Equation
the string is
∂^2 u
∂x^2=^1
c^2∂^2 u
∂t^2, 0 <x<a, 0 <t, (7)
u( 0 ,t)= 0 , u(a,t)= 0 , 0 <t, (8)
u(x, 0 )=f(x), 0 <x<a, (9)
∂u
∂t(x,^0 )=g(x),^0 <x<a, (10)under the assumptions noted plus the assumption that gravity is negligible.
EXERCISES
1.Find the dimensions of each of the following quantities, using the facts that
force is equivalent tomL/t^2 , and that the dimension of tension isF(force):
u,∂^2 u/∂x^2 ,∂^2 u/∂t^2 ,c,g/c^2. Check the dimension of each term in Eq. (5).
2.Suppose a distributed vertical forceF(x,t)(positive upwards) acts on the
string. Derive the equation of motion:∂^2 u
∂x^2 =1
c^2∂^2 u
∂t^2 −1
TF(x,t).
The dimension of a distributed force isF/L. (If the weight of the string is
considered as a distributed force and is the only one, then we would have
F(x,t)=−ρg. Check dimensions and signs.)
3.Find a solutionv(x)of Eq. (5) with boundary conditions Eq. (8) that is
independent of time. (This corresponds to a “steady-state solution,” but
the termsteady-stateis no longer appropriate.Equilibrium solutionis more
accurate.)
4.Suppose that the string is located in a medium that resists its movement,
such as air. The resistance is expressed as a force opposite in direction and
proportional in magnitude to velocity. Thus it affects only Eq. (2). Proceed
to derive the equation that replaces Eq. (7) for this case.3.2 Solution of the Vibrating String Problem
The initial value–boundary value problem that describes the displacement of
the vibrating string,