20.8 EXERCISES
20.14 Solve
∂^2 u
∂x∂y
+3
∂^2 u
∂y^2=x(2y+3x).20.15 Find the most general solution of∂^2 u/∂x^2 +∂^2 u/∂y^2 =x^2 y^2.
20.16 An infinitely long string on which waves travel at speedchas an initial displace-
ment
y(x)={
sin(πx/a), −a≤x≤a,
0 , |x|>a.It is released from rest at timet= 0, and its subsequent displacement is described
byy(x, t).
By expressing the initial displacement as one explicit function incorporating
Heaviside step functions, find an expression fory(x, t)atageneraltimet>0. In
particular, determine the displacement as a function of time (a) atx=0,(b)at
x=a,and(c)atx=a/2.
20.17 The non-relativistic Schr ̈odinger equation (20.7) is similar to the diffusion equa-
tion in having different orders of derivatives in its various terms; this precludes
solutions that are arbitrary functions of particular linear combinations of vari-
ables. However, since exponential functions do not change their forms under
differentiation, solutions in the form of exponential functions of combinations of
the variables may still be possible.
Consider the Schr ̈odinger equation for the case of a constant potential, i.e. for
a free particle, and show that it has solutions of the formAexp(lx+my+nz+λt),
where the only requirement is that
−^2
2 m(
l^2 +m^2 +n^2)
=iλ.In particular, identify the equation and wavefunction obtained by takingλas
−iE/,andl, mandnasipx/,ipy/andipz/, respectively, whereEis the
energy andpthe momentum of the particle; these identifications are essentially
the content of the de Broglie and Einstein relationships.
20.18 Like the Schr ̈odinger equation of the previous exercise, the equation describing
the transverse vibrations of a rod,
a^4∂^4 u
∂x^4+
∂^2 u
∂t^2=0,
has different orders of derivatives in its various terms. Show, however, that it
has solutions of exponential form,u(x, t)=Aexp(λx+iωt), provided that the
relationa^4 λ^4 =ω^2 is satisfied.
Use a linear combination of such allowed solutions, expressed as the sum of
sinusoids and hyperbolic sinusoids ofλx, to describe the transverse vibrations of
a rod of lengthLclamped at both ends. At a clamped point bothuand∂u/∂x
must vanish; show thatthis implies that cos(λL)cosh(λL) = 1, thus determining
the frequenciesωat which the rod can vibrate.
20.19 An incompressible fluid of densityρand negligible viscosity flows with velocityv
along a thin, straight, perfectly light and flexible tube, of cross-sectionAwhich is
held under tensionT. Assume that small transverse displacementsuof the tube
are governed by
∂^2 u
∂t^2
+2v∂^2 u
∂x∂t+
(
v^2 −T
ρA)
∂^2 u
∂x^2=0.
(a) Show that the general solution consists of a superposition of two waveforms
travelling with different speeds.