250 Chapter 3 The Wave Equation
18.Fort<0, water flows steadily through a long pipe connected atx= 0
to a large reservoir and open atx=ato the air. At timet=0, a valve at
x=ais suddenly closed. Reasonable expressions for the conservation of
momentum and of mass are
∂u
∂t=−∂p
∂x,^0 <x<a,^0 <t, (A)
∂p
∂t=−c2 ∂u
∂x,^0 <x<a, allt, (B)
wherepis gauge pressure anduis mass flow rate. If the pipe is rigid,
c^2 =K/ρ, the ratio of bulk modulus of water to its density. Show that
bothpandusatisfy the wave equation. The phenomenon described here
is calledwater hammer.
19.Introduce a functionvwith the definitionu=∂v/∂x,p=−∂v/∂t.
Show that (A) becomes an identity and that (B) becomes the wave equa-
tion forv.
20.Reasonable boundary and initial conditions foruandpareu(x, 0 )=U 0 (constant), 0 <x<a,
p(x, 0 )= 0 , 0 <x<a,
p( 0 ,t)= 0 , allt,
u(a,t)= 0 , t> 0.Restate these as conditions onv. Show that the first and third equations
may be replaced byv(x, 0 )=U 0 x, 0 <x<a,
v( 0 ,t)= 0 , allt.21.Solve the problem in Exercise 20; find a series form forv(x,t).
22.In many problems involving fluid flow, the combination
∂u
∂t+V∂u
∂x
(called theStokes derivative)appears.HereVis the speed of the fluid in
thex-direction. IfVequalsuor otherwise depends onu,thisoperatoris
nonlinear and difficult to work with. Let us assume thatVis a constant,
so that the operator is linear, and define new variablesξ=x+Vt,τ=x−Vt, u(x,t)=v(ξ,τ).