## Transcribed Text

1. Let u(x; t) solve the wave equation utt = c2uxx in x 0 with Robin boundary data
ut + ux = r(t) at x = 0
Here is a constant not equal to c and r(t) is a given function. Show that the boundary
values at x = 0 for the standard Riemann variables R = ut cux are related by
R+(0; t) =
2cr(t) (c + )R(0; t)
c
(Hint: start by expressing ut and ux in terms of R.) Use previous equation to nd
R+(x; t) everywhere for the homogeneous initial-value problem in which u(x; 0) = ut(x; 0) =
0. (Hint: make a sketch of the situation and use invariance of R along left-going char-
acteristics to argue that R = 0 for all x t and t 0.) Show that this is consistent
with
u(x; t) =
c
c
Z tx=c
0
r( )d if t > x=c and u(x; t) = 0 otherwise
2. One-dimensional sound waves may be described by a density disturbance eld 0(x; t)
and a velocity eld u0(x; t) that in linear theory satisfy this coupled system of rst-order
PDEs:
0
t + 0u0
x = 0 and 0u0
t + c20
x = 0:
Here c > 0 is the sound speed and 0 is the constant background density of the resting
air, say.
(a) Show that the non-dimensional variables = 0=0 and u = u0=c satisfy
t + cux = 0 and ut + cx = 0:
(b) Show that both u and satisfy the wave equation with wave speed c.
(c) Show that if Riemann variables R+(x; t) and R(x; t) and chosen as
R = u then (@t c@x)R = 0 holds
(d) Find the functions u(x; t) and (x; t) that solve the initial-value problem
u(x; 0) = A(x) and (x; 0) = 2A(x)
where A(x) is an arbitrary function.
1

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