This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with
x
-dependent coefficients
u
(
x
)
y
tt
−
(
u
(
x
)
y
x
)
x
+
au
(
x
)
y
+|
y
|
p
−2
y
=
f
(
x
,
t
) on (0,
π
)×
under the periodic or anti-periodic boundary conditions
y
(0,
t
)=±
y
(
π
,
t
),
y
x
(0,
t
)=±
y
x
(
π
,
t
) and the time-periodic conditions
y
(
x
,
t
+
T
)=
y
(
x
,
t
),
y
t
(
x
,
t
+
T
)=
y
t
(
x
,
t
). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For
T
=2
π
/
k
(
k
∈
), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with
x
-dependent coefficients.
Global Strichartz estimates for the wave equation with time-periodic potentials