Introduction to Acoustics

first to solve the wave equation (22.118) with initial
conditions [22.7]
u(x, 0) = f (x) and
∂u
(x, 0) = g(x) . (22.119)
∂t
Using the change of variables ξ = x − ct and η = x + ct,
we have
u(x, t) = U(ξ,η) (22.120)
from (22.120), we obtain:
⎧
∂
⎪⎨
2 �
u ∂2U = c2
∂t2 ∂ξ2 − 2 ∂2U ∂ξ∂η + ∂2U ∂η2 �
. (22.121)
⎪⎩
∂2u ∂x2 = ∂2U ∂ξ2 + 2 ∂2U ∂ξ∂η + ∂2U ∂η2 Inserting (22.121) into(22.118) yields � ∂2U � / � ∂ξ∂η �
= 0 which implies
u(x, t) = F(x − ct) + G(x + ct) , (22.122)
where F and G are are two twice-differentiable functions.
F(x − ct) represents a travelling wave moving to
the right (direction of increasing values for x), while
G(x + ct) is a travelling wave moving to the left. Using
(22.119) implies that F and G must fulfill the conditions
⎧
⎨
F(x) + G(x) = f (x)
⎩−cF
′ (x) + cG ′ . (22.123)
(x) = g(x)
Solving (22.123) yields
⎧
F(x) =
⎪⎨
⎪⎩
1 1
f (x) − [−F(0) + G(0)]
2 2
− 1
�x
g(s)ds
2c
0
G(x) = 1 1
f (x) + [−F(0) + G(0)]
2 2
+ 1
. (22.124)
�x
g(s)ds
2c
0
Finally, the solution of the wave equation is written
explicitly
u(x, t) = 1
[ f (x − ct) + f (x + ct)]
2
+ 1
x+ct �
g(s)ds . (22.125)
2c
x−ct
Structural Acoustics and Vibrations 22.3 Strings and Membranes 917
Semi-infinite String. For a semi-infinite string rigidly
fixed at x = 0, we have the boundary condition
u(0, t) = 0. This requires:
F(−ct) + G(ct) = 0 (22.126)
and, finally, with the appropriate change of variables
u(x, t) =−G(x − ct) + G(x + ct)
= F(x − ct) − F(−x − ct) . (22.127)
Equation (22.127) expresses the fact that, due to the
fixed end at x = 0, the left-traveling wave is reflected
with a change of sign and becomes a right-traveling
wave. The validity domain for (22.127) is0≤ x < +∞
and t > 0.
String of Finite Length. In the case of an ideal string
fixed at both ends, the wave approach can still be used,
with the additional boundary condition u(L, t) = 0,
which yields
u(L, t) =−G(L − ct) + G(L + ct)
= F(L − ct) − F(−L − ct) = 0 . (22.128)
Equation (22.128) can be rewritten F(z) = F(z − 2L),
which shows that F (and G) are now periodic functions
with spatial period 2L or, equivalently, temporal
period 2L/c. The validity domain of (22.128) isnow
0 ≤ x ≤ L and t > 0. Equations (22.126)–(22.128) can
be used for step-by-step constructions of the string shape
at successive instants of time.
String Fixed at Both Ends. Eigenmodes
Injecting a harmonic wave of the form u(x, t) = e i(ωt−kx)
in (22.118), we find the dispersion equation D(ω, k)
that governs the relationship between frequency ω and
wavenumber k. Here, we obtain
D(ω, k) = ω 2 − c 2 k 2 = 0 . (22.129)
This equation shows that the ratio between the frequency
and wavenumber is constant, which is a characteristic
property of a nondispersive medium. If we assume
further that the string is rigidly fixed at both ends, the
eigenmodes must satisfy the equation
d2Φ(x) ω2
+ Φ(x) = 0 (22.130)
dx2 c2 with the boundary conditions Φ(0) = Φ(L) = 0 from
which we obtain
Φn(x) = sin knx . (22.131)
Part G 22.3