Introduction to Acoustics

918 Part G Structural Acoustics and Noise
Part G 22.3
The only possible values for the wavenumber are given
by the discrete series
kn = nπ/L so that ωn = nπc/L . (22.132)
Using (22.106), the modal mass is mn = ρSL/2 =
Ms/2, where Ms is the total mass of the string. Recall,
however, that the ratio mn/Ms = 1/2 is purely arbitrary
since the modal masses are defined with an arbitrary
multiplicative constant. The important result here is that
all modal masses are equal and do not depend on the
rank n of the mode.
Application: Plucked String. Modal Approach
The particular case where the string is released from
an initial triangular shape at the origin of time without
initial velocity is now examined. With the assumption
of no stiffness, the initial profile of the string is given by
⎧
hx
⎪⎨ for 0 ≤ x ≤ x0
x0
u(0, t) =
. (22.133)
⎪⎩
h(L − x)
for x0 ≤ x ≤ L
L − x0
The modal method consists of looking for solutions of
the form
u(x, t) = �
Φn(x)qn(t) . (22.134)
n
At the origin of time, we can write
u(0, t) = �
Φn(x)qn(0) . (22.135)
n
The unknowns of the problem are the functions qn(0).
Exploiting again the orthogonality properties of the
eigenmodes, we find
qn(0) =
2hL 2
n 2 π 2 x0(L − x0) sin knx0 . (22.136)
The functions qn(t) are given by the oscillator equations
¨qn + ω 2 n qn = 0 (22.137)
which, in the case of zero initial velocity, leads to qn(t) =
qn(0) cos ωnt. In summary, the transverse displacement
of the string is given by
u(x, t) = � 2hL
n
2
n2π 2x0(L − x0) ×
sin knx0 sin knx cos ωnt . (22.138)
Despite the simplicity of this example, a number of
important issues can be derived from this result:
1. The eigenfrequencies are integer multiples of the
fundamental f1 = c/2L. The solution is periodic
and the fundamental frequency is the inverse of the
period of vibration.
2. The amplitudes of the modal components decrease
as 1/n 2 with the rank n of the modes.
3. Because the magnitude is proportional to sin knx0,
a modal component n can be suppressed by selecting
the plucking point x0 = pL/n, wherep is an
integer < n.
4. The excitation point x0 and observation point x can
be interchanged in (22.138). This is an illustration
of the reciprocity principle.
Moving End
To a first approximation, a vibrating string radiates as
a dipole. Because of its small diameter compared to the
acoustic wavelength, it is a poor radiator. If significant
sound is to be radiated, one end of the string must be
coupled to a component with a considerable vibrating
area (plate, shell, etc.). In Sect. 22.3.2, the general properties
of a heterogeneous string attached to a mass were
obtained. In this section, the simple example of an ideal
string coupled to a spring will be considered to illustrate
the influence of such coupling on the eigenmodes, eigenfrequencies
and modal masses. We consider a string
fixed at point x = 0toaspringofstiffnessK0. The
boundary condition at this end is then
� �
∂u
T = K0 u(0, t) . (22.139)
∂x x=0
The string is rigidly fixed at the other end, so
that u(L, t) = 0. We look for solutions of the form
u(x, t) = Φ(x)cosωt. Following (22.118), the eigenfunctions
Φ(x) must satisfy the equation
d2Φ dx2 + k2Φ = 0 (22.140)
and the boundary conditions. Thus, we derive the equation
for the eigenvalues kn
tan(kn L) =− knT
with kn > 0 . (22.141)
K0
The graphical representation of (22.141) shows that the
kn are no longer integer multiples of k1 (Fig. 22.8). As
a consequence, the free vibration of the string is no
longer periodic. Note that the spring decreases the eigenfrequencies
relative to those of a perfectly rigid end.
The eigenfunctions become
Φn(x) = sin knx + knT
K0
cos knx = sin kn(L − x)
cos kn L
(22.142)