PH3111 - Waves
1) Four identical masses are connected in a chain by three identical springs and are restricted to
move in one dimension (along the direction of the chain). What are (a) the characteristic
frequencies and (b) the corresponding normal modes for small vibrations?
2) In class we showed that in the limit ka << 1, the mechanical impedance, Z, for waves on a set
of masses connected by springs, with spring constant, C, and spacing a, was real.
This came from expanding the force, F,
F = C u(0, t ) (1 + ika +
as well as the relationship between frequency,
ω =
− 1)
, and wavenumber, k,
4C
ka
sin
=
m
2
4C
m
ka
+
2
and keeping only the lowest order terms, then writing the force as
F=−Z
∂u
∂t
Following the same procedure, but keeping the next highest terms (proportional to (ka)2 ),
compute the first correction term and show it is imaginary and proportional to frequency (i.e. it is
a “reactance”).
3) The equation for small, transverse vibrations of a thin rod with uniform cross-section is
∂ 2y
EI ∂ 4 y
=−
∂t2
ρ ∂ x4
where y is the (small) displacement perpendicular to the rod and x is distance along the rod.
A) Show that
y ( x , t ) = cos(ω t + ϕ ) A cosh
ωx
ωx
ωx
ωx
+ B sinh
+ C cos + D sin
v
v
v
v
is the general solution to this equation for vibrations at frequency , where A, B, C, and D are
constants (which will be determined by end conditions) and
v2 = ω
EI / ρ
(Note: E is the Young’s modulus, I depends on the shape of the cross-section, and
of the material – they are all constants.)
is the density
B) Determine the characteristic frequencies (the “normal mode frequencies”) for a thin bar of
length L which is hinged at both ends. The conditions at the ends for a “hinge” are
y=0 ;
∂2y
=0
∂x 2
For a string, the normal mode frequencies are an integer times the lowest frequency (the
“fundamental”). You should not get that result here.
C) (This is a “bonus” question) Determine the (approximate) characteristic frequencies for a thin
bar of length L which is hinged at one end and free at the other. The conditions for a free end are
∂2y
=0 ;
∂x 2
∂3y
=0
∂x 3
(Note: you do not get a nice closed-form solution in this case, but can resort to some simple
numerical techniques to get approximate values).
Displacement
4) Estimate the frequency of the lowest frequency mode for vibration on a string of length L,
fixed at both ends, using Rayleigh’s principle.
Guess that the normal mode looks like a
truncated triangle, as shown below, where x1
1.4
and x2 are equidistant, d, from the center.
1.2
After you find your equation for the frequency
2d
1.0
in terms of d, find the value for d which
minimizes the frequency. How does your result
0.8
x1
x2
compare to the exact solution?
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
x (units of L)
0.8
1.0