Oscillations, waves, optics 2007/2008 ­ problems, part 1.
1. Compose computer program, which finds and writes into the file displacement time dependence
of anharmonic damped oscillations. Reverse force dependence on the displacement
is Fr(x) = -kx(1 + x2
), damping force depends on the velocity as Fo(x) = - ˙x. Spring
constants k, , , system weight m and initial conditions will be parameters of the program.
Find dependence of system on the parameters, especially for harmonic undamped
oscillations, harmonic damped oscillations, positive and negative values of anharmonic force
parameter . Graphs of time dependence of the mentioned states will be part of solution..
2. Find eigen-frequencies and eigen-modes of shown system of transverse oscillations (perpendicular
to the picture plane). All balls have weight m and all spring transverse spring
constant k.
3. A string of length L is tensile between solid points and displaced from its equilibrium position
as shown in the figure. The string is released at time t = 0 . Find function u(x, t)
describing time evolution of string shape. Wave phase velocity in the string is c.
x
h
u
L/4 3L/4 LL/2
4. A simple seismometer consists from weight hanging on a spring fastened to a solid construction
fastened to ground. Weight movement is critically damped and vertical movement of the
weight is registered. Show that measured amplitude of steady oscillations, which are caused
by the ground vertical oscillations H cos t, is given by the relation A/H = (/20)[R()]1/2
,
where 0 is angular frequency of weight eigen-oscillations, R() = 2
2
/[(2
0 -2
)2
+2
2
] is
response function and  is damping constant. Draw graph of A/H as a function of oscillation
frequency .
Oscillations, waves, optics 2007/2008 ­ problems, part 2.
5. A thin non-absorbing layer of thickness d and refractive index n1 is deposited on a nonabsorbing
substrate of refractive index n. Calculate reflectivity at normal incidence of a
system as a function of wavelength. Calculate multiple reflections and suppose that incoming
light is coherent and refractive indeces are independent on the wavelength. Propose a way
to obtain values of n, n1, d from measured spectral dependence. Plot graph of reflectivity
spectral dependence for parameters n = 1.5, n1 = 1.3 a d = 150 nm in the visible light
spectrum.
6. A diffraction screen is put between point monochromatic light source and an observing
screen. The diffraction and observing screens are parallel. The diffraction screen is perforated
with rectangle holes of sides a1, a2. Holes are aligned in a rectangle lattice, their centers
have positions R = n1d1 + n2d2, where 0  n1 < N1, 0  n2 < N2 a n1, n2 are integers.
Numbers N1, N2 describe macroscopic system size. The vectors d1, d2 are perpendicular
and are parallel to rectangle sides. Calculate final amplitude and intensity on the observing
screen. Show, that the result is a product of two terms, one of them depending only on the
holes alignment (geometric factor) and the other only on the hole shape (structure factor).
Plot graph of intensity distribution on the observing screen for suitably chosen parameters.
7. Plot optical schemes of three basic telescopes ­ Galilei, Kepler and Newton. For Galilei and
Kepler telescope solve following problem. The telescope is focused in such a way, that we
can see sharp Moon image by an eye accommodated to infinity. The screen is put in the
distance d from the ocular. How much the ocular with focal distance foc has to be displaced
from the initial position to observe sharp image of Moon on the screen? How big will be the
Moon image, if the objective focal distance is fob? Solve general at first and then with the
values fok = 2 cm, fob = 30 cm, d = 16 cm.
8. Small fish is in the cylindrical fish-tank, with curvature radius R = 40 cm oriented vertically,
at a line connecting center of a fish-tank and observer at distance x = 7,5 cm from center
of a fish-tank and is flowing with velocity v = 3,0 cm.s-1
perpendicular to this connecting
line horizontally. Find image position of the fish, its magnification and the image velocity.
The refractive index of a water is n = 1,33, glass fish-tank can be neglected in the first order
approximation.
Figure 1: Two sharks in a cylindrical fish-tank. Illustrative photo.