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Tlcm 225: Wave Theory Question Paper
Tlcm 225: Wave Theory
Course:Bachelor Of Telecommunication
Institution: Kabarak University question papers
Exam Year:2012
KABARAK
UNIVERSITY
UNIVERSITY EXAMINATIONS
2011/2012 ACADEMIC YEAR
FOR THE DEGREE OF BACHELOR OF TELECOMMUNICATION
TLCM 225: WAVE THEORY
DAY: WEDNESDAY
DATE: 01/08/2012
TIME: 9.00 – 12.00 P.M.
STREAM: Y4S2
INSTRUCTIONS:
1. This paper contains FIVE questions. Answer Question 1 and any other two questions.
2. Question 1 contains 30 marks and the rest contain 20 marks each.
Question 1 (30 marks)
a.) Define sound intensity.
(1 mark)
b.) Give two differences between standing waves and progressive waves.
(2 marks)
c.) State the principle of superposition of waves.
(1 mark)
d.) Two waves are propagating in same long string in opposite directions and their wave
equations are given below
y1 = 6 cos p (x + 4t) and y2= 6 cos p/2 (2x – 8t)
I.) Calculate the frequency, wavelength and speed of each wave.
(4 marks)
II.) Give an equation describing the wave produced by the two waves when they meet at
various points on the string.
(3 marks)
e.) Define Doppler Effect.
(1 mark)
f.) Give an expression for impedance for an inductor and capacitor connected in a series circuit.
(1 mark)
g.) Define a wave.
(1 mark)
h.) 1.) Show that the wave equation y = A sin ?t can also be written as
? t
x ?
y = Asin p
2 ?
-
?
(3 marks)
?T
? ?
2.) Prove that equation in question (1) above is a simple harmonic motion equation and can
be written as;
..
2
+
y = 0
y ?
where symbols have their usual meaning.
(2 marks)
Page 1 of 5
i.) State two factors that determine wave speed of a longitudinal wave.
(2 marks)
j.) From Newton’s law, show that the wave equation can be written as;
2
2
? y
? y
T
- µ
= 0
2
2
?x
?t
where T is the tension and µ is the linear density.
(4 marks)
k.) A spring with a spring constant of 550 Ncm-1 has a mass of 0.4 kg tied at its one end. If it is
pulled downwards to a distance of 0.08 m, determine its;
i.) Angular frequency
(1 mark)
ii.) Linear acceleration
(2 marks)
iii.) Maximum speed of the block.
(2 marks)
Question 2 (20 marks)
a.) A spring 1 mm thick is given a compression force of 350 N at one of its ends. If this
compresses it to a distance of 0.05 m, and it had a length of 0.4 m, determine its density ?
if it had a velocity of 12 ms-1.
(4 marks)
b.) Figure below shows a string tied on one side and loose on the other side.
Fig 1
Show that frequency f of such a structure is given as;
(2n - )
1 .v
f =
4L0
Where L0 is length of string, v is the velocity of the wave and n is the number of nodes.
(3 marks)
c.) Show that the particle pressure that the ear hears can be given by the expression
P = - ßkA cos (?t - kx)
(3 marks)
Where ß is the bulk modulus, k is the wave number and other constants have the usual
meaning.
d.) Determine Eigenvalues and eigenvectors of an oscillating system whose simultaneous
equation are given below and give the interpretation of the Eigenvalues.
..
m
= 4kx + kx
x
1
2
1
(5 marks)
..
m x = 3kx + 2kx
1
2
2
e.) An audience receives a sound wave of pressure 30 Nm-2 (above and below atmospheric
pressure). This is about 100, 000 Nm-2. Find the corresponding maximum displacement if the
frequency is 100 Hz and speed of sound in air is 350 ms-1.
(3 marks)
f.) What is the speed of longitudinal sound wave in a steel wire whose Young’s modulus is 2.0 x
1011 Pa and density of 8.0 x 103 kgm-3?
(2 marks)
Question 3 (20 marks)
a.) Give an expression for total force of a damped simple harmonic motion.
(1 mark)
b.) For the damped oscillator mass m = 250 g, k = 85 Nm and b = 70 gs-1.
i.) What is the period of the motion?
(2 marks)
ii.) How long does it take for the amplitude of the damped oscillations to drop to a one
fourth of its initial value?
(3 marks)
iii.) How long does it take for the mechanical energy to drop to a quarter of its initial
value?
(3 marks)
c.) A string has a linear density of 0.3 kgm-1. Determine its impedance if it has a force of 12 N.
(3 marks)
d.) Fig 2 below shows a connection of an inductor and a capacitor in series. If current I = I0ei?t
and other symbols have their usual meaning;
I
L
C
Va
Fig 2
i.) Determine the equation for reactances.
(3 marks)
ii.) Calculate the impedence Z and hence the current given that V = V0ei?t and Z = Z0eif.
(3 marks)
e.) Give equivalent form of Fourier series in complex exponential form.
(2 marks)
Question 4 (20 marks)
a.) Given a string consisting of two sections joined at x = 0 with a constant tension T along the
whole string, show that the reflection coefficient of amplitude is given as
B
Z - Z
1
1
2
=
A
Z + Z
1
1
2
and transmission coefficient of amplitude
A
2Z
2
1
=
A
Z + Z
1
1
2
Where A1 and B1 are amplitudes of wave one and two respectively and Z1 and Z2 are the
impedences.
(7 marks)
b.) A string vibrating at a frequency of 556.5 Hz has five nodes including the two at the fixed
ends. If the string is 130 cm long, calculate the speed of the traveling wave of the string.
(4 marks)
c.) Define the term periodic motion.
(1 mark)
d.) A pendulum whose bob has a mass of 2 g oscillates at an angular frequency of 6.28 rads-1?
What is its force at 3 cm? Determine also its period T.
(4 marks)
e.) Calculate the intensity of sound of angular frequency 3 rads-1 heard by an ear of surface area
4.1 cm2. Take the velocity of sound in air to be 340 ms-1.
(4 marks)
Question 5 (20 marks)
a.) A particle rotates counterclockwise in a circle of radius 3.00m with a constant angular speed
of 8.00 rad/s. At t = 0, the particle has an x-coordinate of 2.00 m and is moving to the right.
i.) Determine the x-coordinate as a function of time and its phase constant f. (3 marks)
ii.) Find the x-component of the particle’s acceleration and velocity at any time.
(3 marks)
b.)
A
Fig 3
Consider a graphical representation in fig 1 of simple harmonic motion, as described
mathematically as x(t) = A cos(?t + f). When the object is at point A on the graph, what
would be the sign of its velocity and position?
(2 marks)
c.) A stretched string has a linear density of 5.0 gcm-1 and tension of 10 N. A sinusoidal wave on
this string has amplitude of 0.12 mm and frequency of 100 Hz and is traveling in the negative
direction of x. Write an equation for this wave.
(3 marks)
d.) A rocket moves at a speed of 242 ms-1 directly toward a stationary pole (through stationary
air) while emitting sound at frequency f = 1250 Hz.
i.) What frequency f is measured by a detector that is attached to the pole? (3 marks)
ii.) Some of sound reaching the pole reflects back to the rocket as an echo, what
frequency f’ does a detector on the rocket detect for the echo?
(2 marks)
e.) A linear density of a string is 1.6 x 10-4 kg/m. A transverse wave on the string is described by
the equation y = (0.021m) Sin [(2.0m-1)x + (30s-1)t]. What is,
i) the wave speed?
(2 marks)
ii) The tension in the string?
(2 marks)
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