WAVE THOERY 1

SPH 300

WAVE THEORY


WRITTEN BY:
Eng. John Namale




i

,TABLE OF CONTENTS PAGE
LECTURE ONE............................................................................................................................ 1
PERIODIC MOTION AND THEIR SUPERPOSITION ..................................................... 1
LECTURE TWO ........................................................................................................................ 21
FORCE & ENERGY .............................................................................................................. 21
LECTURE THREE .................................................................................................................... 28
COMPLEX VARIABLES REPRESENTATION OSCILLATIONS ................................ 28
LECTURE FOUR ....................................................................................................................... 36
FREE AND FORCED VIBRATIONS .................................................................................. 36
LECTURE FIVE ......................................................................................................................... 70
FOURIER ANALYSIS ........................................................................................................... 70
LECTURE SIX............................................................................................................................ 82
COUPLED OSCILLATORS AND NORMAL MODES ..................................................... 82
LECTURE SEVEN ..................................................................................................................... 97
THE FREE VIBRATIONS OF STRETCHED STRINGS.................................................. 97
LECTURE EIGHT ................................................................................................................... 104
DIFRACTION OF LIGHT AND WATER WAVES ......................................................... 104
LECTURE NINE ...................................................................................................................... 110
REFLECTIONS OF MECHANICAL (SHEAR) WAVES AT A DISCONTINUITY .......110




ii

, LECTURE ONE

PERIODIC MOTION AND THEIR SUPERPOSITION



1.1 Introduction

This deals with periodicity of oscillatory motion of bodies. In this case sinusoidal kind of
motion is considered. Also discussed here is the combination of motions of one or more
bodies especially in one medium.

1.2 Lecture Objectives




By the end of this lecture, you should be able to:
 Differentiate between periodic, oscillatory and simple harmonic motions
 Understand the concept of wave superposition in one dimension
 Derive equations of superposed equations with equal frequencies, different
frequencies (beats) and for many superposed vibrations of same frequency.
 Combine two vibrations at right angle.
 Draw Lissajous figures.
 Compare perpendicular and parallel superposition.




1.3 Periodic motion and their superposition.

Periodicity is a pattern of movement or displacement that repeats itself over and over again. This
pattern may be simple or complex. On the other hand when a body moves back and forth
repeatedly about a mean position, its motion is called Oscillatory.
The motion of the planets around the sun is periodic motion i.e. it takes equal time to go once
around the sun. But the motion is not oscillatory as the planets move in one direction only.
Alternatively oscillatory motion is exemplified in the pendulum motion. The pendulum motion
moves in one direction and then turns back, crosses the mean point (equilibrium) position or
mean position and then goes to the other side, and turns back again. This motion goes on
repeating indefinitely.




1

, The oscillatory motion may also be periodic if it takes equal time to come back to its original
position after completing the to and fro motion once. Such a periodic motion is called harmonic
motion. Thus, a harmonic motions is periodic but not all periodic motion are harmonic.
If a system is disturbed from its position of equilibrium and left to itself, in the absence of
friction the harmonic oscillatory it executes about the mean position of equilibrium, is called
simple harmonic oscillation.

1.3.1 Sinusoidal Vibrations
Our attention will be focused to sinusoidal vibration. For, instance we have a body attached to a
spring, the force exerted on it at a displacement x from equilibrium is
F (x) = - (k1x + k2x2 + k3x3 + …………..
Where k1 ,k2 ,k3 etc are a set of constants, and we can always find a range of values for X within
which the sum of the terms x2, x3,----- xn is negligible, according to some stated criterion (e.g., 1
part in 103, or 1 part in 106) compared to the term – kx, unless k, is equal to zero. If the body of
mass m and that of the spring is negligible, then SHM restoring force on the particle is expressed
as
F = - kx (1.1)
where x denotes the displacement.
But applying Newton’s second law to the motion of the body
F = ma
Where a is acceleration of the particle executing SHM
d 2x
Since a= 2
dt
d 2x
F m (1.2)
dt 2


From (i) and (ii)

d 2x
m  kx
dt 2
d 2x k
or 
x
k dt 2 m
Putting  2
m

d 2x
  2 x
2
dt




2