Complex Notation of Travelling
When dealing with the superposition of
waves, where there are several phase
constants, it is sometimes an advantage
to use the complex notation for the
solution to the classical wave equation:!
2 1 2
(a) Nature of Waves"
(i) Longitudinal waves."
There are waves in which the quantity which is varying does so in
the direction of propagation of the waves.i.e. is the displacement
is in the direction of k . The classic example is a sound wa
Division of Amplitude
You should be able to :
Find the path difference between successive reflected waves
Use time reversal to determine the relationship between different types of
reflections and transmission
Know how to add up successive waves in trans
No one has been able to define the difference between interference and diffraction. It is
just a question of usage, there being no specific important difference between them. The
best we can so roughly speaking is to say that when there are on
It can be shown that, in general, the particles of the surface
execute a vertical elliptical path which approximates to a
circle if the depth of the liquid is greater than the wavelength.!
The concept of wave motion is one of the most
iv) Superposition of Simple Harmonic Motions; Harmonics
This is an elementary example of one of the most
important theorems i.e. Fouriers Theorem.
Consider the superposition of two S.H. vibrations
which have related but not equal frequencie
Let speed on sand =v1, speed in water v2.
If a lifeguard wishes to reach a person in the water and
needs to travel over sand and water. Which is the best route
to follow given that she can run faster than she can swim?
We have seen above that the equation of motion of the
simple pendulum is !
+ sin = 0
The simple harmonic nature of was derived from the
(See Berekeley "mechanics" p.197-199, 225-229 1st Edition, p.224-226,
f) THE DRIVEN HARMONIC OSCILLATOR.!
i) Equation of Motion and its Solution.!
!If a damped harmonic oscillator is coupled to a generator which
drives the oscillator at a certain frequency, we have as the
equation of motion: !
x + bx + kx = F(t).
3) One Body - Two Degrees of Freedom System.!
The systems considered in the last section have two and 4
degrees of freedom, respectively. Lets now consider one body
which can oscillate in 2 dimensions!
The parameters ax, ay, x and y are determined by the
g) The Principle of superposition
An important property of harmonic oscillators is that the
vibrations are additive. If x1(t) is the motion under a
driving force F1(t) and x2(t) is the motion under the driving
force F2(t), then x1(t) + x2(t) would be the
d) The LC- Circuit Oscillator.!
The generality of the second-order D.E. as the equation of motion
of an oscillator is demonstrated by the LC-circuit; an electrical
oscillator. The LC-circuit closely resembles a mass-spring system
in that each has a charac
ii) Superposition of simple harmonic Motion; same
frequency, but at right angles
x 2 y2
+ 2 2
cos = sin 2
Consider a particle to be subjected to two S.H.Ms of
the same frequency but in two different directions at
right angles to each othe
We see that v0 = max when = 0 and from before this
implies that = /2.
It might be thought that the velocity will resonate at the same
frequency as the amplitude, but, as we shall show, this turns
out not to be the case. This is seen a
(i) Power Dissapation.!
Let us calculate the rate at which energy is dissipated by a damped
harmonic oscillator in the weak damping limit ie. 0 >
We have, !
If we assume that ae 2 does not change significantly over one
cycle then the t
School of Physics, Faculty of Engineering
& Information Sciences, 2013
PHYS215 Vibrations, Waves and Optics (6 cp)
Timetable (3hrs Lectures, 3 hrs Labs)