Waves are everywhere:
Earthquake, vibrating strings of a guitar, light from the sun;
a wave of heat, of bad luck, of madness…
Something moving, passing by, bringing a change and
than going away, sometimes without a trace…
Some waves are man-made:
radio waves, stadium waves, microwaves, annoying sound
waves of a physics lecture… amplified by the electronics and
loudspeakers…
A wave is a traveling disturbance that
transports energy but not matter.
• Mechanical waves require
– Some source of
disturbance
– A medium that can be
mechanically disturbed
– Some mechanism
through which adjacent
regions of the medium
influence each other
• All waves carry energy and
momentum
The stadium "wave" travels all around the
stadium. None of the fans travel around the
stadium. They only stand up and sit down.
Well, let’s forget about humans and talk about passively
floating objects and the impact the waves make on them….
Imagine, you are a happy seal
on a buoy. It is nicely warm
and you closed your eyes.
What kind of motion are you
going to experience?
The buoy and the seal are
going to bob on the waves
and its motion can we well
described by a harmonic
oscillation.
Any way to find out that you are actually in sea rather than on
a shaking platform?
NO! An observer recording motion in a
single point in space only sees a
harmonic oscillation and there is no way
to know, whether or not there are waves.
How can he find out that he is
shaking on waves?
To look out and take a photograph (or
to see what happens around him).
The complex nature of a wave:
it is an oscillation in time and a wavy pattern in space!
Let’s do the oscillation part first.
Oscillations have amplitude and frequency.
Any way to measure them without looking out?
You can measure frequency and acceleration…
And use them to derive the amplitude.
http://www.phy.ntnu.edu.tw/java/Pendulum/Pendulum.html
Harmonic oscillation - the motion is sinusoidal
y(t )  A sin( t )  A sin( 2ft)
y - height of the object with respect to its
equilibrium position;
A – amplitude of the oscillations;
  2f – angular frequency, measured in rad/s;
f  1/ T– regular frequency, measured in Hertz
(cycles per second) or s ; T – the period of
-1
oscillations in seconds
What about velocity and acceleration?
dy
v y (t ) 
 A cos(t )  v0 sin( t   / 2)
dt
Harmonic oscillation - the motion is sinusoidal
y(t )  A sin( t )  A sin( 2ft)
dy
v y (t ) 
 A cos(t )  v0 sin( t   / 2)
dt
v0  A – can be defined as the amplitude of oscillations of the velocity
a y (t ) 
dv y
dt
Acceleration
  A sin( t )  a0 sin( t   )
2
a0   A
2
– can be defined as the amplitude of oscillations of the
acceleration
From measurements the maximal acceleration and the period, we can
calculate the frequency and the amplitude of the oscillations.
How is the frequency connected to the
parameters of the system?
1
k
f 

2
m
k
m
k is elastic modulus of the spring;
m is the mass of the oscillating object;
Spring elasticity: the restoring spring force, F, where y
is the deviation from the equilibrium position
F  ky
An oscillation (and a wave!) requires a restoring force – elastic tension
in the spring in this case.
The larger the restoring force (the tougher the spring) the faster it goes
(the higher is the frequency of oscillation)
A large mass (inertia) of the oscillation object slows the oscillations
down (reduces the frequency).
Energy of oscillations – potential and kinetic.
y(t )  A sin( t )  A sin( 2ft)
Potential energy:
Kinetic energy:
1 2 1 2 2
E p  ky  kA sin (t )
2
2
1 2 1
2 2
2
K  mv  m A cos (t )
2
2
• Both energies are proportional to the amplitude squared;
They both reach maxima twice per cycle;
They both are always positive and their sum remains constant;
They oscillate out of phase with each other.
Waves have all the same stuff as the oscillation do:
amplitude, frequency, energy…
But they also have much more, because they propagate in
space…
That’s where the new stuff starts.
l
The two basic new
parameters are:
wave length and
wave speed.
How do we define the wave speed?
L
If two points are a distance L apart and it takes
a wave crest a time t to travel between them,
the wave speed, v, is calculated as:
It is just a regular definition of speed - how
fast the wave crest travels.
L
v
t
How do we calculate the speed of a traveling wave,
knowing its period and wavelength?
l
Two poles placed the wavelength, l, apart.
The oscillations at the two poles are always in phase.
The time it takes a crest to travel between them is the period of
oscillations T (exactly the time between two consecutive crests at a
point!).
Therefore the wave speed, v,
can be calculated as
v
l
T
 lf
L
v   lf
t
Wave motion (a traveling wave)
L
l
There is NO direct connection between the wave speed,
and the velocity of motion of material particles,
dy
v y (t ) 
 A cos(t )
dt
v
l
T
Longitudinal Waves
In a longitudinal wave the particle displacement is parallel to the
direction of wave propagation.
The animation above shows a one-dimensional longitudinal plane
wave propagating down a tube.
The particles do not move down the tube with the wave; they
simply oscillate back and forth about their individual equilibrium
positions. Pick a single particle and watch its motion!
The wave is seen as a motion of compressed regions (i.e. it is a
pressure wave), which move from left to right.