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 man­made:
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(ωt ) = A sin(2πft )
­ height of the object with respect to its y equilibrium position;
A – amplitude of the oscillations;
– angular frequency, ωmeasured in rad/s;
= 2πf
f = 1 – regular frequency, measured in Hertz /T
T – the period of (cycles per second) or s ; ­1
oscillations in seconds
What about velocity and acceleration?
dy
v y (t ) =
= ωA cos(ωt ) = v0 sin(ωt − π / 2)
dt
Harmonic oscillation ­ the motion is sinusoidal
y (t ) = A sin(ωt ) = A sin( 2πft )
dy
v y (t ) =
= ωA cos(ωt ) = v0 sin(ωt + π / 2)
dt
v0 = ωA – can be defined as the amplitude of oscillations of the velocity
a y (t ) =
dv y
dt
Acceleration
= −ω A sin(ωt ) = a0 sin(ωt + π )
2
a0 = ω 2 A – 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 =
ω=
2π
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(ωt ) = A sin( 2πft )
Potential energy:
Kinetic energy:
1 2 1 2 2
E p = ky = kA sin (ωt )
2
2
1 2 1
2 2
2
K = mv = mω A cos (ωt )
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.
λ
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?
λ
Two poles placed the wavelength, λ , 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
λ
v = = λf
T
L
v ≡ = λf
t
Wave motion (a traveling wave)
L
λ
There is NO direct connection between the wave speed, and the velocity of motion of material particles, dy
v y (t ) =
= ωA cos(ωt )
dt
λ
v=
T
Longitudinal Waves In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The animation above shows a one­dimensional 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.