Today’s Lecture
Entropy
Mathematical Description
of
Waves
Entropy
Entropy provides a quantitative measure of disorder.
Consider the isothermal expansion of an ideal gas. If we add heat dQ and let
the gas expand just enough to keep the temperature constant, then from the
first law:
The gas becomes more disordered because there is a larger volume and hence
more randomness in the position of the molecules.
We define the infinitesimal entropy change dS during an
infinitesimal process as:
For an isothermal process the change in entropy is ΔS = Q/T. Higher temperature
implies greater randomness of motion. If the substance is initially cold then adding heat
causes a substantial fractional increase in molecular motion (and randomness). But if
the substance is already hot then adding the same quantity heat adds relatively little
molecular motion that what was already present. Hence Q/T characterizes the increase
in randomness (or disorder) when heat flows into a system.
Entropy
Consider the Carnot
Cycle where we found
If we change the definition of Qc so that it is
the heat added vs heat rejected then
Any closed reversible cycle can be made
up of incrementally small isothermal and
adiabatic cycles. This leads to
This means that the integral
representing the change in entropy,
is path independent!
Entropy
If we take a system around a path that is not closed then its entropy does
change. Since the change in ΔS is path independent, entropy is a state
property, like temperature, and is independent of the path between state 1
and state 2.
Find the entropy change when mixing 100g of aluminum at 300oC with
600g of 20oC water.
First we must find the equilibrium temperature of the aluminum water system
by balancing the heat flow. When we do this we find Teq=29.7oC. The change
in entropy is again found by considering two reversible processes.
Entropy
This definition of entropy is only meaningful for reversible processes. An
irreversible process takes a system out of equilibrium, this means that T
may not be well defined. But entropy is a state variable, it doesn’t care
how you got to a certain state only what the state is!
Consider adiabatic free expansion, an irreversible process, where the
gas is discharged into a vacuum chamber. Neither the temperature nor
internal energy of the gas changes. From the ideal gas law in an
isothermal expansion:
The energy that became unavailable to do work is:
Both of these examples were irreversible processes. Remember entropy is a
state property and as such is independent of how you get to that state. Hence
the results were obtained by reaching the final states via reversible processes.
Entropy
The examples on the previous slide described systems that evolved into systems of
higher disorder. In free expansion and coming to thermal equilibrium, the ability to do
work was lost as the systems evolved into states of higher disorder.
The mixing of colored ink starts from a state of
relative order to a state that is more disordered. A
state of higher entropy. Spontaneous unmixing, a
net decrease in entropy, is never observed!
In general, “When all systems taking part
are included, the change in entropy is
greater than or equal to zero”.
Entropy
In general, “When all systems taking part are
included, the change in entropy is greater than
or equal to zero”.
As an example consider 4 coins that lie on a table
in random orientations.
There is only 1 way that they have either 4 heads
or 4 tails. These are the orientations of highest
order. There are 4 ways that they have either 3
heads and 1 tail or 3 tails and 1 head. These are
the orientations of next highest order. Finally there
are 6 ways in which there are 2 heads and 2 tails.
This is the state of highest disorder!
For any system, the most probable state is the one
with the greatest number of corresponding
microscopic states, which is also the macroscopic
state with the greatest disorder and largest entropy!
Summary of the Second law of Thermodynamics
Cyclic processes:
Second Law of Thermodynamics:
It is impossible to construct a heat engine that extracts
heat from a reservoir and delivers an equal amount of heat.
Irreversible processes decrease the organization of a system. This
is due to statistical reasons.
Efficiency of a reversible
heat engine is
Entropy measures the relative disorder of a system and corresponds to
decreasing energy quality. The entropy difference between two states is
This integral may be done assuming reversible
processes as entropy is a state variable!
Wave Motion
Waves are everywhere:
Earthquakes, 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
then going away, sometimes without a trace…
Some waves are man-made:
radio waves, stadium waves, microwaves, annoying sound
waves of a physics lecture, rap music blaring out of an audio
system in a car…
A wave is a traveling disturbance that
transports energy but not matter.
• Mechanical waves require
– Some source of disturbance
– A medium that can be disturbed
– Some physical connection between or mechanism though
which adjacent portions of the medium influence each other
• All waves carry energy and momentum
What about floating objects and the impact waves
make on them?….
Imagine, you are a 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 shake on the waves
and its motion can we well
described by a harmonic
oscillation.
Any way to find out that you are actually in the sea rather
than on an oscillating platform?
So is there anyway to tell?
The answer is . . .
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.
The observer needs to look out and see what is happening
around him
A wave is an oscillation in time and a wavy pattern in space!
First we will consider oscillations in time
Harmonic oscillation - the motion is sinusoidal
y (t ) = A sin( ωt ) = A sin( 2πft )
y - height of the object with respect to its equilibrium
position;
A – amplitude of the oscillations;
ω = 2πf – angular frequency, measured in radian/s;
f = 1/T – regular frequency, measured in Hertz
(cycles per second) or s-1; T – the period of oscillations
in seconds.
What about velocity and acceleration?
Harmonic Oscillation - Motion is Sinusoidal
y (t ) = A sin( ωt ) = A sin( 2πft )
What about velocity and acceleration?
vo = ωA – can be defined as the amplitude of oscillation’s velocity.
a0 = ω2A – can be defined as the amplitude of oscillation’s acceleration.
From measurements the maximal acceleration/velocity 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 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, kinetic and total.
y (t ) = A sin( ωt ) = A sin( 2πft )
• Both energies are proportional to the amplitude squared;
• They both reach maxima twice per cycle (oscillate at 2ω);
• 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…
More to think about… ☺
λ
The two basic new
parameters are:
wave length and
wave speed.
How do we calculate the speed of a traveling wave?
λ
Consider polls that are placed a wavelength λ apart.
The oscillations there 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
single pole!).
Therefore the wave speed, v,
can be calculated as:
v=
λ
T
= λf
Wave Motion
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
v=
λ
T
How do we calculate the frequency of a traveling wave?
λ
v=
λ
T
= λf
Blue light has shorter
wavelength than red light;
what about their
frequencies?
f =
v
λ
Sound wave and light wave with the same wavelength.
Which has the higher frequency?
Since speed of light is higher than speed of sound, the light wave
has higher frequency.
Harmonic wave - the crests move with a constant speed,
while the material elements oscillate harmonically.
wave speed
material
velocity
Transverse wave – material elements (medium) move
perpendicularly of the direction of wave propagation
The situation becomes more involved in the case of longitudinal waves.
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 the motion of the compressed region (i.e. it is a
pressure wave), which moves from left to right.
Transverse vs. longitudinal wave
Both propagate from left to right, but cause disturbances in
different directions, Δy and Δx.
wavelength, λ
Δx(t ) = Ax cos(ωt )
wavelength, λ
Normally the amplitudes of (harmonic) motion of the particles are
much smaller than the wavelength.
Longitudinal spring waves
Waves on a Spring
Harmonic waves are not the only possible type of waves!
A wave can also have a shape of a propagating pulse.
True for both transverse and longitudinal waves.
A harmonic wave and a pulse are extreme cases.
The intermediate case is a wave train – a finite duration
sinusoidal.
How do we describe a harmonic wave mathematically?
y
x
λ
• Features to incorporate:
in any point in space the wave produces harmonic oscillations of a type:
y(t ) = Ay cos(ωt + ϕ)
ω - angular frequency, ϕ - phase
if we “freeze” the wave in time, we will see a harmonic function in space
y( x) = Ay cos(kx + ϕ)
what is this k ?
if we freeze the wave and move 1 wavelength λ along it, we are supposed
to see the same level of disturbance y
Therefore, it must be kλ
= 2π
so that
y( x + λ) = Ay cos(k ( x + λ) + ϕ) = Ay cos(kx + 2π + ϕ) = y( x)
y
x
λ
y(t ) = Ay cos(ωt + ϕ)
ω - angular frequency
y( x) = Ay cos(kx + ϕ)
ϕ - phase
k is measured in m-1. What is the meaning of it?
If we freeze the wave in time and ride along it, we periodically bump into crests.
k/2π
tells us every how many crests per meter we bump into crests.
ω / 2π = f
tells us every how many times per second we are going to
see a crest if we stay frozen and let the wave propagate.
ω
k
is pretty much the same for space as
k
behaves like a spatial frequency and is usually called the “wave number”
is for time!
y
x
T
λ
is period in time
λ
is period in space
How do we unite the two equations (in time and in space)?
y
x
λ
Considering only one point in space, x0 , means taking ϕ = -kx0.
Freezing it in time, t0 , means taking ϕ = -ωt0.
y
x
λ
-equation of a traveling
harmonic wave
k = 2π / λ
λ = v/ f
ω = 2π / T = 2π f
k = 2π f / v = ω / v
ω = kv
A crest corresponds to a
point, where k(x-vt) = 0.
Therefore position of the crest is given by x =vt, and the crest is
moving at the wave speed v.
y( x, t ) = Ay cos(kx − ωt )
- equation of a harmonic wave
y( x, t ) = Ay cos[k ( x − vt)]
- the same equation rewritten
in a form emphasizing
propagation and wave speed
y( x, t ) = Ay cos[k ( x + vt)]
- what would this one stand for?
v is changed to -v , which means that the wave is propagating in the
negative x direction, from right to left.
In this case location of a crest is given by
cos[k ( x + vt)] = 1
x + vt = 0 ⇒ x = −vt
How can we describe a pulse? (not a harmonic wave)
Generic equation for a wave traveling in
positive x direction with wave speed v:
y ( x, t ) = f ( x − vt)
Here f(x) can be ANY function. The specific function
shape of the wave.
f(x) specifies the
How do we know it is a propagating (traveling) wave?
y (the disturbance) depends on x and t in a VERY SPECIAL WAY:
it only depends on x –
vt.
Therefore the disturbance
a constant, say x0
x − vt = x0
y is the same as long as x – vt equals
⇒ x = vt + x0
A point of constant disturbance, y
at constant wave speed,
= f(x0) , (crest, trough, etc.) moves
Example: a bell-shaped (Gaussian)
curve with a peak at x = 0.
y ( x) = f ( x) = exp( − x )
2
A bell-shaped (Gaussian) curve with
a peak at x = a.
y ( x) = exp[ −( x − a) ]
2
What is the peak is moving along x-axis
with a speed v ?
We can plug in a = vt and get:
y ( x, t ) = f ( x − vt) = exp[ −( x − vt) ]
2
An Example
Ripples on a puddle are propagating at 34cm/s with a frequency of 5.2Hz.
(a) What is the period?
(b) What is the wavelength?
(c) What is the angular frequency?
(d) What is the wave number?
(e) Find the angular frequency from the velocity and wave number.
Another Example
Consider a wave whose displacement is given by y=1.3cos(.69x+.31t).
x and y are measured in centimeters and t in seconds.
(a) What is the period?
(b) What is the wavelength?
(c) What is the amplitude?
(d) What is the velocity?
(e) What is the angular frequency and the wave number?
Yet Another Example
The figure shows a simple harmonic wave at t=0, and later at t=2.6sec.
Write a mathematical description of the wave.
One Final Example
The figure shows a harmonic wave
as a function of x at t=0, and the
same wave as a function of t at
x =0. Write a mathematical
description of the wave including
the sign of the velocity.
At x = 0, it is a negative sine function, hence we choose the negative sign
and v =+2m/s.
Summary of Waves (up to now)
A periodic continuous (SHO)
wave is characterized by its
wavelength and period.
A simple harmonic wave can
be written as:
The minus sign corresponds to a positive velocity and vice versa.
An arbitrary traveling
pulse can be written as: