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Waves – Chapter 14
Superposition and Interference
Waves of small amplitude traveling through the same
medium combine, or superpose, by simple addition.
If two pulses combine to give a larger pulse, this is
constructive interference (left). If they combine to give
a smaller pulse, this is destructive interference (right).
constructive
destructive
constructive
Two waves with distance to
the source different by whole
integer wavelengths Nλ
destructive
Two waves with distance to the
source different by half-integer
wavelengths Nλ
Two-dimensional waves exhibit interference as well. This is
an example of an interference pattern.
A: Constructive
B: Destructive
Superposition and Interference
If the sources are in phase, points where the
distance to the sources differs by an equal
number of wavelengths will interfere
constructively; in between the interference will be
destructive.
Constructive: l = n : , 2 , 3 ....
Destructive: l = (n+1/2)  :  /2, 3  /2, 5  /2...
Interference
Speakers A and B emit sound
waves of  = 1 m, which
interfere constructively at a
donkey located far away (say,
200 m). What happens to the
sound intensity if speaker A is
moved back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell
A
B
L
Interference
Speakers A and B emit
sound waves of  = 1 m,
which interfere
constructively at a donkey
located far away (say, 200
m). What happens to the
sound intensity if speaker A
is moved back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell
If  = 1 m, then a shift of 2.5 m corresponds to 2.5 ,
which puts the two waves out of phase, leading to
destructive interference. The sound intensity will
therefore go to zero.
Follow-up: What if you
move speaker A back
by 4 m?
A
B
L
Standing Waves
A standing wave is fixed in location,
but oscillates with time.
These waves are found on strings with both ends fixed, or
vibrating columns of air, such as in a musical instrument.
The fundamental, or lowest, frequency
on a fixed string has a wavelength twice
the length of the string.
Higher frequencies are called harmonics.
Standing Waves on a String
Points on the string which never move are called
nodes; those which have the maximum movement are
called antinodes.
There must be an integral number of half-wavelengths
on the string (must have nodes at the fixed ends).
This means that only certain frequencies (for fixed
tension, mass density, and length) are possible.
First
Harmonic
Second
Harmonic
Third
Harmonic
First
Harmonic
Second
Harmonic
Third
Harmonic
Musical Strings
Musical instruments are usually designed so that the
variation in tension between the different strings is
small; this helps prevent warping and other damage.
A guitar has strings that
are all the same length,
but the density varies.
In a piano, the strings vary in both length
and density. This gives the sound box of a
grand piano its characteristic shape.
Standing Waves I
A string is clamped at both ends
and plucked so it vibrates in a
standing mode between two
extreme positions a and b. Let
upward motion correspond to
positive velocities. When the
string is in position b, the
instantaneous velocity of points
on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
a
b
Standing Waves I
A string is clamped at both ends
and plucked so it vibrates in a
standing mode between two
extreme positions a and b. Let
upward motion correspond to
positive velocities. When the
string is in position b, the
instantaneous velocity of points
on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
Observe two points:
Just before b
Just after b
Both points change
direction before and after b,
so at b all points must have
zero velocity.
Every point in in SHM, with the
amplitude fixed for each position
Standing Waves in Air Tubes
Standing waves can also be excited in columns of air,
such as soda bottles, woodwind instruments, or
organ pipes.
A sealed end must be at a NODE (N),
an open end must be an ANTINODE (A).
Standing Waves
With one end closed and one open:
the fundamental wavelength is four times the
length of the pipe, and only odd-numbered
harmonics appear.
Standing Waves
If the tube is open at both ends:
both ends are antinodes, and the sequence of
harmonics is the same as that on a string.
Musical Tones
Human Perception: equal steps in
pitch are not additive steps, but
rather equal multiplicative factors
Frequency doubles for octave
steps of the same note
The frets on a guitar are used to
shorten the string.
Each fret must shorten the string
(relative to the previous fret) by the
same fraction, to make equal spaced
notes.
Beats
Two waves with close
(but not precisely the
same) frequencies will
create a time-dependent
interference
Beats
y1  A cos  2 f1t 
y2  A cos  2 f 2t 
 y1  y2  A cos  2 f1t   A cos  2 f 2t 
f1  f 2 
f1  f 2 


 2 Acos  2

 cos  2
2 
2 


Slow
Fast
Beats
Beats are an interference pattern in time, rather than in space.
If two sounds are very close in frequency, their sum also
has a periodic time dependence: f beat = |f1 - f2|, NOT f1  f 2
2
Chapter 15
Fluids
Pressure
Pressure is force
per unit area
Pressure is not the same as force!
The same force applied over a
smaller area results in greater
pressure – think of poking a
balloon with your finger and
then with a needle.
Pressure is a useful concept for discussing fluids,
because fluids distribute their force over an area
On a Frozen Lake
You are walking out
on a frozen lake
and you begin to
hear the ice
cracking beneath
you. What is your
best strategy for
getting off the ice
safely?
a) stand absolutely still and don’t move a muscle
b) jump up and down to lessen your contact time
with the ice
c) try to leap in one bound to the bank of the lake
d) shuffle your feet (without lifting them) to move
toward shore
e) lie down flat on the ice and crawl toward shore
On a Frozen Lake
You are walking out
on a frozen lake
and you begin to
hear the ice
cracking beneath
you. What is your
best strategy for
getting off the ice
safely?
a) stand absolutely still and don’t move a muscle
b) jump up and down to lessen your contact time
with the ice
c) try to leap in one bound to the bank of the lake
d) shuffle your feet (without lifting them) to move
toward shore
e) lie down flat on the ice and crawl toward shore
As long as you are on the ice, your weight is pushing down. What
is important is not the net force on the ice, but the force exerted
on a given small area of ice (i.e., the pressure!). By lying down
flat, you distribute your weight over the widest possible area, thus
reducing the force per unit area.
Atmospheric Pressure
Atmospheric pressure is due to the weight of the
atmosphere above us.
= 1 pascal (Pa)
Various units to describe pressure:
Pascals
pounds per square inch
bars
Atmospheric Pressure
Atmospheric pressure is due to the
weight of the atmosphere above us.
How much is 1 atm ?
Put a 1 atm block on your hand?
4 in2 area -> ~60 lbs!
Hemi-spheres: ~3 inches radius, ~30 in2 area
~450 lbs!
mass of quarter ~ 0.0057 kg
area of quarter ~ 3x10-4 m2
Pressure from weight of one quarter : 180 N/m2
To get 101kPa, one must be buried under a
stack ~560 quarters, or 14 rolls, deep!
Density, height, and vertical force
How does tension change in
a vertical (massive) rope?
How does normal force
change in stack of blocks?
In a fluid, how does force
change with vertical height?
Density
The density of a material is its
mass per unit volume:
Pressure and Depth
Pressure increases with depth in a fluid due to the
increasing mass of the fluid above it.
Pressure and depth
Pressure in a fluid includes pressure on the fluid
surface (usually atmospheric pressure)
Pressure depends only on depth and external pressure
(and not on shape of fluid column)
Equilibrium only when pressure is the same
Unequal pressure will cause liquid flow:
must have same
pressure at A and B
Oil is less dense, so a taller column of oil is
needed to counter a shorter column of water
The Barometer
A barometer compares the pressure due
to the atmosphere to the pressure due to
a column of fluid, typically mercury.
The mercury column has a vacuum above
it, so the only pressure is due to the
mercury itself.
The barometer equilibrates
where the pressure due to the
column of mercury is equal to
the atmospheric pressure.
Patm = ρgh
Atmospheric pressure
in terms of millimeters
of mercury:
The Straw
You put a straw into a glass of water,
place your finger over the top so that
no air can get in or out, and then left
the straw from the liquid.
You find that the straw retains some
liquid. How does the air pressure P
in the upper part compare to the
atmospheric pressure PA?
a) greater than PA
b) equal to PA
c) less than PA
The Straw
You put a straw into a glass of water,
place your finger over the top so that
no air can get in or out, and then left
the straw from the liquid.
You find that the straw retains some
liquid. How does the air pressure P
in the upper part compare to the
atmospheric pressure PA?
a) greater than PA
b) equal to PA
c) less than PA
Consider the forces acting at the bottom of the
straw:
PA – P –  g H = 0
This point is in equilibrium, so net force is zero.
Thus, P = PA –  g H
and so we see that
the pressure P inside the straw must be less
than the outside pressure PA.
H
Pascal’s principle
An external pressure applied to an enclosed fluid is
transmitted to every point within the fluid.
Hydraulic lift
F1 / A1 = P = F2 / A2
Assume fluid is “incompressible”
Pascal’s principle
Hydraulic lift
F1 / A1 = P = F2 / A2
Are we getting “something for nothing”?
Assume fluid is “incompressible”
so Work in = Work out!
Buoyancy
A fluid exerts a net upward force on any object it
surrounds, called the buoyant force.
This force is due to the
increased pressure at the
bottom of the object
compared to the top.
Consider a cube
with sides = L
Archimedes’ Principle
Archimedes’ Principle: An object completely immersed
in a fluid experiences an upward buoyant force equal in
magnitude to the weight of fluid displaced by the object.
Buoyant Force When a Volume V is
Submerged in a Fluid of Density ρfluid
Fb = ρfluid gV
Q: Does buoyant force
depend on depth?
a) yes
b) no
Measuring the Density
The King must know: is his crown true gold?
Get the mass from
W = T1 = mg
Get the volume from
( T1 - T2 ) = V(ρwater g)
Applications of Archimedes’ Principle
An object floats when it displaces an
amount of fluid equal to its weight.
wood
block
brass
block
equivalent
mass of water
equivalent
mass of water
Can Brass Float?
An object made of material that is denser than
water can float only if it has indentations or
pockets of air that make its average density less
than that of water.
brass
block
equivalent
mass of water
An object floats when it displaces an
amount of fluid equal to its weight.
Applications of Archimedes’ Principle
The fraction of an object that is submerged when it
is floating depends on the densities of the object
and of the fluid.
Cartesian Diver
Think of a weighted balloon submerged in water
How will the balloon change when pressure
goes up?
Did its weight change when pressure went up?
So when pressure goes up:
- will it float higher?
- or will it sink?
Wood in Water
Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker and run off). Both beakers are
then weighed. Which scale reads a larger weight?
b
a
c
same for both
Wood in Water
Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker and run off). Both beakers are
then weighed. Which scale reads a larger weight?
The block in 2 displaces an amount of
b
a
water equal to its weight, because it is
floating. That means that the weight
of the overflowed water is equal to the
weight of the block, and so the beaker
in 2 has the same weight as that in 1.
c
same for both
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the
Moon, how would the same block of wood
float in the container of water?
Moon
a
b
c
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the
Moon, how would the same block of wood
float in the container of water?
Moon
A floating object displaces a
weight of water equal to the
object’s weight. On the Moon,
the wooden block has less
weight, but the water itself
also has less weight.
a
b
c
A wooden block is held at the bottom of a bucket filled with water.
The system is then dropped into free fall, at the same time the force
pushing the block down is also removed. What will happen to the
block?
a) the block will float to the surface.
b) the block will stay where it is.
c) the block will oscillate between the
surface and the bottom of the bucket
A wooden block is held at the bottom of a bucket filled with water.
The system is then dropped into free fall, at the same time the force
pushing the block down is also removed. What will happen to the
block?
a) the block will float to the surface.
b) the block will stay where it is.
c) the block will oscillate between the
surface and the bottom of the bucket
Bouyant force is created by a change of pressure with depth.
Pressure is created by the weight of water being held up.
In free-fall, nothing is being held up! No apparent weight!
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
h
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
Vertical force: Fy = (hA)g ρf - (HA)g ρ1
at equilibrium: h0 = Hρ1/ρf
h = h0 - y
Total restoring force: Fy = -(Agρf)y
Analogous to mass on a spring, with κ = Agρf
h
Fluid Flow and Continuity
Continuity tells us that whatever the mass of fluid in a
pipe passing a particular point per second, the same mass
must pass every other point in a second. The fluid is not
accumulating or vanishing along the way.
Volume per
unit time
This means that where the pipe is
narrower, the fluid is flowing faster
Continuity and Compressibility
Most gases are easily compressible; most liquids
are not. Therefore, the density of a liquid may be
treated as constant (not true for a gas).
mass flow is
conserved
volume flow is
conserved
Bernoulli’s Equation
When a fluid moves from a wider area of a pipe to a narrower
one, its speed increases; therefore, work has been done on it.
The kinetic energy of a fluid element is:
Equating the work done to the increase in
kinetic energy gives:
Bernoulli’s Equation
Where fluid moves faster, pressure is lower
Bernoulli’s Equation
If a fluid flows in a pipe of constant diameter, but
changes its height, there is also work done on it
against the force of gravity.
Equating the work
done with the change
in potential energy
gives:
Bernoulli’s Equation
The general case, where both height and speed
may change, is described by Bernoulli’s equation:
This equation is essentially a statement of
conservation of energy in a fluid.
Dynamic lift
v high
P low
v low
P high
Aircraft wing
Applications of Bernoulli’s Equation
If a hole is punched in the side of an open
container, the outside of the hole and the top of
the fluid are both at atmospheric pressure.
Since the fluid inside
the container at the
level of the hole is at
higher pressure, the
fluid has a horizontal
velocity as it exits.