Waves – Chapter 14

Superposition and InterferenceWaves 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 destructive

Two waves with distance to the source different by whole

integer wavelengths Nλ

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

Destructive: l = (n+1/2)

L

A

B

a) intensity increases

b) intensity stays the same

c) intensity goes to zero

d) impossible to tell

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?

L

A

B

If = 1 m = 1 m, then a shift of 2.5 m2.5 m corresponds to 2.52.5, which puts the two waves out of phaseout of phase, leading to

destructive interferencedestructive interference. The sound intensity will therefore go to zero.

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

Interference

Follow-upFollow-up: What if you : What if you move speaker move speaker AA back back

by 4 m?by 4 m?

Standing WavesA standing wave is fixed in location,

but oscillates with time.

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string.

These waves are found on strings with both ends fixed, or vibrating columns of air, such as in a musical instrument.

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

In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape.

A guitar has strings that are all the same length, but the density varies.

Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.

Standing Waves IA 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

b

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.

Standing Waves IA 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

Every point in in SHM, with the amplitude fixed for each position

Standing Waves in Air TubesStanding 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 WavesWith 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

Frequency doubles for octave steps of the same note

Human Perception: equal steps in pitch are not additive steps, but rather equal multiplicative factors

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.

Two waves with close (but not precisely the same) frequencies will create a time-dependent interference

Beats

Beats

1 1

2 2

1 2 1 2

1 2 1 2

cos 2

cos 2

cos 2 cos 2

2 cos 2 cos 22 2

Slow Fast

y A f t

y A f t

y y A f t A f t

f f f fA

Beats

If two sounds are very close in frequency, their sum also has a periodic time dependence: f beat = |f1 - f2|, NOT

Beats are an interference pattern in time, rather than in space.

1 2

2

f f

Chapter 15

Fluids

Pressure

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 not the same as force!

Pressure is force per unit area

Pressure is a useful concept for discussing fluids, because fluids distribute their force over an area

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 LakeOn 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.

On a Frozen LakeOn a Frozen Lake

Atmospheric PressureAtmospheric pressure is due to the weight of

the atmosphere above us.

= 1 pascal (Pa)

Pascals

pounds per square inch

bars

Various units to describe pressure:

Atmospheric PressureAtmospheric pressure is due to the weight of the atmosphere above us.

Hemi-spheres: ~3 inches radius, ~30 in2 area

~450 lbs!

How much is 1 atm ?

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!

4 in2 area -> ~60 lbs!Put a 1 atm block on your hand?

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 DepthPressure 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 BarometerA 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.

Patm = ρghAtmospheric pressure in terms of millimeters of mercury:

The barometer equilibrates where the pressure due to the column of mercury is equal to the atmospheric pressure.

a) greater than PA

b) equal to PA

c) less than PA

The StrawThe StrawYou 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?

Consider the forces acting at the bottom of

the straw: PPAA – – PP – – g H g H = = 00

This point is in equilibrium, so net force is This point is in equilibrium, so net force is

zero.zero.

Thus, PP = = PPAA – – g H g H and so we see that the pressure pressure PP inside the straw must be lessless than the outside pressure outside pressure PPAA.

H

a) greater than PA

b) equal to PA

c) less than PA

The StrawThe StrawYou 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?

Pascal’s principleAn external pressure applied to an enclosed fluid

is transmitted to every point within the fluid.

Hydraulic lift

Assume fluid is “incompressible”

F1 / A1 = P = F2 / A2 F1 / A1 = P = F2 / A2

Pascal’s principle

Hydraulic lift

F1 / A1 = P = F2 / A2 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

Buoyant Force When a Volume V is Submerged in a Fluid of Density

ρfluidFb = ρfluid gV

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.

Q: Does buoyant force depend on depth?

a) yesb) no

Measuring the Density

Get the volume from ( T1 - T2 ) = V(ρwater g)

Get the mass from W = T1 = mg

The King must know: is his crown true gold?

Applications of Archimedes’ Principle

An object floats when it displaces an amount of fluid equal to its weight.

equivalent mass of water

wood block

equivalent mass of water

brass block

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.

An object floats when it displaces an amount of fluid equal to its weight.

equivalent mass of water

brass block

Applications of Archimedes’ PrincipleThe 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?

a b

csame for both

The block in 2 displaces an amount displaces an amount

of water equal to its weightof water equal to its weight,

because it is floating. That means

that the weight of the overflowed weight of the overflowed

water is equal to the weight of the water is equal to the weight of the

blockblock, and so the beaker in 2 has beaker in 2 has

the same weight as that in 1the same weight as that in 1.

Wood in Water

a b

csame for both

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?

Wood in Water II

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?

Earth

Moon

a b c

A floating object displaces a

weight of water equal to the weight of water equal to the

object’s weightobject’s weight. On the

Moon, the wooden block has

less weightless weight, but the water

itself also has less weightalso has less weight.

Wood in Water II

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

Earth

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 h

at equilibrium: h0 = Hρ1/ρf

Total restoring force: Fy = -(Agρf)y

h = h0 - y

Analogous to mass on a spring, with κ = Agρf

Fluid Flow and ContinuityContinuity 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.

This means that where the pipe is narrower, the fluid is flowing faster

Volume per unit time

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 EquationWhen 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 lowP high

v high P low

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.