Old Rillme Tank Document Flashcards by London Peters

Why does the amplitude of a circular wave get smaller the farther away from the source it travels els?

The original energy gets spread out over a larger serconfrence.

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How does this relate to why it is so difficult to hear someone speak across a large space?

Your ear can only caught so much of the energy.

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What is the megnification factor ?

M = image / real

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Shat is the law of reflection?

Angle of insadence = angle of reflection
0i = 0r

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What is the normal?

Is the imaginary line drawn perpendicular at 90 degrees angle to the refkecting surface .

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10l S4

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Waves Traveling in Two Dimension

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10 Waves Travelling in Two

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Dimensions

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10.1 Water Waves

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Waves in a stretched spring or rope illustrate some of the basic

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concepts of wave motion in a one-dimensional medium. We may

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study the behaviour of waves in two dimensions by observing water

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waves in a ripple tank.

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The ripple tank is a shallow

glass-bottomed tank on legs. Water

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is put in the tank to a depth of approximately 2 cm. Light from a

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source above the tank passes through the water and illuminates a

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screen on the table below. The light is made to converge by wave

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crests and to diverge by wave troughs

as illustrated

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and dark areas on the screen. The distance between successive

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bright areas

caused by crests

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waves may be generated on the surface of the water by a point

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source like a finger or a drop of water from an eye-dropper

Straight

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waves may be produced by rolling a dowel in the water.

light from source

water

385

Water waves are approximately

transverse

but the water molecules

move slightly back and forth as well as

up and down. In fact

an individual

particie actualiy moves in a small oval

path

as we see a cork move in water

and as is illustrated here. This

characteristic heips cause ocean waves

to "break" if they become too large or

when they approach a beach. Because

small water ripples are very nearly

transverse

we can use the water in

lakes and the water in ripple tanks in

our study of waves. The ripples move

slowly enough that we are able to

study them directly.

glass bottom

screen

Bright lines occur on the screen where light rays converge.

A wave coming from a point source is circular

whereas a wave

originating from a linear source is straight. As a wave moves away

from its constant frequency source

we observe that the spacing

Straight waves are equivalent to

circular waves a very long way from

the source.

Waves Travelling in Two Dimension

(b)

= (8.0 Hz) (2.0 cm)

= 16 cm/s

2. A page in a student's notebook lists the following information

obtained from a ripple tank experiment with two point sources

operating in phase: n = 3

= 25°

of the waves

using various methods.

Method 1

5 crests = 4^ = 4.2 cm

1 = 4.2 cm

= 1.1 cm

Method 2

d sin ®

(1 - 3)

= (6.0 cm) (sin 25°)

(3 - 7)

= 1.0 cm

Method 3

1(7)(+1)

- (3) -)

= 1.1 cm

Note that the three answers are slightly different because of

experimental error and rounding off

which is to be expected.

Practice

. In a ripple tank

a point on the third nodal line from the centre

is 35 cm from one source and 42 cm from the other source. The

sources are separated by 11.2 cm and vibrate in phase at 10.5 Hz.

Calculate (a) the wavelength of the waves

and (b) the velocity

of the waves.

(2.8 cm

29 cm/s)

2 Two sources 6.0 cm apart

operating in phase

waves. A student selects a point on the first nodal line and

measures from it 30.0 cm to a point midway between the sources

403

Fundamentals of Physics: Combined Edition

right bisector

10'

8к + a = 90° ADEC

+ a = 90° ADSC

:. 8r + a = 0

+ a

or 8: = 0.

In the figure

sin On can be determined from the triangle PBC as

follows:

sin 0. = xx

Since sin 0

= (n -

and sin 0

= sin 0„

then

= (n -

and

" - 1/2) Equation 3 where d is the distance between the sources, x, is the perpendicular distance from the right bisector to the point on the nodal line, L is the distance from the point P to the midpoint between the two sources, and n is the nodal line number. It should be noted that the above relationship is only valid for two point sources in phase. Sample problems 1. Two point sources generate identical waves that interfere in a ripple tank. The sources are located 5.0 cm apart, and the fre- quency of the waves is 8.0 Hz. A point on the first nodal line is located 10 cm from one source and 11 cm from the other. (a) What is the wavelength of the waves? (b) What is the speed of the waves? (a) (PS, - PS, = In - |11 cm - 10 cm| = (1 - =) * × = 2.0 cm

Waves Travelling in Two Dimension

sin 0

= AS

or AS

= d sin 0

Equation 2

But AS

= PaS

and 2 we get

d sin 0n = (n - 5))

sin 0

= (n - 2) ^

EquatioN D

where 0

is the angle for the nth nodal line

and d is the distance between the sources.

This equation allows us to make a quick approximation of the

wavelength for a specific interference pattern. Since sin 0

cannot

be greater than 1

(n - 1/2) A/d cannot be greater than 1. The

largest value of n that satisfies this condition is the number of nodal

lines on either side of the right bisector. By measuring d and count-

ing the number of nodal lines

the wavelength can be approxi-

mated. For example

if d is 2.0 cm and the number of nodal lines

is 4

the wavelength would be determined as follows:

sin • = (n - ¿ ^

But the maximum value of sin 0

is 1

thus (n - b) ^

= 1

or 4 -

and 1 = 0.6 cm

In the ripple tank

it is relatively easy to measure the angle ®„

but this will not be the case for other kinds of waves where both

the wavelength and the distance between the sources are very small

and the nodal lines are close together. Therefore

we must derive

another way to measure the value of sin 0

without measuring 0

itself.

As noted earlier in this section

a nodal line has the shape of a

hyperbola. But at positions on nodal lines relatively far away from

the two sources

these lines are nearly straight

inate from the midpoint of a line joining the two sources.

For a point P. located on a nodal line

far away from the two

sources

the line from P

AS.. Since the right bisector (CB) is perpendicular to S

easily show that 0

= 0n-

right bisector

d -

401

Waves Travelling in Two Dimension

This symmetrical pattern will remain stationary

provided three

factors do not change. These factors are: the frequency of the two

sources

the distance between the sources

sources. When the frequency of the sources is increased

the wave-

length decreases

bringing the nodal lines closer together and in-

creasing their number. If the distance between the two sources is

increased

the number of nodal lines will also increase. Neither of

these factors changes the symmetry of the pattern - there is an

equal number of nodal lines on either side of the right bisector if

the two sources are in phase

and an area of constructive inter-

ference runs along the right bisector. On the other hand

if other

factors are kept constant and the relative phase of the two sources

changes

the pattern shifts (as illustrated in the photographs)

the number of nodal lines remains the same.

399

The interference patters for two point

sources with different phase delays. In

the top photograph

the sources are in

phase. In the bottom photograph

the

phase delay is 180°.

Two-point-source interference patter generated by a computer

graphics program

10.6 The Mathematical Analysis of the Two-

Point-Source Interference Pattern

Like the standing wave interference pattern

the two-point-source

interference pattern is useful because it allows direct measurement

of the wavelength while the interference pattern remains relatively

stationary. By taking a closer look at the two-point interference

pattern we can develop some mathematical relationships that will

be useful in Chapter 14 in analysing the interference of other

kinds of waves.

nterfering waves require equal

amplitudes for total destructive

interference. The resulting pattern is

still observable if the amplitudes vary

slightly

but the nodal lines will not be

as distinct.

rOSy

Fundamentals of Physics: Combined Edition

These areas moved out from the source in symmetrical patterns

producing nodal lines and areas of constructive interference

illustrated. When illuminated from above

the nodal lines appeared

in the ripple tank as stationary

grey areas. Between the nodal lines

were areas of constructive interference that appeared as alternating

bright (double-crest) and dark (double-trough) lines of constructive

interference. In the illustration below

you can see how these al-

ternating areas of constructive and destructive interference are pro-

duced. Note that

although the nodal lines appear to be straight

their paths from the sources are actually curved lines. Mathema-

ticians call these hyperbolae. The pattern of constructive and de-

structive interference does not stay at rest

as in the case of standing

waves in a linear medium such as a long rope. The pattern moves

out from the source.

Tolen

• node

Constructive interference

lines of destructive

interference

(nodal lines)

Constructive interference

Destructive interference

areas of constructive

interference

--0-

VYYYT

The interference patter between two identical sources (S

and Sz).

vibrating in phase

is a symmetrical pattem of nodal lines and areas

of constructive interference in the shape of hyperbolae.

Waves Travelling in Two Dimension

10.5 Interference of Waves in Two Dimensions

As discussed in the previous chapter with waves in a one-dimen-

sional medium

constructive and destructive interference may oc-

cur in two dimensions

sometimes producing fixed patterns of

interference. To produce such a fixed pattern

it is necessary that

the interfering waves have the same frequency (wavelength) and

amplitude as one another. Let us see what patterns of interference

occur between two identical waves when they interfere in a two-

dimensional medium like the ripple tank.

397

Note that a standing wave interference

pattern is produced on the line joining

the two point sources. Note also that

the nodes in the standing wave are

located on the nodai lines of the larger

interference pattern.

The photograph shows two vibrating point sources that were

attached to the same generator and thus had identical frequencies

and amplitudes. Also

they were in phase. Just as was the case

with the one-dimensional medium

the waves passed through one

another unchanged. As successive crests and troughs travelled out

from each source

however

times crest on crest

sometimes trough on trough

crest on trough. Thus

areas of constructive and destructive inter-

ference were produced.

incident waves—

Large opening

Fundamentals of Physics: Combined Edition

As the wavelength increases

the amount of diffraction increases.

In a similar demonstration (see Investigation 10.4)

we can

keep A fixed and change w

to find that the amount of diffraction

increases as the aperture decreases. In both demonstrations

if waves

are to be strongly diffracted they must pass through an opening

that has a width comparable to their wavelength. This means that

if the wavelength is very small

a very narrow aperture is required

to produce any significant diffraction.

?))

nearly straight-line

propagation

incident waves —

Small opening

incident waves —

Shorter wavelength

no change in

opening

The amount of diffraction does not depend on either ^ or w

separately

but on the relationship between the two. Stated math-

ematically

the ratio X/w must have a value of about 1

for any significant diffraction to be apparent.

Perhaps the most common example of the diffraction of waves

occurs with sound. The sounds of a classroom can be heard through

an open door

even though the students are out of sight and behind

a wall. Sound waves are diffracted around the corner of the door-

way primarily because they have relatively long wavelengths. If a

high fidelity sound system is operating in the room

its low fre-

quencies (long wavelengths) will be diffracted around the corner

more easily than will its higher frequencies (shorter wavelengths).

Waves Travelling in Two Dimension

10.4 Diffraction of Water Waves

The ripple tank is probably the best device with which to observe

the diffraction of waves. When periodic straight waves are pro-

duced in a ripple tank

they travel in a straight line as long as the

depth of the water remains uniform. This direction of motion is

indicated by a wave ray drawn at right angles to the wavefront. If

an obstacle is put in the path of these waves

the waves are blocked.

But

if they are allowed to pass by a sharp edge of the obstacle

through a small opening or aperture in the obstacle

the waves

change direction as illustrated in the photograph. This bending is

called diffraction. How much the waves are diffracted depends

on both their wavelength and the size of the opening in the barrier.

As seen in the illustrations

short wavelengths are diffracted slightly.

Longer wavelengths are diffracted to a greater extent by the same

edge or aperture.

short wavelengths

long wavelengths

395

In the case of diffraction through an aperture

the amount of

diffraction is increased when the relative size of the opening is

decreased. This is apparent in the three photographs that follow.

In each case the size of the opening is the same. In the first pho-

tograph

the wavelength (2) is approximately 3/10 of the size of

the opening (w). Only part of the straight wavefronts pass through

to be converted to small sectors of a series of circular wavefronts.

In the second photograph

the wavelength is approximately 5/10

of w and there is considerably more diffraction. But there are still

shadow areas to the left and right

where none of the wave is

diffracted. In the third photograph

the wavelength is approxi-

mately 7/10 of w. Here the small sections of the straight wave that

get through the opening are almost entirely converted into circular

wavefronts. The wave has bent around the corner of the opening

filling almost the complete region on the other side of the aperture.

At higher angles of incidence

there is

reflection as well as refraction. This can

be seen on the right.

Fundamentals of Physics: Combined Edition

is a good approximation of the actual behaviour of waves

since

the dispersion of a wave is the result of minute changes in its speed.

These go unnoticed in most observations

and it is acceptable to

make the assumption that frequency does not affect the speed of

waves in most applications.

When refraction occurs

some of the energy is usually reflected

as well as refracted. In the ripple tank

where the waves are

travelling from deep to shallow water

this behaviour is only

apparent at large angles of incidence (see photograph).

The amount of reflection is more noticeable when a wave

travels from shallow to deep water

where the speed increases

and it becomes more pronounced as the angle of incidence

increases. In fact

as seen in the series of diagrams

angle is reached where the wave is refracted at an angle

approaching 90°. After that

there is no refraction and all the

wave energy is reflected. This total internal reflection of the

waves at high angles of incidence

for waves travelling into a

faster medium

is analogous to the total internal reflection of

light (Section 13.9)

and is discussed further in Chapter 14.

refracted

wavefront

fast medium

slow medium

incident wavefront

reflected

wavefront

Partial refraction

partial reflection

At the critical angle

Total internal reflection

Waves Travelling in Two Dimension

As has been previously noted

the frequency of a wave does not

change when its velocity changes (see Section 9.4). Therefore

since v

/v2 = 1

and the amount of bending would not change for waves of different

frequencies

provided the medium remained the same — for ex-

ample

water of the same depth in both cases.

However

as seen in the photographs

In the first photograph

the low-frequency (long-wavelength) waves

are refracted

as indicated by a rod placed on the screen of the

ripple tank. The rod is exactly parallel to the refracted wavefronts.

In the second photograph

the frequency has been increased (wave-

length decreased)

with the rod left in the same position. Note that

the rod is no longer parallel to the refracted wavefronts. The higher-

frequency waves are refracted in a slightly different direction than

were the low-frequency waves

although the angle of incidence

has remained unchanged. It appears that the amount of bending

and hence the index of refraction

is affected slightly by the fre-

quency of a wave. We can conclude that

since the index of re-

fraction represents a ratio of speeds in two media

the speed of the

waves in at least one of those media must depend on their fre-

quency. Such a medium

in which the speed of the waves depends

on the frequency

is called a dispersive medium.

393

The refraction of straight waves

with a black marker placed parallel

to the refracted wavefronts in the first photograph. in the second

photograph

the refracted wavefronts of the higher frequency waves

are no longer parallel to the marker.

Previously

we made the statement that the speed of waves de-

pends only on the medium. This statement is obviously an ideal-

ization

considering the above evidence. Nevertheless

incident waves—

Large opening

Fundamentals of Physics: Combined Edition

As the wavelength increases

the amount of diffraction increases.

In a similar demonstration (see Investigation 10.4)

we can

keep A fixed and change w

to find that the amount of diffraction

increases as the aperture decreases. In both demonstrations

if waves

are to be strongly diffracted they must pass through an opening

that has a width comparable to their wavelength. This means that

if the wavelength is very small

a very narrow aperture is required

to produce any significant diffraction.

?))

nearly straight-line

propagation

incident waves —

Small opening

incident waves —

Shorter wavelength

no change in

opening

The amount of diffraction does not depend on either ^ or w

separately

but on the relationship between the two. Stated math-

ematically

the ratio X/w must have a value of about 1

for any significant diffraction to be apparent.

Perhaps the most common example of the diffraction of waves

occurs with sound. The sounds of a classroom can be heard through

an open door

even though the students are out of sight and behind

a wall. Sound waves are diffracted around the corner of the door-

way primarily because they have relatively long wavelengths. If a

high fidelity sound system is operating in the room

its low fre-

quencies (long wavelengths) will be diffracted around the corner

more easily than will its higher frequencies (shorter wavelengths).

Waves Travelling in Two Dimension

10.4 Diffraction of Water Waves

The ripple tank is probably the best device with which to observe

the diffraction of waves. When periodic straight waves are pro-

duced in a ripple tank

they travel in a straight line as long as the

depth of the water remains uniform. This direction of motion is

indicated by a wave ray drawn at right angles to the wavefront. If

an obstacle is put in the path of these waves

the waves are blocked.

But

if they are allowed to pass by a sharp edge of the obstacle

through a small opening or aperture in the obstacle

the waves

change direction as illustrated in the photograph. This bending is

called diffraction. How much the waves are diffracted depends

on both their wavelength and the size of the opening in the barrier.

As seen in the illustrations

short wavelengths are diffracted slightly.

Longer wavelengths are diffracted to a greater extent by the same

edge or aperture.

short wavelengths

long wavelengths

395

In the case of diffraction through an aperture

the amount of

diffraction is increased when the relative size of the opening is

decreased. This is apparent in the three photographs that follow.

In each case the size of the opening is the same. In the first pho-

tograph

the wavelength (2) is approximately 3/10 of the size of

the opening (w). Only part of the straight wavefronts pass through

to be converted to small sectors of a series of circular wavefronts.

In the second photograph

the wavelength is approximately 5/10

of w and there is considerably more diffraction. But there are still

shadow areas to the left and right

where none of the wave is

diffracted. In the third photograph

the wavelength is approxi-

mately 7/10 of w. Here the small sections of the straight wave that

get through the opening are almost entirely converted into circular

wavefronts. The wave has bent around the corner of the opening

filling almost the complete region on the other side of the aperture.

At higher angles of incidence

there is

reflection as well as refraction. This can

be seen on the right.

Fundamentals of Physics: Combined Edition

is a good approximation of the actual behaviour of waves

since

the dispersion of a wave is the result of minute changes in its speed.

These go unnoticed in most observations

and it is acceptable to

make the assumption that frequency does not affect the speed of

waves in most applications.

When refraction occurs

some of the energy is usually reflected

as well as refracted. In the ripple tank

where the waves are

travelling from deep to shallow water

this behaviour is only

apparent at large angles of incidence (see photograph).

The amount of reflection is more noticeable when a wave

travels from shallow to deep water

where the speed increases

and it becomes more pronounced as the angle of incidence

increases. In fact

as seen in the series of diagrams

angle is reached where the wave is refracted at an angle

approaching 90°. After that

there is no refraction and all the

wave energy is reflected. This total internal reflection of the

waves at high angles of incidence

for waves travelling into a

faster medium

is analogous to the total internal reflection of

light (Section 13.9)

and is discussed further in Chapter 14.

refracted

wavefront

fast medium

slow medium

incident wavefront

reflected

wavefront

Partial refraction

partial reflection

At the critical angle

Total internal reflection