Physics 11 Unit 1: Waves - Don Bloomfield's Weebly Sitedbloomfield.weebly.com/.../physics_11_waves_unit_notes_part_1.pdf · LECSS Physics 11 Waves Unit Notes Part 1 Revised February

LECSS Physics 11 Waves Unit Notes Part 1

Revised February 7, 2011 Don Bloomfield

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Physics 11 Unit 1 Part 1: General Wave Properties

This unit is essentially divided into two parts:

Part 1: General Wave Properties

Part 2: The Wave Nature of Light

Begin this unit by reading Sections 18.1 and 18.2 of the text. A summary of the key ideas is

given below.

What is a Wave?

A wave is a transfer of energy, in the form of a traveling disturbance, usually through some type of

medium (e.g., water, a spring, air, etc.)

If the source of a wave is a single disturbance in a medium, a wave pulse is created.

If the source of a wave is a vibrating or oscillating object in the medium, a wave train is created.

Describing an Oscillating Object

Several terms may be used to describe the motion of an oscillating object such as a pendulum.

Cycle

An oscillating object exhibits a repetitive pattern of motion. The cycle of a pendulum is its motion

from the rest position, out to its maximum displacement on one side, back through to its maximum

displacement on the other side, and then back to its rest position.

The cycle of a pendulum.

Period, T

The time it takes for an object to complete one cycle is called the period and is given the symbol T.

For the most part, the period is measured in seconds, but other units of time may be used.



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We can devise a very simple equation to calculate the period of any oscillating object using

experimental results.

If we set a pendulum in motion and how long it takes to complete a number of cycles, we can use

simple division to compute the time it takes to complete one cycle.

If the pendulum completes n cycles in a measured time, t, then

In the above equation, the symbol is the Greek letter “delta”. Mathematically, this symbol means

“change in”, so t means “change in time”. In other words, it is the amount of time that passed

during the experiment.

Frequency, f

The frequency of an oscillating object is equal to the number of cycles completed per unit of time.

The unit of time may vary, but for the most part, we will be concerned with the number of cycles

completed per second. When the second is used as the unit of time, the unit of frequency becomes

the Hertz (Hz). This unit is used to describe the frequency of any repetitive occurrence on a "per

second" basis.

We can devise a very simple equation to calculate the frequency of any oscillating object using

experimental results.

In an experiment similar to the one discussed above, the frequency may be calculated as follows:

The Relationship between Period and Frequency

Frequency and period are related very simply. It can be seen from the above discussion that the

period and frequency of any oscillating object are simple mathematical reciprocals of one another.

That is:

n

tT Period,

tf frequency,

n

T

1f and

f

1T



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Amplitude, A

The amplitude of an oscillating object is the measure of the linear distance that the object moves

between the rest position and a position of maximum displacement on either side of its cycle.

Phase

The term phase is used to describe which part of its cycle is in. For example, we sometimes talk

about the "four phases" of the moon as it goes from being a full moon to the “first quarter” to a new

moon to the “last quarter” then back to a full moon.

The cycle of a pendulum can also be divided into four phases:

Phase 1: from the rest position to a point of maximum displacement on one side of the cycle.

Phase 2: from the point of maximum displacement back to the rest position.

Phase 3: from the rest position to a point of maximum displacement on the other side of the

cycle, and,

Phase 4: from this point of maximum displacement to the rest position.

Notice that a pendulum moves through FOUR amplitudes in each cycle of motion.

Sometimes we will use the term "phase" when comparing the motion of two adjacent (side by side)

oscillating objects (such as two pendulums). If two oscillating objects are at the same points in their

cycles at the same instant in time, they are said to be "in phase". If they are at different points in

their cycles, they are "out of phase". If they are at exactly opposite points in their cycles, they are

"exactly out of phase".

Before continuing, examine Sample Problems 1-3 in the Unit 1A Sample Problems Booklet.

Ensure you understand the solutions thoroughly,

At this point you should read through the Investigation: Pulses in a Spiral Spring that starts

on page 340 of the text. Try to determine the answers to the questions listed on page 341. A

summary of the key ideas from this investigation is given below. If you wish to, you can

perform this investigation with the help of your instructor.



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Transverse Waves and Longitudinal Waves

If the medium moves at right angles to the direction of the wave, then the wave is called a

transverse wave. An example is a wave traveling through a stretched piece of rope. Transverse

waves are created when the source vibrates at right angles to the medium.

If the medium vibrates parallel to the direction of the wave, then the wave is called a longitudinal

wave. Sound waves are examples of longitudinal waves.

In no case does the medium move along with the wave. Each part of the medium oscillates in a

localized region only.

Other Important Concepts

A wave that travels along a straight-line medium (such as a stretched spring) is called a one-

dimensional wave. Important things that you need to know about waves that travel along springs

are:

1. As a wave passes through a medium, the medium does not move along with the wave.

2. In the real world, the amplitude of a wave decreases as the wave moves away from the source.

This is due to friction in the medium. Thus, the greater the amplitude of a wave, the greater the

amount of energy it carries. An ideal wave would not lose amplitude. The original amplitude of a

wave is determined by the energy imparted to the medium by the source - the greater the

amplitude of the source, the greater the amplitude of the wave.

3. Because of friction, the speed of a wave usually decreases as it "propagates" (travels). The speed

of an ideal wave would be constant as long as the characteristics of the medium do not change.

For a wave traveling in a spring, it is the amount of tension in the spring that determines the

speed of the wave. Because larger-amplitude waves cause a greater amount of tension in a

spring, they tend to travel slightly faster than smaller-amplitude waves.

NOTE: The speed of a wave is not affected by the frequency or period of the source.

4. When a wave reflects from the fixed end of a spring, it inverts so that a crest (a positive pulse)

becomes a trough (a negative pulse) and vice-versa.

Two-Dimensional Waves

At this point you need to read the Investigation: Water Waves-Transmission and Speed that begins

on page 344 of the text. Try to determine the answers to the questions on page 345. A summary of

the key points is given below. If you wish to, you can perform this investigation with the help

of your instructor.



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A wave that travels across a flat surface, such as the surface of water is called a two-

dimensional wave. Two-dimensional waves, such as water waves, can be straight or circular.

The amplitude of straight water waves would not decrease as the waves propagate if there were

no friction in the medium. The amplitude of circular water waves decrease as they radiate

outward from the source because the energy of the source is spread out over an ever-enlarging

circle. Friction contributes to the decrease in amplitude as well.

The speed of water waves depends upon the depth of the water. Waves travel faster in deep

water and more slowly in shallow water.

Before continuing, read page 346 of the text as well as Section 18.5. A summary of the key

ideas of Section 18.5 is given below.

Describing Wave Trains

When a train of waves travels though a medium, every point in the medium oscillates with a period

and frequency and period equal to that of the source of the waves. Different points in the medium

may be oscillating in phase, out of phase or exactly out of phase.

Frequency of a Wave Train

The frequency of a wave train is defined to be equal to either the frequency of oscillation of any

point in the medium, or the frequency with which one complete wave passes a point in the medium.

In either case, the frequency of the wave train is equal to the frequency of the source that creates it.



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Wavelength of a Wave Train

The wavelength is defined as the length of one crest plus one trough. This is the same as the

distance between the tops of two adjacent crests or the bottoms of two adjacent troughs.

Wavelength is given the symbol (lambda).

Period of a Wave Train

The period of a wave is defined as the time it takes for a single point in the medium to complete one

cycle of oscillation, or as the time it takes for a complete wave to pass a single point in the medium.

In either case, the period of a wave train is equal to the period of the source that it creates it.

Speed of a Wave Train, v

The speed of a wave is determined by the characteristics of the medium it travels through (as

discussed above). Usually, the speed of a wave is measured in units of m/s or cm/s.

A simple equation can be used to calculate the speed of a wave. If the speed of the wave is

constant, and in travels a distance d in an interval of time t, its speed v is given by:

Relationship Between , T, f and v of a Wave Train

For an object moving with a constant speed, a simple relationship exists between its speed and the

distance it travels during a period of time.

In a time equal to the period of the source (and the wave), the wave will travel through a distance

equal to its wavelength. Thus,

The above equation is called the Universal Wave Equation.

t

dv

fv

: tosimplifies which f

vcan write we, f

1T Since

T v



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Before continuing, examine Sample Problems 4-8 in the Unit 1A Sample Problems booklet.

Ensure you understand the solutions thoroughly.

Assignment 1:

A. Complete the Practice Problems on page 339 and 348. As well, complete Review Problems

1-19 and 23 on pages 371 to 377. Numerical answers to these problems are given on page 376.

B. Complete the Fundamental Wave Properties Assignment and submit your solutions to your

instructor for marking.

Reflection of Water Waves

Begin this section by reading through Section 18.6 of the text. Next, you will need to view the

videos "Reflection of Straight Waves from Straight Barriers", "Reflection of Waves from

Concave Barriers" and "Reflection of Circular Waves from Various Barriers. As you view

the videos, try to discover the answers to the questions on page 350 of the text, A summary of

the key points is given below.

A. Reflection of Straight Waves from Straight Barriers

The first segment of the video “Reflection of Straight Waves From Straight Barriers” shows what

happens when a straight wave is incident upon a straight barrier and the wave front is parallel the

barrier. To no surprise, the wave reflects as a straight wave that is also parallel to the barrier.

When a straight wave approaches a straight barrier obliquely (at an angle), things are more

interesting.

As illustrated in the diagram

to the left, the incident wave

and reflected wave form a

"V" shape. The angle between

the incident wave and the

barrier (called the angle of

incidence, i) and the angle

between the reflected wave

and the barrier (called the

angle of reflection, r) are

equal. This is called the First

Law of Reflection.



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A common misconception that students sometimes have is that the angle between the incident wave

and the reflected wave is 90. This misconception may be due to the discussion of wave rays on

page 351 of the text. Wave rays will be discussed later on in these notes, but for now, bear in mind

that the only time the angle between the incident wave and the reflected wave is 90 is when the

angle of incidence is 45. In that case, the angle of reflection will also be 45. Because the angles

on a line add to 180, the angle between the two waves will be 90.

At ALL times, the angle between a wave and the ray that points the direction of the wave is 90.

As you watch the video, it

becomes apparent that the

First Law of Reflection is

obeyed regardless of the size

of the angle of incidence.

You may have noticed in the

video that the reflected waves

are slightly curved at the end.

This is due to friction

between the wave and the

barrier.

B. Reflection of Straight Waves from Concave Barriers

The video “Reflection of Waves from Concave Barriers” shows the reflection of both straight

waves and circular waves from parabolic barriers. You may have studied parabolas in your math



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class, but if not, don’t worry about it. What you need to know is that all parabolas have a special

point called a “focal point” or “focus”.

When a straight wave reflects from a parabolic reflector, it reflects in such a way that every part of

the incident wave converges (“comes together”) as a circular wave to the focus of the parabolic

reflector. After the reflected wave converges, it diverges (“spreads out”) as a circular wave that will

reflect off the barrier as described below (see Reflection of Circular Waves From Concave Barriers

below).

Careful analysis of the angle of incidence and the angle of reflection will show that i = r for each

portion of the incident wave that reflects off of the barrier.

C. Reflection of Circular Waves From Concave Barriers

In this segment of the video, the experimenter is generating circular waves at various positions in

front of a parabolic barrier. The important result is when a circular wave is generated at the focus

of a parabolic reflector. In this situation, the portion of the wave that strikes the barrier reflects as a

straight wave. As one should expect, this result is pretty much the opposite of what occurs when a

straight wave is sent into a parabolic reflector as discussed above.



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Careful analysis will again show that i = r. This is illustrated in the margin of page 352 of the text.

In that analysis, wave rays rather than wavefronts are used. The difference between the two is

discussed on page 351 of the text. Study this section carefully. A further discussion of rays is given

below during our study of the refraction of water waves.

In one segment of this video, the experimenter reflects circular waves off of a semi-circular

reflector. During this, you are prompted to note the “spherical aberration” that occurs. This refers

to the fact that the reflected wave is not perfectly straight along its entire length. Rather, the wave

curved on top and bottom.

Though the experiment is not shown, the dramatic effect of spherical aberration is more important

when straight waves are directed into semi-circular reflectors. In this situation, the energy of the

top and bottom portions of the wave are not reflected to the same point as the middle portion of the

wave. In practical applications, the reflection of radio waves and light from distant objects may be

badly distorted if semi-circular reflectors are used. If you wish to find out more about this situation,

ask your instructor about satellite dishes and the Hubble Space Telescope!

D. Reflection of Circular Waves from Straight Barriers

The first segment of this video shows what happens when a circular wave reflects from a straight

barrier. In this situation, the reflected wave appears to have originated from a point behind the

reflector that is as far behind the reflector as the source of the incident wave is in front of the

reflector.

Again, careful analysis will show that i = r.



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We can draw an analogy to the reflection of light from a plane (flat) mirror. When you stand in

front of a mirror, your reflection appears to be behind the mirror. The distance that your image is

“formed” behind the mirror appears to be equal to the distance between you and the mirror.

This video also shows circular waves reflecting off of circular and elliptical barriers. You need not

be concerned about these situations.

Refraction of Water Waves

Begin this section by reading Section 18.8 of the text. Next, you will need to view the video

"Refraction of Waves in a Ripple Tank". While watching the video, try to determine the

answers to the questions on page 356 of the text. A summary of the key ideas is given below.

As mentioned previously, when water waves travel from deep to shallow water, they slow down

and their wavelength decreases. When they travel from shallow to deep water, they speed up and

their wavelength increases.

If the wave crests are parallel to the boundary it is crossing, the entire wavefront slows down all at

once and no change in direction results. If the wavecrests cross the boundary obliquely (at an angle),

then a change in direction occurs. This change of direction is called refraction.

If a wave crosses the boundary from deep water to shallow water, the direction changes such that

the waves are closer to the boundary and their direction is closer to the normal (see diagram below).

If a wave crosses from shallower water to deeper water, the direction changes so that the waves are

farther from the boundary and their direction is father from the normal.

Whenever refraction occurs there is always some amount of reflection as well. Usually only partial

reflection of the incident wave occurs. However, if waves are traveling from shallow to deep water,

total reflection may occur. See the section below that discusses total reflection.

A Note About Waves and Rays

The angle between the incident waves and the boundary between two depths of water is called the

angle of incidence, i. The angle between the refracted waves and the boundary is called the angle of

refraction, R. These angles can be measured as shown in the diagram below, or by using rays.

A ray is an arrow that shows the direction that a wave is moving. Rays are always drawn at right

angles to the wave-front as shown below.



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A refracted ray shows the direction that a refracted wave is moving. An incident ray shows the

direction that an incident wave is moving.

Let us chose a point on the boundary where an incident wave and a refracted wave meet. We can

draw a line (called a normal line) at right angles to the boundary at this point, as shown in the

diagram below.



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Total Reflection of Water Waves

As mentioned previously, whenever water waves refract, some amount of reflection occurs. When

waves are traveling from deep water into shallow water, there is always some amount of partial

reflection, regardless of the angle of incidence.

Partial reflection also occurs when waves travel from shallow water deep water. But as shown in

the video, total reflection of the incident waves may occur if the angle of incidence is too great.

Whenever waves travel from shallow to deep water, the angle of refraction is always larger than the

angle of incidence. As we increase in the angle of incidence, the angle of refraction gets larger and

larger until it reaches 90. At this point, the direction of the refracted waves is parallel to the

boundary, and each refracted wave-front is perpendicular to the boundary.



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If the angle of incidence is increased beyond this point, we might expect the angle of reflection to

increase beyond 90. This would mean the direction of the refracted waves would be back across

the boundary, back into the shallow water!

What happens instead is that the incident waves are totally reflected by the boundary, just as if they

were reflected off of a straight barrier. This being the case, the First Law of Reflection is obeyed (i

= r).

The Mathematics of Refraction

If we now draw an incident ray at right angles to the incident waves as shown in the diagram on the

bottom of page 12, the angle between this incident ray and the normal line is also equal to the angle

of incidence.

If we draw a refracted ray at right angles to the refracted waves as shown, the angle between this

refracted ray and the normal line is equal to the angle of refraction.

A simple mathematical relationship exists between the wavelength and speed of the incident waves

and the refracted waves. We will use the following notation:

= the wavelength of the incident waves

v1 = the speed of the incident waves

f1 = the frequency of the incident waves

= the wavelength of the refracted waves

v2 = the speed of the refracted waves

f2 = the frequency of the refracted waves

Since v = f, vl = f1 and v2 = f2Therefore:

Because one refracted wave is produced every time an incident wave crosses the boundary, the

frequency of the refracted waves is exactly equal to the frequency of the incident waves (i.e., f1 =

f2). Therefore, the frequencies cancel to leave:

Before continuing, examine Sample Problems 9-12 in the Unit 1A Sample Problems booklet.

Ensure you are able to perform the required calculations.

2

1

2

1

f

f

v

v

2

1

v

v



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Assignment 2:

Complete Investigation 1 from the Student Lab Manual using the photograph provided by

your instructor. Your instructor will also tell you what is required in your Lab Report. This

report must be submitted for marking before continuing.

Before continuing, examine Sample Problems 12-15 in the Unit 1A Sample Problems Booklet.

Ensure you are able to perform the calculations involved. Then complete Assignment 3.

Assignment 3:

A. Complete the Practice Problems on page 357 and Review Problems 26-28 on pages 371-377

of the text. Numerical Answers to the Review Problems are given on page 376.

B. Complete Problems 1-6 on page 11 of the Student Lab Manual. Do these problems on your

own lined paper as they are to be handed in for marking. No marks will be given for correct

answers if no work is shown. The numerical answers to these problems are given on page 225

of the Lab Manual.

As proven in Investigation 1, the ratios of the speeds and wavelengths are also equal to the ratio of

the sine of the angle of incidence and the sine of the angle of refraction. That is:

Diffraction of Water Waves

Begin this section by reading through Section 18.7 of the text. Next you will need to view the

videos "Single Slit Diffraction of Waves" and "Diffraction and Scattering of Waves around

Obstacles. Try to determine the answers to the questions on the bottom of page 353 while

watching the videos. As well, determine how the size of an obstacle placed in the path of a

wave train affects the amount of diffraction. A summary of the key points from the videos is

given below.

Diffraction refers to the bending of a straight wave into a curved wavefront. Diffraction may occur

to differing degrees, depending on the circumstances.

A. Diffraction of Waves Around the Edge of Barriers

As straight waves pass by the edge of a barrier, the edge of the wave that contacts the barrier may

be diffracted into a curved wave.

2

1

v

v

Rsin

sin

i



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The amount of diffraction increases as the wavelength increases. This is the principal behind

"breakwaters" which are present at many beaches.

B. Diffraction Through Openings in a Barrier

As straight waves pass through an opening in a barrier, they may be diffracted to some extent. The

amount of diffraction increases as the wavelength () increases and as the

size of the opening (d) decreases.

Mathematically, as the ratio d/ decreases, the amount of diffraction increases.

C. Diffraction Around Obstacles

As straight waves pass around an obstacle in their path, some interesting things can occur. If the

object is large compared to the wavelength, the waves will be "blocked" (in fact, they will be

reflected) and a "shadow" will appear behind the object.

If the object is small compared to the wavelength, the waves are able to diffract around the edges of

the object. This will cause the divided wave to "re-form" on the far side of the object, if the

wavelength is quite large compared to the size of the object, the waves appear to be unaffected as

they pass by the object.



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Even if the waves completely diffract around an object, a "scattered" circular wave reflects from the

object. The wavelength and frequency of the scattered wave is equal to those of the incident waves.

Interference of Waves

A. Interference of One-Dimensional Waves

To begin this section, read Section 18.9 of the text. If possible, perform the Investigation:

Interference in a One-dimensional Medium that begins on page 359 of the text with the help

of your instructor. It is important that you be able to determine the answers to the questions

on page 360 of the text. A summary of the key ideas is given below.

If a wave pulse is generated at each end of a stretched spring, the waves will travel toward one

another with the same speed, and will eventually "collide".

Unlike the collision of two objects, the collision of two waves does not result in a change in their

motion. Instead, the waves simply "pass through" one another, seemingly unaffected.

When two waves act

simultaneously on the same points

in a medium, they are said to

interfere with one another. The

result of this interference is that the

amplitudes of the waves add

together at each point in the

medium, creating a waveform that

has a different amplitude than

either of the two waves that are

interfering with one another.

The amplitude and shape of the

result of the interference of waves

may be predicted using the

Principle of Superposition. This

principle tells us that the amplitude

of the displacement of any point in

the medium where two or more

waves are acting simultaneously is

exactly equal to the sum of the

amplitudes of the waves acting at

that point.

The illustration to the left shows

two positive wave pulses

approaching one another. The

waves are interfering with one



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another in the middle three illustrations. Notice that at all points where the waves are interfering,

the displacement of the medium is greater than the displacement that would be caused if only one of

the waves was present. What would happen if one pulse was positive and the other was negative?

Constructive Interference occurs when the displacement of the medium is greater than the

displacement that would be caused by either of the interfering waves acting alone. Constructive

interference occurs when two positive pulses or two negative pulses interfere with one another.

Destructive Interference occurs when the displacement of the medium is less than the

displacement that would be caused by either of the interfering waves acting alone.

Destructive interference occurs when a positive pulse interferes with a negative pulse. This is

shown in the diagram to the right.

Before continuing with the next section, study Sample Problem 16 in the Unit 1A Sample

Problem booklet. Ensure you understand how the resulting waveform is produced in each case.



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B. Interference of Water Waves

Begin this section by reading Section 18.11 of the text. The Investigation: Interference of

Water Waves in a Ripple Tank is performed for you in the video "Interference of Water

Waves". View this video and attempt to answer the questions on page 366 of the text. A

summary of the key ideas is given below.

If two identical (same wavelength and amplitude, and vibrating in phase) point sources of water

waves are placed side by side in a ripple tank, an interesting pattern of interference is produced.

In the photograph, there are gray lines radiating outward. These are regions where total destructive

interference occurs (areas where troughs exactly cancel out crests). Points where total destructive

interference occurs are called nodal points and lines of these points are called nodal lines. Midway

between the nodal lines are regions where constructive interference occurs. At the exact centre of

these regions are points where crests add to crests and troughs add to troughs. Thus, the amplitude

at these points is a maximum. Points where maximum constructive interference occurs are called

antinodal points. Lines of these points are called antinodal lines. Because the pattern of

alternating nodal and antinodal lines remains constant (as long as the frequency of the sources does

not change), this pattern is called a standing wave interference pattern.

The number of nodal and antinodal lines changes if either the distance between the sources or the

frequency (and, hence, the wavelength) of the waves is changed. Increasing the distance between



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the sources increases the number of lines, making them closer together. A longer wavelength (lower

frequency) decreases the number of lines, making them farther apart. The mathematical relationship

that allows us to predict the position of the antinodal lines is no longer part of the Physics 11

curriculum.

Assignment 4:

A. Complete Review Problems 24, 30 and 31 on pages 374-377 of the text.

B. Complete the worksheet “Waves Review Part 1” and submit that assignment to your

instructor for marking.

It is now time to write the Waves Properties Test. To prepare for this test, review all the

notes, sample problems and assignments.

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