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The properties and Behaviour of Waves

Vibrations
1
= [ s]
f
time
period 
cycles
T

1
 [ Hz]
T
cycles
frequency 
time
f 
Wave Motion
-
Transverse Wave:
In a transverse wave the particles in the medium vibrate at right angles to the direction in
which the wave travels. The high section of the wave is called a crest and the low section
a trough.
-
Longitudinal Wave:
In a longitudinal wave the particles vibrate parallel to the direction of motion of the wave,
and not at right angles to it. In a longitudinal wave, the regions where the particles are
closer together than normal are called compressions, and the regions where they are
farther apart are called rarefactions.
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
Transmission on Waves
The frequency of the wave is defined as the number of crests and troughs, or complete cycles,
that pass a given point in the medium per unit of time. The frequency of the wave is
exactly the same as that of the source. Once the wave is produced, its frequency never
changes, even if its speed and wavelength do.
-

Wave Equation:
v  f  frequency  wave length  [m / s]
Transmission and Reflection
When a wave travels into a different medium, its speed and wavelength change. At the
boundary between the two media, some reflection occurs. This is called partial reflection,
because some of the energy is transmitted into the new medium and some is reflected back
into the original medium. The phase of transmitted waves is unaffected in all partial
reflections, but inversion of the reflected wave occurs when the wave passes from a fast
medium to a slow medium.
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
Interference of Waves (Principle of Superposition)
When two or more waves act simultaneously on the some particles of a medium, whether in a
simple rope or spring, or in water or air, we speak of wave interference. The resultant
displacement of a given particle is equal to the sum of the displacements that would
have been produced by each wave acting independently. This is called the Principle of
Superposition. When tow or more waves interfere to produce a resultant displacement
greater than the displacement that would be caused by either wave, by itself, we call it
constructive interference. When the resultant displacement is smaller than the displacement
that would be caused by one wave, by itself, we call it destructive interference.
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
Standing Waves – A Special Case of Interference
The amplitude and the wavelength of interfering waves are often different. But if conditions
are controlled so that the waves have the same amplitude and wavelength, yet travel in
opposite directions, the resultant interference pattern is particularly interesting. It is referred
to as a standing wave interference pattern or simply a standing wave. When positive and
negative pulses of equal amplitude and length, travelling in opposite directions, interfere,
there is a point that remains at rest throughout the interference of the pulses. This point is
called a node, or nodal point (N). Midway between the nodes are areas where double crests
and double trough occur. There areas are called loops or antinodes.
In standing waves, the distance between two successive nodes is
1
.
2
e.g. The distance between two successive nodes in a vibrating string is 10 cm. The frequency
of the source is 30 Hz. What is the wavelength of the waves? What is their velocity?
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-
The frequencies at which standing waves can exist in a rope of a given length whose ends
are fixed are the natural frequencies, or resonant frequencies, of the rope.
-
A rope, a stretched spring, and even the air in an air column, such as that used in some
musical instruments, have a large number of resonant frequencies, each of which is a
whole-number multiple of the lowest resonant frequency, called the fundamental.
-
In a vibrating string stretched between two fixed points, nodes must occur at both ends.
In its simplest mode of vibration, the fundamental mode, the string vibrates in one
segment, producing its lowest frequency, the fundamental frequency, fo. If the string
vibrates in more than one segment, the resulting modes of vibration are called overtones.
Since the string can only vibrate in certain patterns, always with nodes at each end, the
frequencies of the overtones are simple (whole numbered) multiples of the fundamental
frequency, such as 2 fo, 3 fo, 4 fo, and so on.
-
The various resonant frequencies of standing waves are often called harmonics, because
in music they harmonize.
Wave Travelling in Two Dimensions
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 between successive crests or successive troughs remains the same as long as the speed of
the wave does not change. This behavior is true for all waves, circular or straight – the
wavelength remains constant if the speed does not change. When the speed decreases, as it does
in shallow water, the wavelength also decreases. That is, wavelength is directly proportional to
the speed (  v). When the frequency of a source is increased, the distance between successive
crests becomes smaller. In other words, waves with a higher frequency have a shorter
wavelength. The wavelength of a wave is inversely proportional to its frequency (  1/ f ). If
these two relationships are combined, the result is the Wave Equation, v = f . This equation
holds for all types of waves.
A continuous crest or trough is referred to as a wavefront. To show the direction of travel, or
transmission, or a wavefront, an arrow is drawn at right angles to the wavefront. This line called
a wave ray. Sometimes we refer to wave rays instead of wavefronts when describing the
behavior of a wave.
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Reflection from a Straight Barrier:
When a straight wave runs into a straight barrier, head on, it is reflected back along its original
path. If a wave encounters a straight barrier at an angle, the wavefront is also reflected at an
angle to the barrier. The angle formed by the incident wavefront and the barrier is equal to
the angles formed by the reflected wavefront and the barrier. These angles are called the
angle of incident and the angle of reflection. When describing the reflection of waves, using
wave rays instead of wavefronts, the angles of incidence and reflection are measured relative to a
straight line perpendicular to the barrier, called the normal. This line is constructed at the point
where the incident wave ray strikes the reflecting surface. The reflection of both straight and
circular waves is predicted by the Law of Reflection. In addition, in all wave reflection there is
no change in the wavelength or the speed of the wave.
Refraction of Waves:
When a wave enters a medium in which it moves more slowly, its wavelength decrease. Since
the frequency remains the same, we can find a relationship between a wave travelling in different
mediums.
v1
f
 1 1
v2
f 2 2
 f1  f 2

v1 1

v2 2
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e.g. A water wave has a wavelength of 2.0 cm in the deep section of a tank and 1.5 cm in the
shallow section. If the speed of the wave in the shallow water is 12 cm/s, what is its speed
in the deep water?
Practice:
1. The speed and the wavelength of a water wave in deep water are 18.0 cm/s and 2.0 cm,
respectively. If the speed in shallow water is 10.0 cm/s, what is the corresponding
wavelength? (1.1 cm)
2. A wave travels 0.75 times as fast in shallow water as it does in deep water. What will the
wavelength of the wave in deep water be, if its wavelength is 2.7 cm in shallow water? (3.6
cm)
When a wave travels from deep water to shallow water, in such a way that it meets the boundary
between the two depths straight on, no change in direction occurs. On the other hand, if a wave
meets the boundary at an angle, the direction of travel does change. This phenomenon is called
refraction. We usually use wave rays to describe refraction. The angle formed by an incident
wave ray and the normal is called the angle of incidence. The angle formed by a refracted
wave ray and the normal is called the angle of refraction.
When a wave travels at an angle into a medium in which its speed decreases, the refracted
wave ray is bent (refracted) towards the normal. If the wave travels at an angle into a
medium in which its speed increases, the refracted wave ray is bent away from the normal.
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Diffraction of Waves:
When periodic straight waves are produced, they travel in a straight line as long as the depth of
the medium remains uniform. The 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, or through a small opening or
aperture in the obstacle, the waves change direction. This bending is called diffraction.
Interference of Waves in Two Dimensions:
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The Production and Properties of Sound
 Every sound wave originates from a vibrating source. The average human is responsive to
frequencies of between 20 Hz and 20000 Hz.
 Frequencies of less than 20 Hz are referred to as infrasonic and those of more than 20000 Hz
are called ultrasonic.
 Accurate measurements of the speed of sound in air have been made at various temperatures
and air pressures. At normal atmospheric pressure and at 0o, it is 332 m/s. If the air pressure
remains constant, the speed of sound increases as the temperature increases. It has been
found that the speed of sound in air changes by 0.6 m/s for each degree Celsius.
Speed of sound in air = (331 + 0.6 T) m/s
(at normal atmospheric pressure)
where T is the temperature in degrees Celsius.
e.g. What is the speed of sound at (a) 20o C and (b) –20o C?
(344 m/s, 320 m/s)
Practice:
1. What is the speed of sound in air when the temperature is (a) –10o C, (b) 24o C, and (c)
35o C? (326 m/s, 346 m/s, 353 m/s)
2. How much time is required for sound to travel 1.4 km through air if the temperature is
30o C? (4.0 s)
 High speeds for supersonic aircraft are given in terms of Mach number rather than kilometres
per hour.
Mach number =
speed of object
speed of sound
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e.g. What is the Mach number of an aircraft travelling at sea level at 0o C with a speed of
1440 km/h? (1.2)
Practice:
1. What is the Mach number of an aircraft travelling at sea level at 0o C with a speed of 900
km/h?
(0.75)
2. A military interceptor airplane can fly at Mach 2.0. What is its speed in kilometres per
hour at sea level, and at 0o C? (2.4  103 km/h)
 Sound wave is longitudinal wave. In sound waves, a compression is an area of higher than
normal air pressure, and a rarefaction is an area of lower than normal air pressure. All of the
terms used to describe waves are used to describe sound waves. The wave equation also
holds.
e.g. The sound from a trumpet travels at a speed of 350 m/s in air. If the frequency of the
note played is 300 Hz, what is the wavelength of the sound wave? (1.17 m)
Practice:
1. An organ pipe emits a note of 50 Hz. If the speed of sound in air is 350 m/s, what is the
wavelength of the sound wave? (7.0 m)
2. If a 260 Hz sound from a tuning fork has a wavelength of 1.3 m, at what speed does the
sound travel? (338 m/s)
3. A sound wave with a wavelength of 10 m travels at 350 m/s. What is its frequency? (35
Hz)
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 Sound intensity, or loudness, is more difficult to measure because the amount of energy
involved is small in comparison with other forms of energy and because the potential range
of sound intensity is great. Sound audible to humans can vary in intensity from the quietest
whisper to a level that is painful to the ear – a difference of a factor of 1013. The unit used to
measure the intensity of sound is the decibel (dB).
The intensity of a sound received by the human ear depends on the power of the source and
the distance between the source and the person. The reading on the decibel scale also
decreases as the distance increases. For example, a reading of 100 dB at 1 m becomes 60 dB
at 100 m.
 Sound waves radiating out from a source are reflected when they strike a rigid obstacle, the
angle of reflection being equal to the angle of incidence. Sound waves conform to the
Laws of Reflection.
A microphone at the focal point of a concave reflector, called a parabolic microphone, is
sometimes used to pick up remote sounds at a sports event or to record bird calls.
Echoes are produced when sound is reflected by a hard surface, such as a wall or cliff. The
echo can be heard by the human ear only if the time interval between the original sound and
the reflected sound is greater than 0.1 s. For practical purposes, the distance between the
observer and the reflecting surface must be greater than 17 m.
The echo-sounder is a device that uses the principles of sound reflection to measure the depth
of the sea. Similar equipment is used in the fishing industry to locate schools of fish. All
such devices are called sonar (sound navigation and ranging) devices.
A similar technique is used in radar (radio detection and ranging).
e.g.A boy yells towards a cliff and hears his echo 3.0 s later. If the speed of sound is 340 m/s,
how far away is the cliff? (510 m)
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Practice:
1. A student stands 90 m from the foot of a cliff, claps her hands, and hears the echo 0.50s
later. Calculate the speed of sound in air. (3.6  102 m/s)
2. A sonar device is used in a lake, and the interval between the production of a sound and
the reception of the each is found to be 0.40 s. The speed of sound in water is 1500 m/s.
What is the depth of the water? (3.0  102 m)
 When the reflecting surface is less than 17 m away, the echo follows so closely behind the
original sound that the original sound appears to be prolonged. This effect is called
reverberation. Reverberations are particularly noticeable in large, empty building such as
cathedrals and concert halls, but they also occur when we shout in a road underpass or sing in
the shower. A certain amount of reverberation may enhance the quality of sound. Excessive
reverberation in a concert hall is undesirable because it interferes with the original sound,
making speech and music indistinct.
The acoustics of a concert hall are the most important concern of the architect, and the most
important property of a concert hall is its reverberation time. This is defined as the time
required for sound of a standard intensity to die away and become inaudible.
 Sound waves can travel around corners because of a property of a property of waves called
diffraction. Diffraction describes the ability of waves to move around an obstacle or
through a small opening. Waves with relatively long wavelengths diffract more than
those with short wavelengths. Lower frequency sound waves have relatively long
wavelengths when compared to the obstacles and openings they commonly encounter.
 The speed with which a sound travels through air can be affected by the temperature of the
air. Sound waves travel faster in warm air than they do in cold air. If they move from air at
one temperature to air at a different temperature, at an angle, they are refracted in such the
same manner as light rays.
On a warm day, sound waves tend to be refracted upward from an observer, which decreases
the intensity of the sound heard by the observer. On the other hand, at night, the cooler air
near the surface of the Earth tens to refract sound waves toward the surface and so they travel
a greater distance. This is particularly true on flat ground or on water.
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The Interference of Sound Waves

Interference of Identical Sound Waves:
 Constructive interference of sound waves (maximum intensity)
 Sound is diminished
 Total destructive interference of sound waves (minimum intensity)

Best Frequency:
 Beat
The periodic changes in sound intensity are called beats.
 Beat Frequency
The number of maximum intensity points that occur per second is called the beat
frequency.
Beat frequency = | f 1 – f 2|
where f 1 and f 2 are the frequencies of the two sources.
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If the two sources have the same frequency, no beats are heard.
If the tension of the sting increases, the frequency increases. If the tension decreases, the
frequency decreases.
e.g. A tuning fork with a frequency of 256 Hz is sounded, together with a note played on a piano.
Nine beats are heard in 3 s. What is the frequency of the piano note? (253 Hz or 259 Hz)
Practice:
1. A tuning fork with a frequency of 400 Hz is struck with a second fork, and 20 beats
are counted in 5.0 s. What are the possible frequencies of the second fork? (396 Hz,
404 Hz)
2. A third fork with a frequency of 410 Hz is struck with the second fork in question 1,
and 18 beats are counted in 3.0 s. What is the frequency of the second fork? (404 Hz)

Vibrating Strings:
For a string, the pitch, or frequency, or a vibration string is determined by four factors – its
length, its diameter, and the density of the material it is made of.
 The frequency increases when the length decreases. The frequency varies inversely with
length.
1
f 
L
 The frequency increases when the tension increases. The frequency varies directly as the
square root of the tension.
f  T
 The frequency increases when the diameter decreases. The frequency varies inversely
with the diameter.
1
f 
d
 The frequency increases when the density decreases. The frequency varies inversely as
the square root of the density.
1
f 

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e.g. The D string of a violin is 30.0 cm long and has a natural frequency of 288 Hz. Where must
the violinist place his finger on the string to produce B (383 Hz)? (7.5 cm from the end of
the string)
e.g. A piano string with a pitch of A (440 Hz) is under a tension of 140 N. What tension would
be require to produce a high C (523 Hz)? (198 N)
Practice:
1. A 1.0 m string has a frequency of 220 Hz. If the string is short-ended to 0.80 m, what
will its frequency become? (275 Hz)
2. A string under a tension of 150 N has a frequency of 256 Hz. What will its frequency
become if the tension is increased to 300 N? (362 Hz)
3. Two strings have the same diameter, length, and tension. One is made of brass
(density = 8.70  103 kg/m) and the other is made of steel (density = 7.83 103 kg/m).
If the frequency of the brass string is 440 Hz, what is the frequency of the steel string?
(463 Hz)
4. Two copper strings of equal length and tension have diameters of 0.80 mm and 1.0
mm respectively. If the frequency of the first is 200 Hz, what is the frequency of the
second? (160 Hz)
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 Quality of Sound
In a vibrating string stretched between two fixed points, nodes occur at both ends.
Different frequencies may result, depending on how many loops and nodes are
produced. In its simplest mode of vibration, the fundamental mode, the string
vibrates in one segment, producing its lowest frequency or pitch, called the
fundamental frequency, f o. If the string vibrates in more than one segment, the
resulting modes of vibration are called overtones. Since the string can only vibrate in
certain patterns, always with nodes at each end, the frequencies of the overtones are
simple multiples of the fundamental frequency, called harmonics, such as 2f o, 3f o,
and so on.
The quality of a musical note depends on the number and relative intensity of the
overtones it produces, along with the fundamental. It is the element of quality that
enables us to distinguish between notes of the same frequency and intensity coming
from different sources.

Mechanical Resonance:
Every object has a natural frequency at which it will vibrate. Resonance is the response
of an object that is free to vibrate to a periodic force with the same frequency as the
natural frequency of the object. We call such resonance mechanical because there is
physical contact between the periodic force and the vibrating object. It can be
demonstrated with a series of pendulums suspended from a stretched string.
Mechanical resonance must be taken into account in the designing of bridges, airplane
propellers, helicopter rotor blades, turbines for steam generators and jet engines,
plumbing systems, and many other types of equipment.

Resonance in Air Columns:
Sound waves from one source can cause an identical source to vibrate in resonance.
-
Closed Air Column:
If an air column is closed at one end and open at the other, it is referred to as a closed air
column. When a vibrating tuning fork is held over the open end of such a column and
the length of the column is increased, it is found that the loudness increases sharply at
very specific lengths. If a different tuning fork is used, the same phenomenon is
observed except the maxima occur at different lengths. To explain this behaviour we
must recall the behaviour of standing waves.
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For a closed air column such as a tube, a node is
formed at the bottom of the column, and since the air
is free to move at top of the tube, a loop forms there.
When the resonance first occurs, the column is ¼  in
length, since a single loop and node are formed. The
next possible lengths with a node at one end and a
loop at the other are ¾ , 5 4 , and so on. Thus, the
resonant lengths in a closed air column occur at ¼ ,
¾ , 5 4 , and so on. The resonant length of the
wooden box that is open at one end and attached to a
tuning fork is ¼ .
e.g. The first resonant length of a closed air column
occurs when the length is 18 cm.
(a) What is the wavelength of the sound? (0.72 m)
(b) If the frequency of the source is 512 Hz, what
is the speed of sound? (370 m/s)
Practice
1. The first resonant length of a closed air column occurs when the length is 30 cm. What will
the second and third resonant lengths be? (90 cm, 150 cm)
2. The third resonant length of a closed air column is 75 cm. Determine the first and second
lengths. (15 cm, 45 cm)
3. What is the shortest air column, closed at one end, that will resonate at a frequency of 440 Hz,
when the speed of sound is 352 m/s. (20.0 cm)
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-
Open Air Column:
Resonance can also be produced in open air columns or pipes.
If a standing wave interference pattern is created by reflection
at a free end, a loop occurs at the free end. Since an open pipe
is open at both ends, loops are formed at both ends. The first
length at which resonance occurs is ½ . Succeeding resonant
lengths will occur at , 3 2 , 2 , and so on.
e.g. An organ pipe, open at both ends, produces a musical
note at its first resonant length.
(a) What is the wavelength of the note produced? (7.2 m)
(b) What is the frequency of the pipe, if the speed of
sound in air is 346 m/s? (48 Hz)
Practice
1. The second resonant length of an air column that is open at both ends is 48 cm. Determine
the first and third resonant lengths. (24 cm, 72 cm)
2. An organ pipe, open at both ends, resonates at its first resonant length with a frequency of
128 Hz. What is the length of the pipe if the speed of sound is 346 m/s? (1.35 m)
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
Sonic Booms
A static, or stationary, source radiates sound waves in concentric spheres. An airplane
radiates spheres of sound waves from successive positions. Because the aircraft was moving,
the wavefronts were farther apart behind it than they were in front of it. When an airplane is
flying at the speed of sound, the wavefronts in front of it pile up, producing an area of very
dense air, or intense compression, called a sound barrier. Unless the air craft has been
designed to “cut” through this giant compression, it will be buffeted disastrously. At
supersonic speeds, the spheres of sound waves are left behind the aircraft. These interfere
with each other constructively, producing large compressions and rarefactions along the sides
of an imaginary double cone extending behind the airplane from the front and the rear. This
intense acoustic pressure wave sweeps along the ground in a swath whose width is
approximately five times the altitude of the aircraft and is usually referred to as a sonic boom.
The sonic boom is heard as two sharp cracks, like thunder or a muffled explosion.
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
Doppler Effect
The general principle of the Doppler Effect is that, when a source generating waves
approaches an observer the frequency of the source apparently increases, and when the
source moves away from the observer the frequency apparently decreases.
v
 vobserver 
f   f  sound
 v

 sound  vobject 
The upper signs apply if source and/or observer move toward each other; the lower signs
apply if they are moving apart.
e.g. The siren of a police car at rest emits at a predominant frequency of 1600 Hz. What
frequency will you hear if you are at rest and the police car moves at 25.0 m/s (the speed
sound = 343 m/s)
(a) toward you. (1726 Hz)
(b) away from you. (1491 Hz)
Practice
1. A train moving at a speed of 40 m/s sounds is whistle, which has a frequency of 500 Hz.
Determine the frequencies heard by a stationary observer as the train approaches and
recedes from the observer. (566 Hz, 448 Hz; Speed of Sound = 343 m/s)
2. An ambulance travels down a highway at a speed of 33.5 m/s. It siren emits sound at a
frequency of 400 Hz. What is the frequency heard by a passenger in a car traveling at
24.6 m/s in the opposite direction as the car approaches the ambulance and as the car
moves away from the ambulance? (475 Hz, 338 Hz; Speed of Sound = 343 m/s)