PHYSICS: FORM FOUR: Topic 1 – WAVES

A wave is a disturbance that travels through a medium from one location to another location.

Waves are classified into;

• transverse waves: Waves which have their direction of motion perpendicular to the vibration of the particles e.g water waves.
• longitudinal waves: Waves have their direction of motion parallel to the direction of the vibration of the particles e.g sound waves.

Consider a slinky wave as an example of a wave. When the slinky is stretched from end to end and is held at rest, it assumes a natural position known as the equilibrium or rest position.

The coils of the slinky naturally assume this position, spaced equally far apart. To introduce a wave into the slinky, the first particle is displaced or moved from its equilibrium or rest position. The particle might be moved upwards or downwards, forwards or backwards; but once moved, it is returned to its original equilibrium or rest position.

The act of moving the first coil of the slinky in a given direction and then returning it to its equilibrium position creates a disturbance in the slinky. We can then observe this disturbance moving through the slinky from one end to the other. If the first coil of the slinky is given a single back-and-forth vibration, then we call the observed motion of the disturbance through the slinky a slinky pulse.

A pulse is a single disturbance moving through a medium from one location to another location.

However, if the first coil of the slinky is continuously and periodically vibrated in a back-and-forth manner, we would observe a repeating disturbance moving within the slinky that endures over some prolonged period of time. The repeating and periodic disturbance that moves through a medium from one location to another is referred to as a wave.

A medium is a substance or material that carries the wave.

You have perhaps heard of the phrase news media. The news media refers to the various institutions (newspaper offices, television stations, radio stations, etc.) within our society that carry the news from one location to another. The news moves through the media.

The wave medium is not the wave and it doesn’t make the wave; it merely carries or transports the wave from its source to other locations.

In the case of our slinky wave, the medium through that the wave travels is the slinky coils. In the case of a water wave in the ocean, the medium through which the wave travels is the ocean water. In the case of a sound wave moving from the church choir to the pews, the medium through which the sound wave travels is the air in the room.

Consider the transverse wave below:

1. A transverse wave is a wave in which the particles of the medium are displaced in a direction perpendicular to the direction of energy transport.
2. The crest of a wave is the point on the medium that exhibits the maximum amount of positive or upward displacement from the rest position.
3. The trough of a wave is the point on the medium that exhibits the maximum amount of negative or downward displacement from the rest position.
4. The amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position.
5. The wavelength of a wave is simply the length of one complete wave cycle. If you were to trace your finger across the wave in the diagram above, you would notice that your finger repeats its path. A wave is a repeating pattern. It repeats itself in a periodic and regular fashion over both time and space. And the length of one such spatial repetition (known as a wave cycle) is the wavelength. The wavelength can be measured as the distance from crest to crest or from trough to trough. In fact, the wavelength of a wave can be measured as the distance from a point on a wave to the corresponding point on the next cycle of the wave.
6. A longitudinal wave is a wave in which the particles of the medium are displaced in a direction parallel to the direction of energy transport. A longitudinal wave can be created in a slinky if the slinky is stretched out horizontally and the end coil is vibrated back-and-forth in a horizontal direction.
7. A compression is a region in the longitudinal waves where the wave particles are closest to each other. It is a high pressure region and characterized by high concentration of wave particles.
8. A rarefaction is the opposite of compression. It is region where wave particles are far apart from each other. it is a low pressure region.
9. The frequency, (f) of a wave refers to how often the particles of the medium vibrate when a wave passes through the medium. It the number of waves/oscillations passing a point per second. Given this definition, it is reasonable that the quantity frequency would have units of cycles/second, waves/second, vibrations/second, or something/second. The SI unit for frequency is the Hertz(abbreviated Hz) where 1 Hz is equivalent to 1 cycle/second.
10. Period, (T) refers to the time that it takes to do something. When an event occurs repeatedly, then we say that the event is periodic and refer to the time for the event to repeat itself as the period. The period of a wave is the time for a particle on a medium to make one complete cycle (oscillation). Period, being a time, is measured in units of time such as seconds, hours, days or years. The period of orbit for the Earth around the Sun is approximately 365 days; it takes 365 days for the Earth to complete a cycle.
11. The speed of an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time.The SI unit of speed is m/s.

Wave equation

Previously we classified the waves by considering the movement of the particles. And now you’re going to find out the 2 types of waves according to the media of propagation.

1. Mechanical Waves
2. Electromagnetic Waves

Mechanical Waves

Electromagnetic Waves

Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves.

The laws of reflection

Characteristics of Reflection of waves:

1. It obeys the Law of Reflection.
2. The wavelength, λ of the reflected wave is the same as that of the incident waves.
3. The frequency,fof the reflected waves is the same as that of the incident waves.
4. Therefore the speed, v of the reflected waves is the same as that of the incident waves.

Ripple Tank

Refraction is the change in direction of propagation of a wave due to a change in its transmission medium.

The phenomenon is explained by the conservation of energy and conservation of momentum. Due to change of medium, the phase velocity of the wave is changed but its frequency remains constant. This is most commonly observed when a wave passes from one medium to another at any angle other than 0° from the normal.

Refraction of light is the most commonly observed phenomenon, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth.

Snell’s law

Refraction of a light ray.

Interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude.

Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency.

Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves.

Constructive Interference

Destructive Interference

Diffraction refers to a change in direction of waves as they pass through an opening or around a barrier in their path.

In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle. These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance.

Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves. Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle and the principle of superposition of waves.

The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves.

When waves are put together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes.

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.A slit which is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit

Diffraction of water waves is observed in a harbor as waves bend around small boats and are found to disturb the water behind them. Diffraction of sound waves is commonly observed; we notice sound diffracting around corners, allowing us to hear others who are speaking to us from adjacent rooms.

Diffraction is observed of light waves but only when the waves encounter obstacles with extremely small wavelengths (such as particles suspended in our atmosphere).

Applications of Refraction

1. Refraction has many applications in optics and technology. A lens uses refraction to form an image of an object for many different purposes, such as magnification.
2. A prism uses refraction to form a spectrum of colors from an incident beam of light.
3. Refraction also plays an important role in the formation of a mirage and other optical illusions.

Solution

v_diamond= 1.24 × 10⁸ms^-1

c= 3.0 × 10⁸ms^-1

(from data sheet)

Refractive index of diamond for this colour of light = 2.42

The ear is the organ that detects sound. It not only receives sound, but also aids in balance and body position. The ear is part of the auditory system.

Often the entire organ is considered the ear, though it may also be considered just the visible portion. In most mammals, the visible ear is a flap of tissue that is also called the pinna (or auricle in humans) and is the first of many steps in hearing.

Vertebrates have a pair of ears placed somewhat symmetrically on opposite sides of the head. This arrangement aids in the ability to localize sound sources.

The human ear is divided into three parts:

The outer ear comprises the pinna (auricle), which is made of a convoluted plate of flexible cartilage that extends as a nearly closed tube one-third of the way down the ear canal. This outer third, which is about eight millimetres (one-third of an inch) long, has small hairs that point outwards to form a line of defence against small animals creeping in. The roots of the hairs produce oils and these mix with the secretions from nearby sweat-like glands to form the basis of wax.

The deep two-thirds of the ear canal (16 millimetres/two-thirds of an inch long) has a bony wall lined with thin and rather fragile skin which is devoid of glands. At the far end of the ear canal and stretched across it is the eardrum (tympanic membrane), which forms the boundary between the outer and middle ears.

The eardrum is a circle of thin skin about eight to nine millimetres (one-third of an inch) in diameter. Despite its name, it is not flat like the skin of a drum, but is slightly conical with the curved sides sloping inwards. The eardrum has three layers.

The inner ear is probably the most remarkably intricate piece of the body. It makes hearing possible by converting sound into electrical impulses that then travel along the hearing nerve (the acoustic nerve or auditory nerve) to the brain. The inner ear also plays a major role in balance. The balance portions of the inner ear (vestibular labyrinth) can detect acceleration of the head in any direction whether in a straight line (linear) or twisting and turning (angular). The electrical signals that arise in response to head movement pass along the balance nerve (vestibular nerve), which in due course joins with the hearing nerve to form a single bundle (stato­acoustic, vestibulo-acoustic or eighth nerve, nerve VIII) which then enters the brain.

The portion of the inner ear that actually hears is the cochlea. This is a hollow coiled tube set in the very dense bone called the bony labyrinth (part of the petrous [rock­like] temporal bone). This tube is filled with fluid, which is much the same as general body fluid (lymph) and that which surrounds the brain (cerebrospinal fluid – CSF). This inner-ear fluid is called perilymph. Inside the perilymph is another coiled triangular-shaped tube called the cochlear duct (scala media), which contains the all-important ‘hair cells’ – these convert sound into electricity. These hair cells are arranged in two groups that follow the coils of the cochlear duct and spiral upwards from base to apex. There is a single row of inner hair cells (IHCs), which lie closer to the core of the cochlea (modiolus), and three or four rows of outer hair cells (OHCs), which are further away. In a healthy young human ear there are about 3,500 IHCs and about 12,000 OHCs. Each hair cell has a cluster of small rigid hairs (stereocilia), which project from the thicker upper surface of the cell into the special fluid that fills the cochlear duct. This fluid is called endolymph and is remarkable in that it has a strongly positive electrical charge associated with it – about 80 millivolts – and is rich in potassium, a metallic element.

The hair cells in their rows are grouped together with their supporting cells in the organ of Corti. This is a small ridge that sits on a thin, very flexible membrane called the basilar membrane. The basilar membrane forms the floor of the triangular cochlear duct. The sloping roof is another very thin membrane (Reissner’s membrane) and the side wall is a thickened region rich in blood vessels (the stria vascularis). This structure is responsible for maintaining the composition of the rather unusual and very important endolymph.

Adjacent to the base of the hair cells are the nerves that carry impulses to the brain (the afferent nerves). At least 90 per cent of these nerves come from the inner hair cells, despite their smaller number. Each inner hair cell has about 10 nerve endings attached to it and there are, therefore, about 30,000 nerve fibres in the acoustic nerve.

The hearing nerves travel inwards, along with the balance and facial nerves, through a canal in the inner part of the skull (variously called the internal auditory meatus [IAM], internal auditory canal [IAC] or porus acousticus) to reach the brain stem. This part of the brain deals with lots of automatic functions such as pulse, blood pressure, general alertness, balance, and so on.

About half of the hearing nerves from each ear cross over to the other side of the brain stem and then, on both sides, the nerves pass up the brain stem through the mid­brain, eventually to reach ‘conscious­ness’ in what is called the cortex of the brain. For hearing, this conscious region is located in the temporal lobe portion of the brain, which lies on each side of the head just above the ear.

An echo occurs when a sound wave is reflected and hence arrives to the listener after some time delay after the direct sound. Echo is simply a reflected sound.

When we are in an empty space surrounded by mountains and we shout loudly, we see that faint sound resembling the original sound coming back after some delay of time. This perception of the reflected wave is nothing but the echo. This is experienced in remote places, in rooms that are big and empty, in caves, buildings and in big forests.

Echo is derived from the Greek word which means Sound.The echo is produced due to hitting of the sound waves with the obstacles which makes the sound to reflect back

Application of Echo:

Reverberations are the collection of reflected sounds from the surfaces in an enclosure like an auditorium. It is a desirable property of auditoriums to the extent that it helps to overcome the inverse square law drop off of sound intensity in the enclosure.

However, if it is excessive, it makes the sounds run together with loss of articulation – the sound becomes muddy, garbled.

A reverberation is the same as echo but the distance here is less.The distance between the source of the sound and the obstacle by which it is reflected is less in Reverberation.Here, in reverberation the time delay is less than 0.1 second.

The reflected wave reaches the observer in less than 0.1 second. Now as the delay in time is less than the original sound is still in memory, the delay between perception of sound and the original sound is very-very less.

How is an Echo Different from the reverberation?

Is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude

This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The most common cause of standing waves is the phenomenon of resonance, in which standing waves occur inside a resonator due to interference between waves reflected back and forth at the resonator’s resonant frequency.

For waves of equal amplitude traveling in opposing directions, there is on averageno net propagation of energy.Traveling waves have high points called crests and low points called troughs (in the transverse case) or compressed points called compressions and stretched points called rarefactions (in the longitudinal case) that travel through the medium.

Standing waves don’t go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are called nodes. There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called antinodes.

The locations at which the amplitude is minimum are called nodes.

The locations where the amplitude is maximum are called antinodes.

Fundamental note is the lowest resonant frequency of a vibrating object. Most vibrating objects have more than one resonant frequency and those used in musical instruments typically vibrate at harmonics of the fundamental.

A harmonic is defined as an integer (whole number),n multiple of the fundamental frequency. Vibrating strings, open cylindrical air columns, and conical air columns will vibrate at all harmonics of the fundamental.

An overtone is any frequency higher than the fundamental frequency of a sound. Using the model of Fourier analysis, the fundamental and the overtones together are called partials.Harmonics, or more precisely, harmonic partials, are partials whose frequencies are integer multiples of the fundamental (including the fundamental which is 1 times itself). These overlapping terms are variously used when discussing the acoustic behavior of musical instruments.

There are integer multiples of a certain frequency (fundamental), that are called harmonics, partial tones (partials) or overtones.It is important to note that the term ‘overtones’ does not include the fundamental frequency. The first overtone is therefore already the second harmonic or the second partial. The term overtone should never be mixed with the other terms, as the counting is unequal.The term harmonic has a precise meaning – that of an integer (whole number) multiple of the fundamental frequency of a vibrating object.

A closed ended instrument has one end closed off, and the other end open. An example would be an instrument like some organ pipes (although in some designs they are open), or a flute.

Although you blow in through the mouth piece of a flute, the opening you’re blowing into isn’t at the end of the pipe, it’s along the side of the flute. The end of the pipe is closed off near the mouth piece. Remember that it is actually air that is doing the vibrating as a wave here.

• The air at the closed end of the pipe must be a node (not moving), since the air is not free to move there and must be able to be reflected back.
• There must also be an antinode where the opening is, since that is where there is maximum movement of the air.
• The simplest, smallest wave that I can possibly fit in a closed end pipe is shown in Figure below.
• Notice how even though it has been flipped left-to-right and it looks squished and stretched a bit to fit, this is still ¼ of a wavelength.
• Since this is the smallest stable piece of a wave I can fit in this pipe, this is the Fundamental, or 1^st Harmonic.

Since the length of the tube is the same as the length of the ¼ wavelength I know that the length of this tube is ¼ of a wavelength… this leads to our first formula: L = ¼ λ

• “L” is the length of the tube in meters. On it’s own this formula really doesn’t help us much.
• Instead, we have to solve this formula for λ and then combine it with the formula v=fλ to get a more useful formula:

When the wave reaches the closed end it’s going to be reflected as an inverted wave (going from air to whatever the pipe is made of is a pretty big change so this is what we would expect). It would look like Figure above.

This does not change the length of the wave in our formula, since we are only seeing the reflection of the wave that already exists in the pipe.

What does the next harmonic look like? It’s the 3^rd Harmonic.

• I know this name might seem a little confusing (I’m the first to agree with you!) but because of the actual notes produced and the way the waves fit in, musicians refer to the next step up in a closed end pipe instrument as the 3^rd harmonic… there is no such thing as a 2^nd harmonic for closed end pipes.
• In fact, all of the harmonics in closed end pipes are going to be odd numbers.

Remember that we have to have an antinode at the opening (where the air is moving) and a node at the closed end (where the air can’t move). That means for the 3^rd harmonic we get something like Figure above. This is ¾ of a wavelength fit into the tube, so the length of the tube is… L = ¾ λ

This is the third harmonic of the closed end pipe. The formula for the frequency of the note we will hear is…

Do you notice a pattern forming in the formulas? Hopefully, because for both open and closed end pipes, we will only give you the formulas for the fundamentals lengths. You need to remember how to get the rest.

If we drew in the reflection of the third harmonic it would look like Figure below.

One more to make sure you see the pattern. The 5^th Harmonic (fig. above)

There is one full wavelength in there (4/4) plus an extra ¼ of a wavelength for a total of 5/4. The length of the pipe is… L = 5/4 λ And the note produced by the 5^th Harmonic is found using the formula…f = 5v/4L

Figure below shows the reflection of a 5^th Harmonic for a closed end pipe.

An open ended instrument has both ends open to the air. An example would be an instrument like a trumpet. You blow in through one end and the sound comes out the other end of the pipe.

• The fundamental (first harmonic) for an open end pipe needs to be an antinode at both ends, since the air can move at both ends.
• That’s why the smallest wave we can fit in is shown in Figure 11.
• This looks different than the ½ wavelength that I showed you in Figure 3, but it is still half of a full wavelength.
• That means the length of the tube and frequency formulas are…L = ½ λ

f = v/2L

The next note we can play is the 2^nd harmonic.

• Yep, open end pipes have a 2^nd harmonic… they can have any number harmonic they want, odd or even.
• Again, it kind of looks weird, but trace it out and you’ll see that there is exactly one wavelength here.
• The length and frequency formulas are…L = 2/2 λ, f= 2v/2L

I’m not going to show you what the 3^rd harmonic looks like. Instead, try drawing it yourself and see what you get.

As a hint to help you, the formulas for the length and frequency are…L = 3/2 λ, f = 3v/2L

Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum. Radio waves have frequencies from 300GHz to as low as 3kHz, and corresponding wavelengths ranging from 1 millimeter (0.039in) to 100 kilometers (62mi).

Like all other electromagnetic waves, they travel at the speed of light. Naturally occurring radio waves are made by lightning, or by astronomical objects. Artificially generated radio waves are used for fixed and mobile radio communication, broadcasting, radar and other navigation systems, communications satellites, computer networks and innumerable other applications.

Radio waves are generated by radio transmitters and received by radio receivers.

Sources of radio waves

• alternating electric currents flowing in antennae
• oscillators
• planets, comets, stars and galaxies

Microwaves have a short wavelength of between 10⁻⁴m to about 0.1.

Sources

• Are produced by oscillation of charges in special antennae mounted on dishes.
• Magnetrons.

They have the frequency 0f between 10 and 10 Hz. They lie between the visible light and microwaves. They have the heating effect.

Sources: Vibration of atoms and molecules due to their thermal energy. All hot bodies emit infrared radiations.

Is the narrow range of EW frequencies to which human eyes are sensitive.

Infrared radiation is invisible to the human eye. However any nocturnal animal can “see” infrared radiations. We can sense infrared radiations as heat. Devices used to detect infrared radiations include black bulb thermometers, photographic films, thermistors and phototransistors.

We see things because they either emit visible light or because they reflect visible light from another source.

Ultraviolet is light is detected by Photographic films; Fluorescent materials