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# Waves/Sine Waves

Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology

### Sine Waves

A particularly simple kind of wave, the sine wave, is illustrated in figure 1.2. This has the mathematical form:
    $h(x)=h_{0}\sin(2\pi x/\lambda ),$ (2.1)
where
    h is the displacement (which can be either longitudinal or transverse),
    $h_{0}$ is the maximum displacement, sometimes called the amplitude of the wave,
    λ is the wavelength.tory Physics fig 1.2.png|Figure 1.2: Sine wave]] -->
Figure 1.2: Definition sketch for a sine wave, showing the wavelength λ and the amplitude $h_{0}$ and the phase φ at various points.
So far we have only considered a sine wave as it appears at a particular time. All interesting waves move with time. The movement of a sine wave to the right, a distance $d$ may be accounted for by replacing $x$ in the above formula by $x-d$. If this movement occurs in time $t$, then the wave moves at velocity $c=d/t$. Solving this for $d$ and substituting yields a formula for the displacement of a sine wave as a function of both distance $x$ and time $t$:
    $h(x)=h_{0}\sin[2\pi (x-ct)/\lambda ].$ (2.2)
The time for a wave to move one wavelength is called the period of the wave: $T=\lambda /c$. Thus, we can also write
    $h(x)=h_{0}\sin[2\pi (x/\lambda -t/T)].$(2.3)
Physicists actually like to write the equation for a sine wave in a slightly simpler form. Defining the wavenumber as $k=2\pi /\lambda$ and the angular frequency as $\omega =2\pi /T$, we write
    $h(x)=h_{0}\sin(kx-\omega t).$(2.4)
We normally think of the frequency of oscillatory motion as the number of cycles completed per second and is given by $f=1/T$. It is related to the angular frequency omega by $\omega =2\pi f$. The angular frequency is used because it is directly analogous to the wavenumber, see above. Converting between the two is not difficult. Frequency is measured in units of hertz, abbreviated Hz; $1{\mbox{Hz}}=1{\mbox{ s}}^{-1}$ and angular frequency $\omega$ is in units of radians per second. (Not converting between frequency and angular frequency is a common mistake in many calculations)
The argument of the sine function is by definition an angle. We refer to this angle as the phase of the wave, $\phi =kx-\omega t$. The difference in the phase of a wave at fixed time over a distance of one wavelength is $2\pi$, as is the difference in phase at fixed position over a time interval of one wave period.
As previously noted, we call $h_{0}$, the maximum displacement of the wave, the amplitude. Often we are interested in the intensity of a wave, which is defined as the square of the amplitude, $I=h_{0}^{2}$.
The wave speed we have defined above, $c=\lambda /T$, is actually called the phase speed. Since $\lambda =2\pi /k$ and $T=2\pi /\omega$, we can write the phase speed in terms of the angular frequency and the wavenumber:
    $c={\frac {\omega }{k}}{\mbox{(phase speed)}}.$ (2.5)
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology
# Waves/Sound Waves

Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology

#### Sound Waves

Sound is a longitudinal compression-expansion wave through matter. The wave speed is
    $c=(\gamma RT_{abs})^{1/2}$ (2.11)
where $\gamma$ and $R$ are constants and $T_{abs}$ is the absolute temperature. The absolute temperature is measured in Kelvins and is numerically given by
    $T_{abs}=T_{C}+273.16^{\circ }$ (2.12)
where $T_{C}$ is the temperature in Celsius degrees. The angular frequency of sound waves is thus given by
    $\omega =ck=(\gamma RT_{abs})^{1/2}k.$ (2.13)
The speed of sound in air at normal temperatures is about $340{\mbox{ m}}{\mbox{ s}}^{-1}$.
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology
# Waves/Total Internal Reflection

Waves : Geometrical Optics 1 - 2 - 3 - 4 - 5 - 6 Problems

### Total Internal Reflection

When light passes from a medium of lesser index of refraction to one with greater index of refraction, Snell's law indicates that the ray bends toward the normal to the interface. The reverse occurs when the passage is in the other direction. In this latter circumstance a special situation arises when Snell's law predicts a value for the sine of the refracted angle greater than one. This is physically untenable. What actually happens is that the incident wave is reflected from the interface. This phenomenon is called total internal reflection. The minimum incident angle for which total internal reflection occurs is obtained by substituting $\theta _{R}=\pi /2$ into equation (3.2), resulting in
    $\sin(\theta _{I})=n_{R}/n_{I}\quad {\mbox{(total internal reflection)}}.$ (4.3)
Waves : Geometrical Optics 1 - 2 - 3 - 4 - 5 - 6 Problems
# Waves/Types of Waves

Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology

### Types of Waves

In order to make the above material more concrete, we now examine the characteristics of various types of waves which may be observed in the real world.

#### Ocean Surface Waves

Figure 1.3: Wave on an ocean of depth $H$. The wave is moving to the right and the particles of water at the surface move up and down as shown by the small vertical arrows. The particles move up, and gravity is the 'restoring' force. Water waves are also longitudinal, the water particles moving forward, then back. The restoring forces are more complex, but involve the inertia of the mass of water surrounding. Think of water waves as superimposed transverse and longitudinal waves. The net particle paths are nearly circular. This is typical of waves that travel along the boundary (interface) between two substances, in this case water and air.
These waves are manifested as undulations of the ocean surface as seen in figure 1.3. The speed of ocean waves is given by the formula
    $v=\left({\frac {g\tanh(kH)}{k}}\right)^{1/2},$ (2.6)
where $g=9.8{\mbox{ m}}{\mbox{ s}}^{-2}$ is a constant related to the strength of the Earth's gravity, $H$ is the depth of the ocean, and the hyperbolic tangent is defined as 2.7
    $\tanh(x)={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}.$ (2.7)
Figure 1.4: Plot of the function $\tanh(x)$. The dashed line shows our approximation $\tanh(x)\approx x$ for $\vert x\vert <<1$.
As figure 1.4 shows, for very small x, we can approximate the hyperbolic tangent by $\tanh(x)\approx x$, while for very large x it is positive 1 for positive x and negative 1 for negative x. This leads to two limits: Since $x=kH$, the shallow water limit, which occurs when $kH<<1$, yields a wave speed of
    $v\approx (gH)^{1/2},{\mbox{(shallow water waves)}},$ (2.8)
while the deep water limit, which occurs when $kH>>1$, yields
    $v\approx (g/k)^{1/2},{\mbox{(deep water waves)}}.$ (2.9)
Notice that the speed of shallow water waves depends only on the depth of the water and on $g$. In other words, all shallow water waves move at the same speed.
On the other hand, deep water waves of longer wavelength (and hence smaller wavenumber) move more rapidly than those with shorter wavelength. Waves for which the wave speed varies with wavelength are called dispersive. Thus, deep water waves are dispersive, while shallow water waves are non-dispersive.
For water waves with wavelengths of a few centimeters or less, surface tension becomes important to the dynamics of the waves. In the deep water case the wave speed at short wavelengths is actually given by the formula
    $c=(g/k+Ak)^{1/2}$ (2.10)
where the constant $A$ is related to surface tension and depends on the surfaces involved. For an air-water interface near room temperature, $A\approx 74{\mbox{ cm}}^{3}{\mbox{ s}}^{-2}$.
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology
# Waves/Waves in One Dimension

Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology

## The Mathematics Of Waves

We start our discussion of waves by taking the equation for a very simple wave and describing its characteristics. The basic equation for such a wave is
    $y=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)$
where $y$ is the height of the wave at position $x$ and time $t$. This equation describes a fairly simple wave, but most complex waves are just sums of simpler ones. If we freeze this equation in time at $t=0$, we get
    $y=a\ \sin \left({\frac {2\pi x}{\lambda }}+\alpha \right)$
which looks like this: [TODO - Add a Graph]
From the graph we can see that each of the three parameters has a meaning. $a$ is the amplitude of the wave, how high it is. $\lambda$ is the wavelength, the distance from a part of the wave in one cycle to the same part of the wave in the next cycle. $\alpha$ is the phase of the wave, which shifts the wave to the left or right. The wavelength is a distance, and is usually measured in meters, millimeters or even nanometers depending on the wave. Phase is an angle, measured in radians.
Now that we have mapped out the wave in space, let's instead set $x=0$ and see how the wave changes over time
    $y=a\ \sin(-2\pi ft+\alpha )$
Amplitude $a$ and phase $\alpha$ remain, but the wavelength is gone and a new quantity has appeared: $f$, which is the frequency, or how rapidly the wave moves up and down. Frequency is measured in units of inverse time: in a fixed period of time, how many times does the wave move up and down? The unit usually used for this is the hertz, or inverse second.
Now let's combine these two pictures and see how the wave moves. Figure 3 is a diagram of how the wave looks when you plot it in both space and time. The straight lines are the places where the simple wave reaches a maximum, minimum, or zero (where it crosses the x axis).
We can look at the zeros to determine the phase velocity of the wave. The phase velocity is how fast a part of the wave moves. We can think of it as the speed of the wave, but for more complicated waves it is only one type of speed - more on that in later sections.
We can get an equation for the zeros by setting our equation to zero.
    $0=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)$
    $0={\frac {2\pi x}{\lambda }}-2\pi ft+\alpha$
    $x=f\lambda t-{\frac {\alpha \lambda }{2\pi }}$
You see here that we have the equation for a straight line, describing a point that is moving at velocity $f\lambda$. This gives us the equation for the phase velocity of the wave, which is
    ${\mbox{velocity}}={\mbox{frequency}}\times {\mbox{wavelength}}\quad v=f\lambda$
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology
# Waves/Single slit Diffraction

Waves : 2 and 3 Dimension Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 Problems

### Diffraction Through a Single Slit

How does all of this apply to the passage of waves through a slit? Imagine a plane wave of wavelength $\lambda$ impingent on a barrier with a slit or opening of width $w$. The barrier transforms the plane wave with infinite extent in the lateral direction into a beam with initial transverse dimensions equal to the width of the slit. The subsequent development of the beam is illustrated in figures 2.13 and 2.14, and schematically in figure 2.15. In particular, if the slit width is comparable to the wavelength, the beam spreads broadly as in figure 2.13. If the slit width is large compared to the wavelength, the beam doesn't spread as much, as figure 2.14 illustrates. Equation (2.22) gives us an approximate quantitative result for the spreading angle if $w$ is interpreted as the width of the slit.
One use of the above equation is in determining the maximum angular resolution of optical instruments such as telescopes. The primary lens or mirror can be thought of as a rather large “slit”. Light from a distant point source is essentially in the form of a plane wave when it arrives at the telescope. However, the light passed by the telescope is no longer a plane wave, but is a beam with a tendency to spread. The spreading angle $\alpha _{max}$ is given by equation (2.22), and the telescope cannot resolve objects with an angular separation less than $\alpha _{max}$. Replacing $w$ with the diameter of the lens or mirror in equation (2.22) thus yields the telescope's angular resolution. For instance, a moderate sized telescope with aperture $1{\mbox{ m}}$ observing red light with $\lambda \approx 6\times 10^{-7}{\mbox{ m}}$ has a maximum angular resolution of about $3\times 10^{-7}{\mbox{ radians}}$.
# Word formation/Clipping

Clipping occurs whenever a word is shortened but nevertheless still retains its original meaning and belongs to the same word-class.

## Productivity

Clipping-productivity is governed by the condition of derivatives being phonologically well-formed, with morpheme-boundary cuts no longer being a requirement (this helps distinguish certain clipping products from back-formation where such mechanism exists). This being said, with the use of la langue one is more likely to derive cello from violoncello rather than *ncell since it does not represent a pronunciation that is common or even correct in the English language. Another principle that that applies is a product of clipping must be either one- or two-syllable long; also — first or primarily-stressed syllables are retained.
This indeed is the case: refrigerator — fridge, advertisement — ad, penitentiary — pen.
In some cases, the syllable structure may be altered (e-xa-mi-na-tion - e-xam). Because of the language's tendency to adopt ways of expressing oneself in the most effective manner — that is convey as much meaning as possible in as little as possible — products of clipping tend to gain ground in slang words and become a tool used in relaxed, informal, every-day communication, which further solidifies their position in the language. They also indicate speaker's familiarity towards the concept that is presented through the word.
What also has been noted to happen is once a product of clipping is derived, it may rival with the lexeme it was derived from and because of so undergo semantic changes. This is contradictive to the rule of meaning-retention, and once again proves that word-formation often produces lexemes that are uneasy to define. The examples include fanatic — fan, brandywine — brandy, caravan — van. These show that discrepancies between the derivatives can exist, which can be further proved by the example of fanatic — fan — fanboy where both fanatic and the compound fanboy are semantically distant from each other.

## Clipping types

There are four types of possible clipping processes, depending on which part of the word undergoes structural changes: back-clipping (temperature — temp, rhino — rhinoceros, gym — gymnasium), fore-clipping (helicopter — copter, telephone — phone, plane — aeroplane), mixed clipping (influenza — flu, refrigerator — fridge) and clipping-compounds (paratrooper — parachute + trooper).

## References

* ↑ Szymanek, Bogdan (1998). Introduction to Morphological Analysis. p. 97.
* ↑ Štekauer, P (2000). Rudiments of English Linguistics. p. 111.
# Word formation/Back-formation

Back-formation, also known as back-derivation, forms new lexemes by means of suffix-like segment extraction. This means that not only suffixes can be deleted from the stem, but also structures that behave alike, as long as the cut is done at a boundary between corresponding morphemes. Taking these two possibilities into account, one may be presented not only with such examples as lecher — to lech, peddler — to peddle, sculptor — to sculpt, but also the lexeme to lase being a derivative of laser where the {-er} segment is a by-product of deconstructing an acronym rather than the agentive/instrumental suffix -er. This type of word formation in English is by far dominated by creating derivatives that belong to the grammatical class of verb. Back-derivation is often an outcome of linguistic productivity based on paradigms — it is safe to make an assumption that pairs edit — editor and exhibit — exhibitor were created in a similar manner by employing strategies speakers of the language know because of their linguistic competence.

## References

* ↑ Szymanek, Bogdan (1998). Introduction to Morphological Analysis. p. 93.
* ↑ Stekauer, Pavol (2000). Rudiments of English Linguistics. p. 109.
# Word formation/Conversion

In contrast to reduplication which because of its nature is rather doubtful in being clearly labelled as a tool of non-concatenative morphology branch, conversion is an easily-defined word-formation process.
Conversion takes place when a given word changes its word-class, hence becoming a new one. Because this involves no extension at the level of the word's internal structure, this process is also called zero-derivation or zero-affixation. It is argued that even though there is no visual representation of meaning-derivation, the so-called zero morpheme is added to the base as a justification for the change: [cook]v → [[cook]V + O ]N.
What can be exposed to modification, however, is the way the word is stressed (compare 'import [n.] and im'port [v.]). Despite this being the only noticeable change, there are several ways to find out which one is the derivative of the other. These include the date of the word's first appearance, frequency of the word's occurrence in use and complexity of meaning. Because the word that appeared first tends to carry less content meaning-wise, the one whose meaning is extended is regarded to be a derivative. Although conversion seems to be a relatively free process and can produce derivatives of virtually any word-class, four certain types of this phenomenon seem to prevail in English: noun-to-verb (a pilot [n.] — to pilot [v.]), verb-to-noun (to cook [v.] — a cook [n.]), adjective-to-noun (professional [a.] — a professional [n.]) and adjective-to-verb (empty [v.] — to empty [v.]). Also, one can distinguish between partial conversion and total conversion. Whereas total conversion deals with the transformation of the word into a new one following all its principles of grammatical suitability, partial conversion derivatives can differ from the original in a way (eg. advice — to advise, house — to house: /s/ is exchanged for /z/).

## References

* ↑ Bauer, Laurie (1983). English Word-Formation. p. 226.
* ↑ Plag, I (2007). Introduction to English Linguistics. p. 100.
# Word formation/Reduplication

Reduplication is a process that takes place when the root or stem is reduplicated and added to the existing one. The added morpheme might or might not be further modified and on the basis of this division one can distinguish partial and complete variants or reduplication. The latter of the two does not change the structure of the reiterated word.
Examples found in the Indonesian language:
* kitab — 'book',
* kitab kitab - 'various books',
* anak — 'child',
* anak anak — 'various children'.
While these clearly show that the additional morpheme not only denotes 'various X' but also suggests plural, conversely English girly-girly and goody-goody represent derivatives whose meaning implies using irony or sarcasm.
Partial reduplication, in turn, takes advantage of morphemes that have been modified to a degree. This is where pseudomorphemes are introduced — such that have been modified in a way they remain as sound clusters that no longer carry meaning and serve as a tool of expressing stylisation or emotional involvement of the speaker: shillyshally, zig-zag. The presence of pseudomorphemes is an indicative of the word being an outcome of reduplication rather than compound-creation where meaningful morphemes are used.

## References

* ↑ Szymanek, Bogdan (1998). Introduction to Morphological Analysis. p. 72.
# Word formation/Acronymisation

The products of acronymisation are called acronyms. These word-structures are made of several other words' initial segments that make up a whole and serve as a meaningful shortening of a longer phrase. This definition is also true for abbreviations.
The key to distinguish the two is the manner of pronunciation of either derivatives. Whereas acronyms are pronounced as a complete word, the other type rather as a series of separate letters. This being said, one will pronounce the acronym UNESCO (United Nations Educational, Scientific, and Cultural Organisation) as [ju ˈnesk əʊ], and UFO which is an abbreviation as [ˌjuː ef ˈəʊ].
# Word formation/Internal modification

Internal modification occurs when a phoneme (or a group of phonemes) in a word is replaced by another one and thus creates a new item. Although the process itself is not very productive in English, a variety of changes can be introduced with it, like word-class change or tense change. There are several possibilities of swapping the phonological segments — these include replacing vowels, consonants or both the same time (mixed modifications). Replacing all the sound segments result in a phenomenon called suppletion, eg. człowiek (sg.) — ludzie (pl.). In English it is common to observe internal modification in the past verb forms (wind — wound, steal — stole, make — made) and in irregular plural forms of nouns (mouse — mice, woman — women, foot — feet). In contrast, the Polish language makes use of internal modification in creating augmentatives: nos — nochal, kluska — klucha.

## References

* ↑ Szymanek, Bogdan (1998). Introduction to Morphological Analysis. p. 76.
# Word formation/Blending

Blending, or contamination, is the name of the process which occurs when at least two lexemes are shortened and joined together disregarding their initial boundaries. What is created in such a process can be called a blend, though other names are also found to operate — those being portmanteau or telescoped words.
A characteristic feature of contamination, and also the one that enables a speaker to coin blends freely is that the semantic content of the derivative most of the time carries some of the meaning of its constituents. There are few rules that govern blend-creation. The speaker is not restricted in any way from using part of any length of either of the constituents (the obvious cases would be the ability to pronounce such a word), thus distinguishing elements that add up to the derivative is not always an easy task. This leads to a conclusion that blends found in English are constructed according to various patterns.
Among others, one can find words that are constructed of two constituents, but only one of them is shortened, or even the two overlap each other:
| Lexeme 1 | Lexeme 2 | Blend |
| breath | analyser | brethalyser |
| cable | telegram | cablegram |
| slang | language | slanguage |