This document discusses different types of mechanical waves and their properties. It defines mechanical waves as oscillations that transfer energy through a medium. Key points include: - Mechanical waves can be transverse (perpendicular to direction of travel) or longitudinal (parallel to direction of travel). - They transport energy through the medium and require a medium, like air or water, to propagate. - Examples of mechanical waves include water waves, sound waves, and waves on a string or rope. - Harmonic waves have a sinusoidal shape described by a mathematical function involving amplitude, wavelength, frequency, and phase.

1. Waves : A disturbance or variation that transfers energy progressively from
point to point in a medium .

3. If the displacement of the individual atoms or molecules is perpendicular to
the direction the wave is traveling, the wave is called a transverse wave.
If the displacement is parallel to the direction of travel the wave is called a
longitudinal wave or a compression wave.

4. A mechanical wave is a wave that is an oscillation of matter, and therefore transfers
energy through a medium.
Waves can move over longer distances but the medium oscillates in SHM about an
equilibrium point. Therefore, the oscillating material does not move far from its initial
equilibrium position.
Mechanical waves transport energy. This energy propagates in the same direction as the
wave. Any kind of wave (mechanical or electromagnetic) has a certain energy.
Mechanical waves require medium to propagate.
Mechanical waves can be produced only in media which possess elasticity and inertia.
Example : Rock thrown into water will create mechanical waves which will propagate
outward in all directions.
Mechanical waves can also travel through a
rope/cord

5. Water waves are surface waves, a mixture of longitudinal and transverse waves.
Most ocean waves are produced by wind, and the energy from the wind offshore is
carried by the waves towards the shore.
Wind-driven waves, or surface waves, are created by the friction between wind and
surface water.

6. Sound waves are longitudinal waves that travel through a medium like air or water. When
we think about sound, we often think about how loud it is (amplitude, or intensity) and
its pitch (frequency).

7. • Electromagnetic (EM) waves propagate through space and can propagate through
any medium.
• They are a natural phenomenon

9. Matter waves are not electromagnetic waves.
Matter waves are generated by the motion of particles. If the particles are at rest,
then there is no meaning of matter waves associated with them.
The only function of the wave is to pilot or to guide the matter particles as shown
and hence it is called as pilot wave.

10. If the individual atoms and molecules in the medium move with simple harmonic
motion, the resulting periodic wave has a sinusoidal form. We call it a harmonic
wave or a sinusoidal wave.
Harmonic waves
Consider a transverse harmonic wave traveling in the positive x-direction. Harmonic
waves are sinusoidal waves. The displacement y of a particle in the medium is given as
a function of x and t by
y(x,t) = Asin(kx - ωt + φ)
where k is the wavenumber, k = 2π/λ, and ω = 2π/T = 2πf is the angular frequency of
the wave. φ is called the phase constant.

11. At a fixed time t the displacement y varies as a function of position x as
y = Asin(kx) = Asin*(2π/λ)x]
The phase constant φ is determined by the initial conditions of the motion. If at t = 0 and x
= 0 the displacement y is zero, then φ = 0 or π. If at t = 0 and x = 0 the displacement has its
maximum value, then φ = π/2. The quantity kx - ωt + φ is called the phase.
At a fixed position x the displacement y varies as a function of time as
y = Asin(ωt) = Asin*(2π/T)t+ with a convenient choice of origin.
For the transverse harmonic wave y(x,t) = Asin(kx - ωt + φ) we may write
y(x,t) = Asin[(2π/λ)x - (2πf)t + φ+ = Asin[(2π/λ)(x - λft) + φ+
or, using λf = v and 2π/λ = k,
y(x,t) = Asin[k(x - vt) + φ].
This wave travels into the positive x direction. Let φ = 0. Try to follow some point on the
wave, for example a crest. For a crest we always have kx - vt = π/2. If the time t increases,
the position x has to increase, to keep kx - vt = π/2.

12. For a transverse harmonic wave traveling in the negative x-direction we have
y(x,t) = Asin(kx + ωt + φ)= Asin(k(x + vt) + φ).
Interference
Two or more waves traveling in the same medium travel independently and can pass
through each other. In regions where they overlap we only observe a single
disturbance. We observe interference.
When two or more waves interfere, the resulting displacement is equal to the vector
sum of the individual displacements. If two waves with equal amplitudes overlap in
phase, i.e. if crest meets crest and trough meets trough, then we observe a resultant
wave with twice the amplitude. We have constructive interference.
If the two overlapping waves, however, are completely out of phase, i.e. if crest meets
trough, then the two waves cancel each other out completely. We have destructive
interference.

13. Standing waves
Consider two waves with the same amplitude, frequency, and wavelength that are
travelling in opposite directions on a string.
Using the trigonometric identities sin(a + b) = sin(a)cos(b) + cos(a)sin(b) we write the
resulting displacement of the string as a function of time as
y(x,t) = Asin(kx - ωt) + Asin(kx + ωt) = 2Asin(kx)cos(ωt).
This wave is no longer a traveling wave because the position and time dependence
have been separated. All sections of the string oscillate either in phase or 180o out
of phase. The section of the string at position x oscillates with amplitude
2Asin(kx). No energy travels along the string. There are sections that oscillate with
maximum amplitude and there are sections that do not oscillate at all. We have a
standing wave.

14. Impedance of a string
Impedance denoted by Z, tells us how much resistance the medium offers to the
passage of the wave.
This the ratio of the transverse force to transverse velocity
When we say transverse, it is transverse to the direction of motion of the wave. Ie.
If the wave is travelling is positive x direction, then the transverse force is the one
that is perpendicular to the wave.
where rho is the linear density and c the wave velocity
Impedance is the property of the medium
Impedance an also be written as …..

15. Any medium through which waves propagate will present an impedance to those
waves.
If the medium is lossless, and possesses no resistive or dissipation mechanism, this
impedance will be determined by the two energy storing parameters, inertia and
elasticity, and it will be real.
The presence of a loss mechanism will introduce a complex term into the impedance.
A string presents such an impedance to progressive waves and this is defined,
because of the nature of the waves, as the transverse impedance (Z)

16. Wave Packet

18. Superposition of waves and Fourier method

19. The Superposition Principle And Fourier
Using Superposition Principle below, let's see how a complex wave can be described.
Sin waves A & B Using superposition principle , C = A + B
From the diagrams above we know that C = A + B.
Here, A = 0.5 * sin(2wt), and B = 0.2 * sin(16wt).
So, if f(t) represents the complex wave, then:
Note, the 'w' is the "angular frequency", usually given in radians per second. 'w = 2*pi*f0', where f0 is the
fundamental frequency of the wave. Notice that wave A has a frequency twice the fundamental ( 2wt ) and wave B
has 16 times the frequency of the fundamental (16wt).

20. Phase Shift
Phase shifts must also be handled, because a sinusoid can be shifted along the x-axis. If wave A above
were shifted by, say, 90 degrees, or pi/2, then the results would look as follows:
two waves, A and B, one phased shifted, A.
Phase shifting doesn't affect the fundamental frequency. It only affects the wave's
shape.
Using superposition principle , C = A + B

21. The DC Components
A wave can also have a constant or DC component or signal that shifts a sinusoid up or
down the y-axis so that it no longer oscillates around y = 0. The term "DC" comes from
"direct current". It's an artifact of electronics, due to the fact that Fourier is often used in
dealing with electrical signals. However, DC in Fourier does not have to be an electrical
signal. It's just the constant part of any signal, regardless of whether or not the medium
is electric, electromagnetic, pressure, etc.
A DC Component, 'A', and a sine wave, 'B'.
C = A + B.
The equation for the complex wave, C, above would be:

22. The General Form of the Fourier Series
First a brief summary of what we've learned so far. The Fourier Series applies only to
periodic waves. All of the components of a periodic waves are integer multiples of the
fundamental frequency. We also know that each component has its own phase and
amplitude. We also must account for a DC component if it exists. Assembling these facts,
here is the general form of the Fourier Series:
where a0 is the DC component, and w = 2*pi*f and f = the fundamental frequency.
A More Common Representation of the Fourier Series
More often the Fourier Series is represented by a sum of sine and cosine
waves (and often as complex notation, eiwt.

23. Periodic Waves and Fourier Transform
A periodic waveform consists of one lowest frequency component and multiples of
the lowest frequency component. The lowest frequency component is called
fundamental and the components that are multiples of fundamental frequency are
called harmonics.
French mathematician Jean-Baptiste Joseph Fourier demonstrated that any periodic
waveform is composed of a fixed term, plus an infinite series of cosine terms, plus
an infinite series of sine term. The frequency of these infinite sine and cosine terms
are multiples of the one fundamental frequency.
Mathematically we represent this statement as
f(t) = A0+A1cos(wt) + A2cos(2wt) + A3cos(3wt) + - - - - - - + Ancos(nwt)
+A1sin(wt) + A2sin(2wt) + A3sin(3wt) + - - - - - - + Ansin(nwt)
A0 represents the DC component of the periodic wave. For most periodic waves, the
values of the coefficients of the cosine and the sine terms diminish rapidly as the order
of the harmonics increases.

24. The figure illustrates a periodic wave (named total wave) formed of a
perfect sine wave of amplitude 1, a second harmonics of amplitude 0.2, and
a third harmonics of amplitude 0.15
Figure : A periodic wave composed from sum of the fundamental, 2nd Harmonics and 3rd
Harmonics
By varying the relative amplitudes of the fundamental and harmonics we can get more and more
shapes.
Notice that the rising edge of total waveform in the above diagram is faster than the rising edge of
fundamental sine wave. This is because all the harmonics start rising at the same time in this case.
In order to preserve a higher rising edge signal and its shape we will need to preserve more
harmonics of the wave as it propagates. In other words, it will require more bandwidth.

25. A symmetrical square wave is composed of the fundamental wave and the odd harmonics of sine waves.
The amplitude of the harmonics is inversely proportional to the frequency.
f(t) = A[sinwt + (1/3) sin3wt + (1/5) sin5wt + - - - - - - + (1/n)sin nwt].
The figure below shows the waveform produces by taking into account the fundamental, the first
harmonics and the third harmonics into account. Notice that the rise time of the square wave is faster
than the fundamental and the harmonics. The ripples of the wave would have decreased had we taken
more harmonics.
Figure : A square wave formed with fundamental, 3rd harmonics and 5th harmonics.
The square wave in the figure above (marked wave) is not a perfect square wave. Why? Because we
have formed it with only up to fifth harmonics. If we take more and more harmonics (7th , 9th etc.) and
add them, the resulting wave will be more and more close to a perfect square wave.

26. Fourier series of a square wave
The Fourier series is ………..

27. Bandwidth Theorem
This is applicable for wave groups made of many frequency components , each of
amplitude a lying within a narrow frequency range Dw.
Dw.Dt = 2p .......(1)
(width in frequency domain) x (width in time domain) = 2p
We know w = 2pn
Therefore eqn (1) becomes
D 2pn . Dt = 2p
Dn . Dt = 1 .......... (2)
This relation is applicable to all waves including particle waves in quantum mechanics.
This band width theorem implies that any wave phenomenon that occurs over a time
Dt will have a frequency spread, Dn given by
Dn =
1
Dt
𝐻𝑧
If Dt is small, then Dn will be large

28. The theorem also states that a single pulse of time duration Dt is the result of the
superposition of frequency components over the range Dw.
Shorter the period Dt of the pulse, the wider the range Dw of the frequencies
required to represent it.
Example
Lets assume that at one instance we clap our hands and at other instance we
cough.
Clap has smaller time width Dt than the cough. Therefore a clapping sound has
much larger frequency spread than the cough.
Our ear very sensitive to different frequencies and so can easily distinguish
between these two sound packets.
We thus realize band width theorem in regular life, an understanding of
superposition of waves.