1. 11. Waves
Any disturbance that propagates with a well-defined velocity is called a wave.
Waves can be classified in the following three categories:
Mechanical waves: can exist only within a material medium such as air, water, etc. The medium’s mass and elasticity influence the propagation properties of mechanical waves.
2. Electromagnetic waves: involve propagating disturbances in the electric and magnetic field governed by Maxwell’s equations and do not require a material medium. Common examples are radio waves, light, x-rays, gamma rays.
3. Matter waves: associated with all microscopic particles such as electrons, protons, neutrons, atoms, etc (de Broglie rule).
Transverse
wave – the
disturbance is
perpendicular
to the wave
propagation
velocity.
Longitudinal
wave – the
disturbance is
parallel to the
wave propag.
velocity. A
Figures from HRW,2
A wave in a stretched string
A sound wave in a pipe

2. where k – wave number, λ – wavelength, T – period of oscillations
The disturbance can be described by function giving its magnitude at point x and instant t. If the oscillations of a wave’s source are given by the harmonic function then one can write
The motion of a given element at a position x and instant t is the same as the motion of a source at x = 0 and an earlier instant (v - the wave speed).
(11.1)
where k – wave number, λ – wavelength,
T – period of oscillations
A snapshot (fixed instant of time) of a
transverse wave in a string
(in this case ψ(x,t) = y(x,t) in eq.(11.1)).
A wavelength λ measured from an
arbitrary position x1 is indicated. A

3. 11.1. The phase velocity of a wave
The argument of the sine function in eq.(11.1) is called the phase:
(11.2)
To keep a constant phase with increasing time t, it follows from (11.2) that x must
also increase what means that phase (11.2) describes a wave travelling in the positive direction of coordinate x.
Calculating the differential of a phase φ(x,t) one obtains:
(11.3)
For a constant phase dφ=0, then
or (11.4)
The quantity vφ (or v) is the phase velocity
(the speed of propagation of a given phase).
Eq.(11.4) can be more generally written as
what is called a relation of dispersion. The phase velocity can be a function of k (or λ) what means that a wave being a superposition of waves with different k udergoes dispersion. In this case it changes a shape during propagation, as the constituent waves move with different velocities.
A wave moving
with velocity v
(two snapshots
are shown). A

4. 11.2. The wave equation
In one dimension the wave equation can be written as
partial differential equation (11.5)
Eq.(11.5) is fulfilled by the superposition of waves with constant velocity v.
Now we prove an assumption that eq.(11.5) is obeyed by the superposition of waves moving in +x and –x directions
(11.6)
Calculating the derivatives one obtains
(11.6a)
(11.6b)
Substituting (11.6a) and (11.6b) into (11.5) we get
what means that eq.(11.5) is obeyed by expression (11.6). A

5. 11.3. The principle of superposition
The wave equation is linear. For this type of equations the following theorem obeys:
if ψ1 and ψ2 are solutions of the equation, the function
c1 ψ1 + c2 ψ2 is also a solution (c1 and c2 are constants).
The direct consequence of the linearity of the wave
equation is the principle of superposition:
Two different waves propagate independently of each
other through a medium; the resulting disturbance
at any point in space at any instant is the superposition
(sum) of the disturbances due to each wave.
If one wave is characterised by a displacement
y1(x,t) and the second by a displacement y2(x,t),
the net wave after overlapping is obtained by
algebraic summation
Overlapping waves do not alter each other.
Figure from HRW,2 A
The superposition of two pulses
travelling in opposite directions.

6. Interference of waves
The waves with a constant in time phase shift interfere (all types of waves). The
sources sending these waves are called coherent sources. Typical examples are the
sources of radio waves or microwaves. Natural sources of light are incoherent. In this
case to observe the phenomenon of diffraction one has to use a diaphragm with slits
(Young’s experiment, 1801).
The small slit S0 in the first diaphragm makes the
sources S1 and S2 coherent. The waves from sources
S1 and S2 are of equal phases, frequencies and
amplitudes. At point P the waves have different
phases and this difference depends on the position
of point P on the screen. As a result one obtains the
sequence of interference fringes.
The superposition of waves at point P gives
(11.7)
The phase difference for these waves is
(11.8)
therefore depends on the path length difference r2 – r1. A

7. Interference of waves, cont.
For the conditions:
we have the greatest possible amplitude – fully constructive
interference. The above condition is equivalent to
or (11.9)
what means that the difference in path length is an integer number of wavelength.
fully destructive interference. This is equivalent to
or (11.10)
in this case the diference in path length is an odd number of half of wavelength.
For other conditions we have
an intermediate interference.
Example
Interference of two sinusoidal waves
y1=Asin(kx-ωt) and
y2= Asin(kx-ωt+φ) what gives
y = y1+y2 = 2Acos(φ/2)sin(kx-ωt+φ/2).
The resultant waves for three
different phase shifts φ are shown. A
Figure from HRW,2

8. Interference of waves, cont.
The positions of interference extrema in Young’s experiment are function of the
diffraction angle θ, because the path difference for two rays is equal
(11.11)
According to (11.10) the main maxima (bright fringes)
are then obtained for diffraction angles
(11.12)
To determine the light intensity in two-slit interference
we can use phasor (wskaz) diagrams
Interferens fringes for two very
narrow (a << λ) slits. The phase
shift, taking into account (11.11)
can be determined as
The wave disturbance can be
represented as a projection of
the vector amplitude rotating
around an origin with the
angular frequency w of the wave. A
From the condition that one obtains (11.13)
where (11.13a)

9. Interference of waves, cont.
The intensities of interference fringies in double-slit experiment, calculated from
eqs. (11.13) and (11.13a) are shown in the figure below
I0 - intensity for one source
2I0 - intensity for two incoherent sources
4I0 - intensity for two coherent sources
(vawe intensity is a power passing through
the unit surface area)
Standing waves
We analyze the interference of two waves travelling in opposite directions. From the
superposition principle one obtains
(11.14)
We obtained the equation which is not a travelling wave butf an oscillatory motion
with the amplitude A’(x) varying with position x. This is called a standing wave.
The position dependent amplitude is zero for kx = nπ (nodes) and has a maximum
value for
kx = (n + ½) π (antinodes). A

10. Standing waves, cont. A standing wave can be set up by reflecting
Figure from HRW,2
Five snapshots at
indicated times of a
wave travelling to the
left, travelling to the
right and the
superposition of both
waves.
antinode
node
A standing wave can be set up by reflecting
a travelling wave from the obstacle.
When reflecting wave loses energy, the standing
wave has no point nodes. In this case the SWR
(standing wave ratio) is defined
Ar- amplitude of a reflected wave
Ai- amplitude of an incident wave A
Ten snapshots of a standing wave.
Both adjacent nodes and adjacent
antinodes are separated by λ/2.

11. Standing waves, resonance
When we are trying to set up a standing wave in a string clamped on both sides, only at selected frequencies the interference produces a standing wave . These frquencies are called resonant frequencies of the system and the phenomenon itself is known as a resonance.
Generally the resonance occurs when the standing wave
satisfies the boundary conditions of the system.
For the case shown in the figure these conditions are:
the amplitudes at point A and B must be zero (nodes) as at
these points the string is clamped.
The lowest resonant frequency possible in the system is
shown in fig.(a).
The second standing wave has three nodes and frequency
fig.(b)
Generally for the system of a string clamped on both sides
one obtains A B A
Figure from HRW,2

12. Sound waves
Mechanical longitudinal waves that propagate in solids, liquids, and gases are called sound waves. Seismic waves propagate in the Earth’s crust, sound waves generated by a sonar system propagate in the sea, an orchestra creates sound waves that propagate in the air, using ultrasound a medical diagnosis can be made.
A sound wave in a long air-filled tube. It consists of periodic expansions and compressions of the air.
Pressure variations can be described as follows
In the isotropic medium, i.e. the medium in which sound propagates with the same speed for all directions, the wavefronts are spheres centered at the point source S.
A wavefront is the locus of the points of a sound wave that have the same displacement.
Rays are lines perpendicular to the wavefronts and they
point along the direction in which the wave propagates.
The single arrows indicate the rays. The double arrows indicate the motion of the molecules of the medium in which sound propagates.

13. The speed of sound
The speed of sound in a homogeneous isotropic medium is equal
where:
B – bulk modulus
r - density
Applying overpressure Dp on an object of volume V, results in a change of volume DV.
The bulk modulus depends on the fractional change in volume produced by a change in pressure and is defined by
more compressible media and media with higher density exhibit lower speed of sound Δp
Examples of the speed of sound (m/s) for some media
air (0o C) water (0o C) aluminum
air (20o C) water (20o C) steel
helium seawater (20o C) 1522

14. Intensity and sound level
Each vawe transports energy. As a result some power P passes through area A intercepting the sound. Sound intensity I is defined as
units: W/m2
For the point sound source and isotropic medium the wavefronts are spheres. The sound intensity at a distance r from the point source is then given by
The intensity of point sound source decreases with the square of the distance from this source.
The sound varies in an enormous range of intensities. The auditory sensation of human ear is proportional to the logarithm of sound intensity I. In order to mimick the response of the human ear one defines sound level as a logarithm of intensity
I0 – reference intensity equal W/m2
b is measured in decibels (dB)
Reference intensity is near to the lower limit of the human range of hearing. Every time the sound intensity increases by an order of magnitute, b increases by 10 dB.

15. Musical sound
Standing waves of sound set up in a string or in air-filled pipe correspond to the resonant frequencies of the string or pipe. Resonant oscillations have large amplitude and by interaction with the ambient air produce the sound vaves with the same frequency.
For a pipe of length L with two ends open the following wavelengths can be generated
n – harmonic number
For n = 1 we have the first harmonic (fundamental mode) which is characteristic of a given note.
For a pipe of length Lwith one open end the resonant frequencies are determined by
The fundamental and higher harmonics are generated in a given instrument simul-
tanously. The content of
higher harmonics determines
the sound timbre (barwa).
Figure from HRW
Standing wave patterns for (a) the pipe with both ends open where any harmonic (odd and even) can be generated and (b) the pipe with one end open where only odd harmonics can be set up. At the closed end a node is formed and at the open end – an antinode.