Summary of Waves
1. Mathematical description of a wave: y(x, t) = A cos2
x t , T
sinusoidal wave moving in the positive x-direction. = 2f =
2 T
,
is the angular frequency in terms of the frequency and period. k=
2
,
is the wave number in terms of the wavelength. The wave function can also be written as y(x, t) = A cos kx t , The velocity of a particle on the wave is given by vy(x, t) =
y t
= A sin kx t
and the acceleration is given by ay(x, t) =
y
2
t
2
= 2A cos kx t = 2y(x, t)
2. Speed of a transverse wave in a string When a force F keeps one end of string of length ℓ and mass m under tension, then the speed of transverse waves traveling in the string is given by v=
F
where is the linear mass density given by
m
The average power in a sinusoidal wave is given by Pavg =
1 2
F A
2
2
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3. Principle of Superposition When two waves overlap, the resultant displacement of any one point on the wave is the sum of the displacements the points would have had if only one of the waves were present at that point.
4. Standing waves in a string When a wave traveling along a string is reflected at the fixed end of the string, the incident wave and the reflected can be superimposed to form a standing wave.
The points of maximum displacement are called anti-nodes while the points of zero displacement are called nodes. The wave function of a standing wave is given by y(x, t) = (ASW sin kx) sint, where ASW = 2A. Nodes occur at while antinodes occur at x = n(/2), n = 0,1,2,3….. x = (n+½)(/2), n = 0,1,2,3…..
For a string fixed at both ends, the frequencies of the standing waves formed depend on the length, L, of the string and are given by fn = n
v 2L v 2L nf 1 , n = 1,2,3…..
f1 =
is called the fundamental frequency, while the series are together called harmonic series. The second harmonic also called the first overtone, etc.
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5. Sound Waves (a) Pressure amplitude Since sound waves are longitudinal waves, we consider pressure fluctuations of the particles instead of particle displacements as in transverse waves. The pressure of any particle at any instant of time is given by p(x, t) = BkA sin(kx t), The maximum pressure is given by pmax = BkA where B is the bulk modulus.
for sound waves.
(b) Speed of sound in gases:
v B
, where is the density of the gas , (only for ideal gases)
v
γRT M
where is the ratio of the specific heat capacities, R is the universal gas constant, T is the temperature on the Kelvin scale and M is the molar mass of the gas. (c) Sound Intensity The intensity І of sound is the rate at which energy is transported per unit area and is given by given by І= or І =
p max 2 v
2
1 2
B A
2
2
When we compare two sound intensities we determine the sound intensity level, which is given by
10 log
I I0
,
measured in decibels (dB)
and І0 = 10-12 W/m2, is called the threshold of hearing.
(d) Range of frequency of sound waves
20 Hz Infrasonic Range Audible Range 20 kHz Ultrasonic Range
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(e) Open and closed pipes The normal mode frequencies for pipe open at both ends (called an open pipe) are given by fn =
nv 2L
( n = 1,2,3,…)
open pipe
Note that the open ends are displacement antinodes which correspond to pressure nodes. And for a pipe closed at one end (stopped pipe) fn =
nv 4L
( n = 1,3,5,…) stopped pipe
The closed end corresponds to a displacement node which is also a pressure antinode.
open pipe
closed (stopped) pipe
(f) Resonance A system will resonate when the frequency of the driving force is equal to one of the normal modes of oscillation of the system. Resonance is usually accompanied by an increased amplitude. (g) Interference of Sound Waves Constructive interference occurs when the path difference traveled by the two sets of identical waves, d , is given by d = n , n =0, 1 ,2 ,…. Destructive interference occurs when the path difference traveled by the two sets of identical waves, d , is given by d = (n + ½ ) , n =0, 1 ,2 ,….
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(h) Beats When two sounds of slightly different frequencies occur at the same time the resulting sound is a beat which has a time-varying amplitude.
If the two frequencies are f1 and f1 with f1 > f1, then the beat frequency is given by fbeat = f1 f2 (i) The Doppler Effect The Doppler effect is the perceived shift in the frequency of sound as a result of the relative motion of the source of sound and the listener.
shorter wavelength ( increased f) vS Longer wavelength ( decreased f)
The Doppler shifted frequency is given by
fL =
v vL fS v vS
where v is the speed of sound, vS is the speed of the source, vL is the speed of the listener and fS is the frequency of the source. The direction from listener to source is taken as the positive direction for all vectors, besides v, the velocity of sound which is always taken as positive.
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6. Electromagnetic Waves and Polarization Electromagnetic waves consist of two components, an electric and a magnetic component traveling at right angles to each other through space. There is a broad spectrum of electromagnetic waves as shown in the diagram below:
All electromagnetic waves travel at the same speed which is c = 3x10 8 m s-1. 7. Polarization When the particles on a wave have displacements only along one axis, we say that the wave is linearly polarized.
It must be noted that only transverse waves can be polarized. Since light is an electromagnetic wave it has components propagating in many different planes. If the light is passed through a Polaroid filter, it can be linearly polarized as shown below:
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We can pass light through two polarizing filters as shown below:
The first polarizer is rotated at angle with respect to the second which is called an analyzer. Only light which lies along the polarizing axis of the polarizer will pass the polarizer. (The components marked E in the diagram.) When E passes through the analyzer, only the Ecos component will emerge. Thus the light has been linearly polarized. When the light passes through the polarizer, it will emerge with only half of its intensity i.e. І2 = ½ І1, where І1 is the intensity of the incident light. The intensity of the light passing through the analyzer can be found by using Malus’ Law: І3 = І2 cos2. It must be noted that polarization of light proves that light is a transverse wave.
8. Interference of Light We have already learnt from sound waves that constructive and destructive interference occur when path difference = d = m , m = 0, 1 , 2 ,…. constructive interference path difference = d = (m + ½ ) , m =0, 1 , 2 ,…. destructive interference
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Young’s Double Slit experiment
Monochromatic light (of one frequency) is incident on a pair of double slits spaced d apart. The interference fringes are observed on a screen placed a distance R away fro the double slits. Bright fringes occur on the screen where constructive interference takes place: d sin = m , m = 0, 1 , 2 ,…. constructive interference
Dark bands occur on the screen where destructive interference takes place: d sin = (m + ½ ) , m = 0, 1 , 2 ,…. destructive interference
The position of the m th bright fringe from the central bright fringe is given by ym = R tan For small angles , we have the approximation sin tan . Then we have ym = R tan = R sin = R (m /d) ym = R
m d
, m = 0, 1 , 2 ,…. for constructive interference.
The intensity of the bright fringes is given by І = І0 cos2(/2), Where І0 is the intensity of the incident light, and is the phase angle given by =
2 d sin
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9. Interference in Thin films Two plates of glass are separated by a thin wedge or film of air. Consider light reflected from the two surfaces surrounding the air wedge as shown in the diagram. The thickness of the air wedge at the point shown is t. If neither or both of the reflected waves from the two surfaces have a half cycle reflection phase shift, then constructive interference occurs for 2t = m ( m = 0, 1, 2,…) If one of the two waves has a half-cycle reflection phase shift, then the above equation is the condition for destructive interference. If neither or both of the reflected waves from the two surfaces have a half cycle reflection phase shift, then destructive interference occurs for 2t = (m + ½ ) ( m = 0, 1, 2,…) This is also the condition for constructive interference when one of the waves has a half cycle phase shift. Note that when the transmitted wave moves more slowly than the incident wave (moving from a medium of lower refractive index to a medium of higher refractive index) then a half-cycle phase change occurs upon reflection.
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10. Diffraction Single slit diffraction
The condition for a dark fringe on the screen is Sin =
mλ a
m = 1 , 2 ,…. for dark fringes from a single slit.
The intensity of the bright bands is given by
sin[ a (sin ) / ] I I0 a (sin ) /
2
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