Waves
Types of waves
Transverse: displacement of individual particles is perpendicular to direction of wave propagation.
Examples: light, radio waves [all electromagnetic waves]
Longitudinal: displacement of individual particles is parallel to direction of wave propagation
Example: sound waves
water waves: drop a pebble into still water, see series of concentric circles
this is a complicated combination of transverse and longitudinal waves.
Wave frequency, wavelength, and wave speed
= wavelength
wavelength is distance over which a wave repeats itself
unit = m
period T = time for one complete cycle
frequency f = 1/T
wavespeed v = distance/time = /T = f
speed of light c = 3.0X108 m/sec
Problem 1: Sound waves travel in air at 343 m/sec. The lowest frequency we can hear is 20 Hz and the
highest is 20 kHz. What are the wavelengths for these frequencies?
Waves on a string:
Speed of wave on a string depends on the tension force , F, and the mass of the string
For a string of length L and mass M,
 = mass per unit length = M/L
The string is held taut.
If the tension is increased, the wavespeed is increased
If the mass of the string is increased, the wavespeed is decreased.
Wavespeed v = √[F/]
Problem2) a 5.0 meter rope with mass 0.52 kg is pulled taut with tension 46N. find the speed of waves
on the rope.
Problem 3: a 12 m rope is pulled tight with tension 92 N. when one end of the rope is struck, it takes
0.45 sec for the disturbance to propagate to the other end. What is the mass of the rope?
Problem 4: if the tension is doubled in problem 3, how long will it take for the disturbance to travel to
the other end?
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When a wave or pulse traveling down a rope reaches a fixed boundary it reflects and inverts without
changing speed.
When the wave or pulse reaches a movable boundary it reflects and does not invert.
Sound waves:
Sound propagates in air at 343 m/sec.
Sound propagates in liquids and solids at much faster speeds
In water: 1284 m/sec
In steel : 5960 m/sec
Why?
Problem 5) you drop a stone into a wishing well that is 7.35 meters deep
How long does it take before you hear the splash?
Problem 6) Five seconds after a brilliant flash of lightning thunder shakes the house. How far away was
the lightning?
Frequency of a sound wave:
We can detect sound waves from 20 to 20,000 Hz
Below 20 Hz = infrasonic
Above 20 kHz = ultrasonic
V = f
Sound intensity I depends on the energy E that passes through an area A per unit time t.
I = E/At
Power = energy/time = E/t so I = P/A
Intensity I = P/A = power per unit area
Units watts/meter2
Intensity at distance r from a point source of power P:
I = P/[4r2]
Comparing intensity I1 and I2 for a point source of Power P at distances r1 and r2 from the source
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I2 = [r12/r22] I1
Problem 7: a loudspeaker puts out 0.15 watts of sound through a square area of 2.0 m on each side.
What is the intensity of the sound?
Problem 8: A baseball is hit so hard that a fan in the bleachers 140 meters from home plate hears it with
an intensity of 3.80 X10-7 W/m2
What is the intensity heard by the first base umpire 27.4 meters from home plate?
Human perception of sound is over a wide range of frequencies.
We perceive comparative “loudness” of 2 sounds
Io is the lowest detectable intensity. Io = 10-12 W/m2
Io is intensity of lowest sound and I is intensity of second sound. If we perceive I2 to be twice as loud as
Io, then I2 is actually 10 times the intensity of Io.
Loudness is measured in bels or decibels. B = 10 db log [I/Io]
A db is a dimensionless quantity.
If a sound seems twice as loud, it is 10db more intense.
Doppler effect: we have all heard the change in pitch in the sound of a train whistle as the train
approaches and passes us.
Doppler effect occurs whenever a source of sound is moving wrt an observer.
Doppler effect happens when
the source is moving and the observer is stationary
Or the source is stationary and the observer is moving
Or both the source and the observer are moving.
The equation for the generalized Doppler is :
f ’ is the apparent frequency
f ’ = f [ v +/- uo]
use + if observer is approaching source
[V-/+ us]
use – if source is approaching observer
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Remember: f increases as  decreases wavelength decreases as source approaches observer or as
observer approaches source. f’> f when numerator ↑ [ use + sign]
f’> f when denominator ↓ [use – sign]
of BOTH of these
Where
f’ = frequency heard by observer
f = frequency of sound
v = velocity of sound
uo = velocity of observer
us = velocity of sound
u = velocity of observer
us, uo and v are all positive quantities.
In numerator: use + if observer moves toward source
Use – if observer moves away from source
In denominator: use – if source moves towards observer
Use + if source moves away from observer.
Problem 9: a train moving at 9.18 m/sec sounds a 136 Hz horn.
a) What frequency is heard by an observer standing near the tracks as the train
approaches?
b) What frequency is heard by an observer approaching the train at 1.4 m/sec?
c) What frequency is heard by an observer moving in same direction as the train at 1.2
m/sec?
Problem 10: a street musician sounds the A string on his violin, producing a tone of 440 Hz. What
frequency does a bicyclist hear as he approaches at 11.0 m/sec. What frequency does he hear as he
recedes from the musician at 11.0 m/sec?
Problem 11. A train sounds its whistle as it approaches a tunnel in a cliff. The whistle produces a sound
of
650.0 Hz. The train travels at 21.2 m/sec
a) find frequency hear by an observer standing near the tunnel entrance.
b) the sound from the whistle reflects back from the cliff to the train
engineer. What frequency does the engineer hear?
Superposition and interference
Superposition: combination of 2 or more waves to form a resultant
wave.
When waves travelling in the same medium meet they pass through
each other , unchanged.
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If their amplitudes are in the same direction, they positively interfere. If the amplitudes are opposite,
they destructively interfere.
constructive
destructive
http://www.goshen.edu/physix/204/gco/2slit.php
Circular waves: constructive interference: where crest meets crest or trough meets trough
Destructive interference: where crest meets trough
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When waves combine they form an interference pattern
If two waves are synchronized they are in phase.
Conditions for interference for waves emitted in phase from 2 sources separated a distance D apart.
The waves from each source reach an observer. If the difference in distance the waves travel is an integer
multiple, the waves interfere constructively. If the difference in distance the waves travel is an odd
multiple of half a wavelength apart, the waves interfere destructively
Constructive:Path Difference=nλ
Destructive: Path Difference=(n+12)λ
sources in phase
sources in phase
When the waves are emited OPPOSITE PHASE these are reversed
Destructive :Path Difference=nλ
sources opposite phase
constructive: Path Difference=(n+12)λ sources opposite phase
Problem 12: 2 speakers placed distance D = 4.30 meters apart emit sound with frequency 221 Hz. The
speakers are in phase with one another. A person listens from a location 2.80 meters directly in front of
speaker 1. Does the person hear constructive or destructive interference?
Speaker
1
d1
D
Speaker
2
d2
observer
Steps for solution:
1) List given:
2)
3)
4)
5)
D = 4.30 m
d1 = 2.80 m
f = 221 Hz.
Calculate d2.
determine the wavelength  of the sound given the speed of sound = 343 m/sec
find the path difference:  d = d2-d1.
Determine number of wavelengths in  d
Remember: Conditions for in-phase interference:
Constructive:Path
Destructive: Path
Difference=nλ
Difference=(n+1/2) λ
Remember: Conditions for opposite phase interference:
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destructive:Path
constructive: Path
Difference=nλ
Difference=(n+1/2) λ
Problem 13: the speakers shown have opposite phase. They are separated by a distance 5.20 m and emit
frequency 104 Hz. A person stands 3.00 meters in front of the speakers and 1.30 meters to one side of the
center line between them. What type of interference occurs at the person’s location?
Speaker
1
Speaker
2
D
d1
3.00
d2
1.30
observer
Standing waves: wave that oscillates in time but remains fixed in location.
Resulting from constructive interference of a wave with itself.
Standing waves in strings:
Wave reflects from boundary and forms standing wave under certain conditions:
Length of string L must be a multiple of half wavelength of the fundamental frequency.
l = 2L
Fundamental frequency f1 = v/2L wave velocity =√[ T/]
Fundamental = 1st harmonic
f1 = v/1
1 = 2L
f2 = v/ 2 = v/L = 2 f1 2 = L
f3 = v/3 = v/ [2L/3] = 3v/2L = 3f1 etc
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summary: standing waves in a string
f1 = v/1 = v/2L
fn = nf1 = nv/2L
n = 1/n = 2L/n
Problem 14: one of the harmonics on a string 1.30 meters long has a frequency of 15.60 Hz. The next higher
harmonic has a frequency 23.40 Hz. Find a) the fundamental frequency and b) the speed of waves in this string.
Vibrating columns of air:
Standing waves in open and half-closed columns
Open air column: air column length is ½ wavelength of fundamental
f1 = Vsound/2L Vsound = 343 m/sec
Harmonics of open air column
open air column: both ends act as antinodes, same as harmonics on a string, n=1,2,3….
Closed air column: length is ¼ wavelength of fundamental
closed air column: closed end is note, open end is antinode, so only odd number harmonics: n= 1,3,5 …
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Beats are interference patterns. The number of beats is the slight difference in frequency of two interfering
waves. The intensity varies because of the interference
fbeat = f1 –f2 
ELECTROMAGNETIC WAVES
Electricity and magnetism are closely related. Electricity and magnetism can be considered different
aspects of the same thing.
Electric and magnetic fields can work together to produce traveling waves called electromagnetic waves.
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electromagnetic waves are produced by an alternating current in a wire.
Electromagnetic (EM) waves are produced by an alternating current in a wire. As the charges in the
wire oscillate back and forth, the electric field around them oscillates as well, in turn producing an
oscillating magnetic field. This magnetic field is always perpendicular to the electric field, and the EM
wave propagates perpendicular to both the E- and B-fields. This gives us a right-hand-rule relating
the directions of these 3 vectors:
1) Point the fingers of your right hand in the direction of the E-field
2) Curl them toward the B-field.
3) Stick out your thumb - it points in the direction of propagation of EM wave.
the relationship between wavelength , frequency and speed is given by
f=c/
c =speed of light = 3.0X108 m/sec
problem 15: the distance between the Earth and the Sun is 1.50X1011 m. how long does it take for light
to travel this distance?
Speed of light c = 1
√
Doppler effect for EM waves
f ’ = f [1 +/- u ]
[
c ]
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problem 16: An FM radio station broadcasts at a frequency of 88.5 MHz. if you drive your car toward the
station at 32.0 m/sec, what change in frequency do you observe?
first determine what is moving or stationary.
THE ELECTROMAGNETIC SPECTRUM
When white light passes through a prism, it spreads out into a rainbow of colors: red to violet
All travel at same speed, differ only in f and 
c= f/

wavelight of visible light usually expressed in nanometers. 1 nm = 10-9 m
sometimes the wavelength of light is given in angstroms. 1 angstrom = 10 -10 m = 10nm

problem 17: find the frequency of red light with wavelength 700 nm and violet light with wavelength
400 nm.
The electromagnetic spectrum includes the entire range of
radiations, which are measured either as waves or
frequencies.
low frequency: AM radio
low energy
high frequency : gamma rays
high energy
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light waves interfere destructively or constructively.
Constructive : Waves in phase add to give larger amplitude
Destructive: Waves ½ out of phase interfere destructively, cancel each other.
For two sources of EM radiation connected to same transmitter, , radiating at frequency f, and
wavelength ,
measure the path length l1 from antenna 1 and l2 from antenna 2 , to an observer.
If the difference between the path lengths is amultiple of , constructive interference
If the difference between the path lengths is a ½ multiple, destructive interference
Constructive interference: l2-l1 = m m = 0,1,2
Destructive interference: l2-l1 = mm-1/2 l m = 1,2,3
This is similar to interference in water waves.
Problem 18: two friends tune their radios to the same frequency and pick up the same signal
transmitted simultaneously by a set of antennas. The person who is at point Po is equidistant from the
two antennas receives a strong signal. The person at a point Q1 receives a very weak signal. Find the
wavelength of the radio waves if d = 7.50 km and y = 1.88 km. assume that Q1 si the first point of
minimum signal as one moves away from Po in the y direction.
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