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Determine the combined intensity of two waves: perfect constructive interference

If two identical waves, each having an intensity of 1 . 00 W/m 2 size 12{1 "." "00"`"W/m" rSup { size 8{2} } } {} , interfere perfectly constructively, what is the intensity of the resulting wave?

Strategy

We know from Superposition and Interference that when two identical waves, which have equal amplitudes X size 12{X} {} , interfere perfectly constructively, the resulting wave has an amplitude of 2 X size 12{2X} {} . Because a wave’s intensity is proportional to amplitude squared, the intensity of the resulting wave is four times as great as in the individual waves.

Solution

  1. Recall that intensity is proportional to amplitude squared.
  2. Calculate the new amplitude:
    I X 2 = 2 X 2 = 4 X 2 . size 12{I rSup { size 8{'} } prop left (X rSup { size 8{'} } right ) rSup { size 8{2} } = left (2X right ) rSup { size 8{2} } =4X rSup { size 8{2} } } {}
  3. Recall that the intensity of the old amplitude was:
    I X 2 . size 12{I rSup { size 8{'} } prop X rSup { size 8{2} } } {}
  4. Take the ratio of new intensity to the old intensity. This gives:
    I I = 4 . size 12{ { {I} over {I rSup { size 8{'} } } } =4} {}
  5. Calculate to find I size 12{I'} {} :
    I = 4 I = 4 . 00 W/m 2 . size 12{I'=4I=4 "." "00"`"W/m" rSup { size 8{2} } } {}

Discussion

The intensity goes up by a factor of 4 when the amplitude doubles. This answer is a little disquieting. The two individual waves each have intensities of 1 . 00 W/m 2 size 12{1 "." "00"`"W/m" rSup { size 8{2} } } {} , yet their sum has an intensity of 4 . 00 W/m 2 size 12{4 "." "00"`"W/m" rSup { size 8{2} } } {} , which may appear to violate conservation of energy. This violation, of course, cannot happen. What does happen is intriguing. The area over which the intensity is 4 . 00 W/m 2 size 12{4 "." "00"`"W/m" rSup { size 8{2} } } {} is much less than the area covered by the two waves before they interfered. There are other areas where the intensity is zero. The addition of waves is not as simple as our first look in Superposition and Interference suggested. We actually get a pattern of both constructive interference and destructive interference whenever two waves are added. For example, if we have two stereo speakers putting out 1 . 00 W/m 2 size 12{1 "." "00"`"W/m" rSup { size 8{2} } } {} each, there will be places in the room where the intensity is 4 . 00 W/m 2 size 12{4 "." "00"`"W/m" rSup { size 8{2} } } {} , other places where the intensity is zero, and others in between. [link] shows what this interference might look like. We will pursue interference patterns elsewhere in this text.

Two speakers are shown at the top of the figure at left and right side. Rarefactions are shown as dotted curves and compression as dark curves. The interference of the sound waves from these two speakers is shown. There are some red spots, showing constructive interference, are shown on the interfering waves.
These stereo speakers produce both constructive interference and destructive interference in the room, a property common to the superposition of all types of waves. The shading is proportional to intensity.

Which measurement of a wave is most important when determining the wave's intensity?

Amplitude, because a wave’s energy is directly proportional to its amplitude squared.

Section summary

Intensity is defined to be the power per unit area:

I = P A size 12{I= { {P} over {A} } } {} and has units of W/m 2 size 12{"W/m" rSup { size 8{2} } } {} .

Conceptual questions

Two identical waves undergo pure constructive interference. Is the resultant intensity twice that of the individual waves? Explain your answer.

Circular water waves decrease in amplitude as they move away from where a rock is dropped. Explain why.

Problems&Exercises

Medical Application

Ultrasound of intensity 1 . 50 × 10 2 W/m 2 size 12{1 "." "50" times "10" rSup { size 8{2} } `"W/m" rSup { size 8{2} } } {} is produced by the rectangular head of a medical imaging device measuring 3.00 by 5.00 cm. What is its power output?

0.225 W

The low-frequency speaker of a stereo set has a surface area of 0 . 05 m 2 size 12{0 "." "05"`m rSup { size 8{2} } } {} and produces 1W of acoustical power. What is the intensity at the speaker? If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity 0 . 1 W/m 2 size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} ?

To increase intensity of a wave by a factor of 50, by what factor should the amplitude be increased?

7.07

Engineering Application

A device called an insolation meter is used to measure the intensity of sunlight has an area of 100 cm 2 and registers 6.50 W. What is the intensity in W/m 2 size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} ?

Astronomy Application

Energy from the Sun arrives at the top of the Earth’s atmosphere with an intensity of 1.30 kW/m 2 . size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} How long does it take for 1.8 × 10 9 J size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} to arrive on an area of 1 . 00 m 2 size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} ?

16.0 d

Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?

2.50 kW

Engineering Application

(a) A photovoltaic array of (solar cells) is 10.0% efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is 700 W/m 2 , size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} what area should your array have to gather energy at the rate of 100 W? (b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 ¢ per kilowatt-hour.

A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally 2.00 × 10 –5 W/m 2 , size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} but is turned up until the amplitude increases by 30.0%, what is the new intensity?

3.38 × 10 –5 W/m 2

Medical Application

(a) What is the intensity in W/m 2 size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} of a laser beam used to burn away cancerous tissue that, when 90.0% absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s? (b) Discuss how this intensity compares to the average intensity of sunlight (about 700 W/m 2 size 12{0 "." 1`"W/m" rSup { size 8{2} } } {} ) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, College physics -- hlca 1104. OpenStax CNX. May 18, 2013 Download for free at http://legacy.cnx.org/content/col11525/1.1
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