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16.5 Interference of waves  (Page 5/12)

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y 1 ( x , t ) = A sin ( k x − ω t + ϕ ) , y 2 ( x , t ) = A sin ( k x − ω t ) .

When these two waves exist in the same medium, the resultant wave resulting from the superposition of the two individual waves is the sum of the two individual waves:

y R ( x , t ) = y 1 ( x , t ) + y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) + A sin ( k x − ω t ) .

The resultant wave can be better understood by using the trigonometric identity:

sin u + sin v = 2 sin ( u + v 2 ) cos ( u − v 2 ) ,

where u = k x − ω t + ϕ and v = k x − ω t . The resulting wave becomes

y R ( x , t ) = y 1 ( x , t ) + y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) + A sin ( k x − ω t ) = 2 A sin ( ( k x − ω t + ϕ ) + ( k x − ω t ) 2 ) cos ( ( k x − ω t + ϕ ) − ( k x − ω t ) 2 ) = 2 A sin ( k x − ω t + ϕ 2 ) cos ( ϕ 2 ) .

This equation is usually written as

y R ( x , t ) = [ 2 A cos ( ϕ 2 ) ] sin ( k x − ω t + ϕ 2 ) .

The resultant wave has the same wave number and angular frequency, an amplitude of A R = [ 2 A cos ( ϕ 2 ) ] , and a phase shift equal to half the original phase shift. Examples of waves that differ only in a phase shift are shown in [link] . The red and blue waves each have the same amplitude, wave number, and angular frequency, and differ only in a phase shift. They therefore have the same period, wavelength, and frequency. The green wave is the result of the superposition of the two waves. When the two waves have a phase difference of zero, the waves are in phase, and the resultant wave has the same wave number and angular frequency, and an amplitude equal to twice the individual amplitudes (part (a)). This is constructive interference. If the phase difference is 180 ° , the waves interfere in destructive interference (part (c)). The resultant wave has an amplitude of zero. Any other phase difference results in a wave with the same wave number and angular frequency as the two incident waves but with a phase shift of ϕ / 2 and an amplitude equal to 2 A cos ( ϕ / 2 ) . Examples are shown in parts (b) and (d).

radians. The crests of the blue wave coincide with the troughs of the red wave and vice versa. The green wave is absent. Figure d is labeled delta phi equal to 3 pi by 2 radians. Here, the red and blue waves each have an amplitude of 10 m and the green wave has an amplitude of 15 m. It has the same wavelength as the other two waves. The crests of the green wave are formed where the crests of the red and blue waves intersect each other.
Superposition of two waves with identical amplitudes, wavelengths, and frequency, but that differ in a phase shift. The red wave is defined by the wave function y 1 ( x , t ) = A sin ( k x − ω t ) and the blue wave is defined by the wave function y 2 ( x , t ) = A sin ( k x − ω t + ϕ ) . The black line shows the result of adding the two waves. The phase difference between the two waves are (a) 0.00 rad , (b) π / 2 rad, (c) π rad, and (d) 3 π / 2 rad .

Summary

  • Superposition is the combination of two waves at the same location.
  • Constructive interference occurs from the superposition of two identical waves that are in phase.
  • Destructive interference occurs from the superposition of two identical waves that are 180 ° ( π radians ) out of phase.
  • The wave that results from the superposition of two sine waves that differ only by a phase shift is a wave with an amplitude that depends on the value of the phase difference.

Conceptual questions

An incident sinusoidal wave is sent along a string that is fixed to the wall with a wave speed of v . The wave reflects off the end of the string. Describe the reflected wave.

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A string of a length of 2.00 m with a linear mass density of μ = 0.006 kg/m is attached to the end of a 2.00-m-long string with a linear mass density of μ = 0.012 kg/m . The free end of the higher-density string is fixed to the wall, and a student holds the free end of the low-density string, keeping the tension constant in both strings. The student sends a pulse down the string. Describe what happens at the interface between the two strings.

At the interface, the incident pulse produces a reflected pulse and a transmitted pulse. The reflected pulse would be out of phase with respect to the incident pulse, and would move at the same propagation speed as the incident pulse, but would move in the opposite direction. The transmitted pulse would travel in the same direction as the incident pulse, but at half the speed. The transmitted pulse would be in phase with the incident pulse. Both the reflected pulse and the transmitted pulse would have amplitudes less than the amplitude of the incident pulse.

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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