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A long, tight spring is held by two students, one student holding each end. Each student gives the end a flip sending one wavelength of a sinusoidal wave down the spring in opposite directions. When the waves meet in the middle, what does the wave look like?

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Many of the topics discussed in this chapter are useful beyond the topics of mechanical waves. It is hard to conceive of a mechanical wave with sharp corners, but you could encounter such a wave form in your digital electronics class, as shown below. This could be a signal from a device known as an analog to digital converter, in which a continuous voltage signal is converted into a discrete signal or a digital recording of sound. What is the result of the superposition of the two signals?

Figure a shows the graph of a red wave with sharp corners. The y value is 0 at x=0. At x=3, the y value rises to 2 and stays constant till x=5. Here, it rises to 3 and stays constant till x=8. Here, it dips down to -2 and stays constant till x=9. Here it rises to 0 and stays constant. Figure b shows the graph of a blue wave with sharp corners. The y value is 0 at x=0. At x=3, the y value rises to 2 and stays constant till x=7. Here it dips to -1 and stays constant till x=9. Here it rises to 0 and stays constant.

Figure shows the graph of a wave. Its y value is 0 at x=0. At x=3, the y value rises to 4 and stays constant till x=5. Here, it rises to 5 and stays constant till x=7. Here, it dips to 2 and stays constant till x=8. Here, it dips to -3 and stays constant till x=9. Here, it rises to 0 and stays constant.

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A string of a constant linear mass density is held taut by two students, each holding one end. The tension in the string is constant. The students each send waves down the string by wiggling the string. (a) Is it possible for the waves to have different wave speeds? (b) Is it possible for the waves to have different frequencies? (c) Is it possible for the waves to have different wavelengths?

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Problems

Consider two sinusoidal waves traveling along a string, modeled as y 1 ( x , t ) = 0.3 m sin ( 4 m −1 x + 3 s −1 t ) and y 2 ( x , t ) = 0.6 m sin ( 8 m −1 x 6 s −1 t ) . What is the height of the resultant wave formed by the interference of the two waves at the position x = 0.5 m at time t = 0.2 s?

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Consider two sinusoidal sine waves traveling along a string, modeled as y 1 ( x , t ) = 0.3 m sin ( 4 m −1 x + 3 s −1 t + π 3 ) and y 2 ( x , t ) = 0.6 m sin ( 8 m −1 x 6 s −1 t ) . What is the height of the resultant wave formed by the interference of the two waves at the position x = 1.0 m at time t = 3.0 s?

y ( x , t ) = 0.76 m

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Consider two sinusoidal sine waves traveling along a string, modeled as y 1 ( x , t ) = 0.3 m sin ( 4 m −1 x 3 s −1 t ) and y 2 ( x , t ) = 0.3 m sin ( 4 m −1 x + 3 s −1 t ) . What is the wave function of the resulting wave? [ Hint: Use the trig identity sin ( u ± v ) = sin u cos v ± cos u sin v

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Two sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle ϕ . What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [ Hint: Use the trig identity sin u + sin v = 2 sin ( u + v 2 ) cos ( u v 2 )

A R = 2 A cos ( ϕ 2 ) , ϕ = 1.17 rad

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Two sinusoidal waves are moving through a medium in the positive x -direction, both having amplitudes of 6.00 cm, a wavelength of 4.3 m, and a period of 6.00 s, but one has a phase shift of an angle ϕ = 0.50 rad . What is the height of the resultant wave at a time t = 3.15 s and a position x = 0.45 m ?

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Two sinusoidal waves are moving through a medium in the positive x -direction, both having amplitudes of 7.00 cm, a wave number of k = 3.00 m −1 , an angular frequency of ω = 2.50 s −1 , and a period of 6.00 s, but one has a phase shift of an angle ϕ = π 12 rad . What is the height of the resultant wave at a time t = 2.00 s and a position x = 0.53 m?

y R = 1.90 cm

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Consider two waves y 1 ( x , t ) and y 2 ( x , t ) that are identical except for a phase shift propagating in the same medium. (a)What is the phase shift, in radians, if the amplitude of the resulting wave is 1.75 times the amplitude of the individual waves? (b) What is the phase shift in degrees? (c) What is the phase shift as a percentage of the individual wavelength?

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Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is y R ( x , t ) = 0.70 m sin ( 3.00 m −1 x 6.28 s −1 t + π / 16 rad ) . What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

ω = 6.28 s −1 , k = 3.00 m −1 , ϕ = π 8 rad, A R = 2 A cos ( ϕ 2 ) , A = 0.37 m

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Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is y R ( x , t ) = 0.35 cm sin ( 6.28 m −1 x 1.57 s −1 t + π 4 ) . What are the period, wavelength, amplitude, and phase shift of the individual waves?

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Consider two wave functions, y 1 ( x , t ) = 4.00 m sin ( π m −1 x π s −1 t ) and y 2 ( x , t ) = 4.00 m sin ( π m −1 x π s −1 t + π 3 ) . (a) Using a spreadsheet, plot the two wave functions and the wave that results from the superposition of the two wave functions as a function of position ( 0.00 x 6.00 m ) for the time t = 0.00 s . (b) What are the wavelength and amplitude of the two original waves? (c) What are the wavelength and amplitude of the resulting wave?

a.
Figure shows a blue wave labeled y1, a red wave labeled y2 and a black wave labeled yR on the same graph. The red and the blue waves have the same wavelength and amplitude, but are out of phase. The black wave has the same wavelength as the other two, but is greater in amplitude. ;
b. λ = 2.0 m , A = 4 m ; c. λ R = 2.0 m , A R = 6.93 m

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Consider two wave functions, y 2 ( x , t ) = 2.00 m sin ( π 2 m −1 x π 3 s −1 t ) and y 2 ( x , t ) = 2.00 m sin ( π 2 m −1 x π 3 s −1 t + π 6 ) . (a) Verify that y R = 2 A cos ( ϕ 2 ) sin ( k x ω t + ϕ 2 ) is the solution for the wave that results from a superposition of the two waves. Make a column for x , y 1 , y 2 , y 1 + y 2 , and y R = 2 A cos ( ϕ 2 ) sin ( k x ω t + ϕ 2 ) . Plot four waves as a function of position where the range of x is from 0 to 12 m.

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Consider two wave functions that differ only by a phase shift, y 1 ( x , t ) = A cos ( k x ω t ) and y 2 ( x , t ) = A cos ( k x ω t + ϕ ) . Use the trigonometric identities cos u + cos v = 2 cos ( u v 2 ) cos ( u + v 2 ) and cos ( θ ) = cos ( θ ) to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?

y R ( x , t ) = 2 A cos ( ϕ 2 ) cos ( k x ω t + ϕ 2 ) ; The result is not surprising because cos ( θ ) = sin ( θ + π 2 ) .

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Questions & Answers

A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
what is a scalar quantity
Peter Reply
scalar: are quantity have numerical value
muslim
is that a better way in defining scalar quantity
Peter
thanks
muslim
quantity that has magnitude but no direction
Peter
upward force and downward force lift
adegboye Reply
upward force and downward force on lift
adegboye
hi
Etini
hi
elo
hy
Xander
Hello
Jux_dob
hi
Peter
Helo
Tobi
Yes what about it?
Daniel
what's the answer? I can't get it
Rachel Reply
what is the question again?
Sallieu
What's this conversation?
Zareen
what is catenation? and give examples
sununu
How many kilometres in 1 mile
Nessy
1.609km in 1mile
Faqir
what's the si unit of impulse
Iguh Reply
The Newton second (N•s)
Ethan
what is the s. I unit of current
Roland Reply
Amphere(A)
imam
thanks man
Roland
u r welcome
imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
Pallavi Reply
Practice Key Terms 6

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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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