How Do You Calculate Resultant Displacement for Two Interfering Waves?

[roam]

Homework Statement

Two displacement waves with the same period T = 3.0 s are described mathematically by:

y₁ = (10.0 m) cos[2πt/T + π]
y₂ = (9.0 m) cos[2πt/T + 2π]

What is the resultant displacement at time t = 6.0 s?

The Attempt at a Solution

I tried adding up

y₁ + y₂ = 19 cos [2πt/T + 3π]

But the answer I get is wrong. I also tried the following formula from my texybook which is supposed to give the resultant of two traveling sinusoidal waves:

$$y=2Acos \left( \frac{\phi}{2} \right) sin \left( kx-\omega t + \frac{\phi}{2} \right)$$

I don't know what the phase constant (φ) is, so I assume it is zero since the waves are in phase, then cos(φ/2)=cos(0)=1.

When I plug in the numbers the formula then becomes

y=6sin(3π+2π6/3)

Again, this produces the wrong answer, the correct answer should be -1.0 m. What's wrong with my calculations?

 

[ehild]

Homework Statement

Two displacement waves with the same period T = 3.0 s are described mathematically by:

y₁ = (10.0 m) cos[2πt/T + π]
y₂ = (9.0 m) cos[2πt/T + 2π]

What is the resultant displacement at time t = 6.0 s?

The Attempt at a Solution

I tried adding up

y₁ + y₂ = 19 cos [2πt/T + 3π]

:

You can not do that! Try to think instead of plugging into equations you do not understand.

Resolve the cosine terms. What is the relation between cos(x) and cos(x+2π)? between cos(x) and cos(x+π)?


ehild

 

[roam]

You can not do that! Try to think instead of plugging into equations you do not understand.

Resolve the cosine terms. What is the relation between cos(x) and cos(x+2π)? between cos(x) and cos(x+π)?


ehild

Does this mean that they are out of phase? I think if one wave has phase constant φ=(2N)π and the other wave has φ=(2N+1)π, where N is any integer, then the two waves are not in phase, and therefore destructive interference occurs.

What do you mean by "resolving" the cosine terms?

The "superposition principal" states that the resultant value of the wave functions of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves. This is what I was trying to do, in my problem do I need to somehow subtract thw two given wave functions?

 

[ehild]

What do you mean by "resolving" the cosine terms?

Well, my English is poor, especially in maths expressions. I meant to express both cos[2πt/T + π] and cos[2πt/T + 2π] with cos (2πt/T). Think of the definition of the cosine function.

And you are right, y1 and y2 are out of phase.

If you still do not get it, just plug in 6.0 s for t, calculate both y1 and y2 and add them.

ehild

 

[rdl]

As a scientist, it is important to approach problems with a systematic and logical approach. In this case, it seems that the incorrect answer may be due to a misunderstanding of the concept of phase and how it relates to the displacement waves. The phase constant, φ, represents the initial position of the wave, and in this case, it is not zero since the waves have different initial positions. Therefore, the correct formula to use would be y = 2Acos(φ/2)sin(kx-ωt+φ/2). By plugging in the given values, the correct answer of -1.0 m can be obtained. It is also important to carefully check the units and make sure they are consistent throughout the calculations. Additionally, it may be helpful to draw a diagram or visualize the waves in order to better understand the problem and approach it more accurately.