Optic/Wave Questions: Answers & Solutions

[zumbo1]

In all but one of these problems all I need is the answers checked. I am uncertain how to do number 5 part b.

1.A wave is traveling 50 m/s to the left(this is a graph) and its wavelength is 10 meters. What is the frequency of the traveling wave?

Answer:v=λ*f; f=5 hertz. Frequency can not be negative, right?

2.What is the speed and direction of the following waves.
ψ(y,t)=A(y-t)^2.Answer: The wave is traveling 1 to the right.

ψ(x,t)=A(Bx+Ct+D)^2. Answer: The wave is traveling at C to the left.

ψ(z,t)=Ae^(Bz^2+BC^2t^2-2BCzt) Answer: Ae^(-B*(z-Ct)^2) The wave is traveling C to the right.

3.A wave of the form y(x,t)=100*sin(2πx-4πt) and you have two detectors to measure the disturbance at points x1=2 and x2=10. What will be the magnitude of the disturbance at x2 the instant t1, when y(x1,t1)=100.
Answer:Knowing that sin(π/2)=1 therefore π/2=4π-4πt; (1-t) = 1/8; t=7/8. Then I plugged t1 = 7/8 into the orginal equation with x2 so y(x2, t1) =~ 70.71.

4.Determine the imaginary part of
Note: The general equation I used for the following part was e^(iθ)=cosθ+i*sinθ and e^(-iθ)=cosθ-i * sinθ
a)z=5*e^(iky)*e^(-iωt)*e^(iε)
Answer: This is what I get for the Imaginary part after some simplifying 5(i*sin(ε)cost(ωt)cos(kx) + i*cos(ε)cos(ωt)sin(kx) + i*cos(ε)sin(ωt)cos(kx) - i*sin(εs)sin(ωt)sin(kx))

b)z=((Ae^(iωt))/(Be^(ikx)))*e^(iε) Answer:I factored out the coeffieceint A/B so I was left with A/B((e^(iωt)* e^(iε)) /e^(ikx) ). Then, I assumed that the entire thing would end up being imaginary, would I be correct in assuming that.

c)z=(Ae^(iωt) + Ae^(-iωt))/2. Answer: When I expapanded this the imaginary parts canceled each other out. So, there would be no imaginary part in this one.

5.Find the magnitude if the complex quantities:
a)ψ(x,t)=e^(ikx)*e^(-iωt)*e^(iε)
Answer: I thought this one was a little to simple, wouldn't it just be equal to 1.
b)ψ(y,t)=2*e^(iky)*e^(iωt) + 4*e^(iky)*e^(-iωt)
Answer: This is the one I am stuck on. This is what I did 2e^(iky)*(e^(iωt)+2e^(-iωt)); 2e^(iky)*(cos(ωt)+i*sin(ωt)+2cos(ωt)-2i*sin(ωt));
2e^(iky)*(3cos(ωt)-isin(ωt))
Then I just didnt know what to do, can you walk me threw how to do this one?

6.There is a wave y=sin(2π(4t-5x+2/3))Find the ...
a)the amplitude, Answer: 1

b)the wavelength, Answer: knowing that this wave is in the form ψ(x,t) = Asin(kx-ωt+ε) then k=10π and ω=8π, then using the formula λ=2π/k=1/5

c)the frequency ω=2πf; f=4hertz

d)the intial phase angle Answer: I guess this would just be the number in the sin if all the variables were zero so it would be 4π/3?

e)the displacement at time t=0 and x=0. Answer: This would just be sin(4π/3)=~.866. Displacement is always postive right?


Thank you for the help.

 

[Jhero]

For number 5 part b, you are correct in your approach. To find the magnitude, we can use the formula |z| = √(a^2 + b^2) where a and b are the real and imaginary parts, respectively. So for this problem, we have a=6cos(ωt) and b=-2sin(ωt). Plugging these values into the formula, we get |z| = √(6^2 + (-2)^2) = √40 = 2√10. Therefore, the magnitude of the complex quantity is 2√10.