Determining phase shift of two frequencies

[teetar]

Homework Statement

I'm given two frequencies: 4,000 Hz, and 5,000 Hz.
They are completely in-shift at time ##t=0##.
I am to find the time it takes them to get completely out of phase.

Homework Equations

The Attempt at a Solution

I've not gotten waves very well thus far in physics. My teacher said being completely out of phase occurs at a phase shift of one half wavelength, or, at π rads. Where do I start?

 

[Charles Link]

You need to write each wave as ## E(t)=A \, cos(2 \pi f t) ## where ## f ## is the frequency, because ## f=1/T ## where ## T ## is the period (time of one cycle). The number * inside the cos(*) will define the phase of the wave at time t. I've probably already given you more information than we are supposed to, but hopefully you can work it from here.

 

[teetar]

You need to write each wave as ## E(t)=A \, cos(2 \pi f t) ## where ## f ## is the frequency, because ## f=1/T ## where ## T ## is the period (time of one cycle). The number * inside the cos(*) will define the phase of the wave at time t. I've probably already given you more information than we are supposed to, but hopefully you can work it from here.

So, is it as simple as saying that the waves are completely out of shift when 10,000πt - 8,000πt = π, meaning that t = 1/2,000? I must be oversimplifying this without fully understanding the fundamental ideas behind it. Am I evaluating the change in the cosine equal to π rads (cos(10,000πt) - cos(8,000πt) = π)? Or am I still missing the obvious answer?

 

[Charles Link]

So, is it as simple as saying that the waves are completely out of shift when 10,000πt - 8,000πt = π, meaning that t = 1/2,000? I must be oversimplifying this without fully understanding the fundamental ideas behind it. Am I evaluating the change in the cosine equal to π rads (cos(10,000πt) - cos(8,000πt) = π)? Or am I still missing the obvious answer?

You did it correctly and the answer is in seconds (t=.0005 seconds).

 

[teetar]

You did it correctly and the answer is in seconds (t=.0005 seconds).

Thanks for the help! I definitely need to spend more time studying waves.

If anyone passes by this thread and knows of any good resources, could you reply to me with them, or PM me them? I looked all over Google trying to answer this question and didn't find many relevant sources.

 

[rude man]

Phase is the total argument of sin or cos which evolves over time.
So if you have x1 = sin(2πf₁t) and x2 = sin(2πf₂t) then the phases are 2πf₁t and 2πf₂t and the equation to solve is 2πf₁t = 2πf₂t + π with f₁ > f₂.
There are an infinite number of answers but you want to find the shortest time t satisfying the equation.