Understanding phase shift w/ time of 2 different frequencies

[uniqueguy]

Hi all,

I hope this question is of an acceptable format to ask about here. I'm just having trouble trying to get a conceptual understanding of the following.

I've been told that if I'm given two different frequencies that are completely in phase at time equal to zero, then I can find their phase shift after a certain period of time with the following equation: $2\pi f_1t-2\pi f_2t$

I'm not certain where this comes from or what the intuition behind using it is. Can anyone help me through trying to understand this equation and its use in this example situation?

I appreciate any help with this matter!
-Uniqueguy

 

[Matternot]

The first thing would be to ask yourself what the phase of a wave means. from your equation

2πf1t−2πf2t

for the phase difference you've shown how you might calculate the phase of a oscillation, but what does it mean? Drawing a graph of position against time for an oscillation might come in handy when thinking about this.

 

[anorlunda]

An exactly analogous scenario is two cars traveling down the road. One moves at speed 60 and the other at speed 61. Write an equation to express the distance between the two cars. Now, if the two cars travel on a circular course rather than a straight line, their position on the course is like the phase.

Edit: An even more familiar analogy is the distance between the hour hand and the minute hand on a clock.

 

[uniqueguy]

An exactly analogous scenario is two cars traveling down the road. One moves at speed 60 and the other at speed 61. Write an equation to express the distance between the two cars. Now, if the two cars travel on a circular course rather than a straight line, their position on the course is like the phase.

Edit: An even more familiar analogy is the distance between the hour hand and the minute hand on a clock.

Looking at your analogous scenario has given me a much better idea of what I'm looking at.

The frequency is the number of wavelengths of a wave in one second. By multiplying by a certain period of time, we get the total number of wavelengths that the wave propagated in that time. And so, by multiplying by $2\pi$, we're saying that each wavelength is $2\pi$ radians, and getting the total number of radians that the wave travels. By comparing these values, we can get the phase difference.

I've still got just a slight conceptual block when considering the multiplication of $2\pi$. In my above understanding, we're equating $2\p$ radians to one wavelength, hence the multiplication. How do we know that one wavelength is equivalent to $2\pi$ radians?

Thanks for your help!
-Uniqueuy

 

[BvU]

https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html

 

[Matternot]

How do we know that one wavelength is equivalent to $2\pi$ radians?

If you draw a graph of y=sin(x), where x is the phase in radians, think about when this function repeats itself.

 

[anorlunda]

Looking at your analogous scenario has given me a much better idea of what I'm looking at.

The frequency is the number of wavelengths of a wave in one second. By multiplying by a certain period of time, we get the total number of wavelengths that the wave propagated in that time. And so, by multiplying by $2\pi$, we're saying that each wavelength is $2\pi$ radians, and getting the total number of radians that the wave travels. By comparing these values, we can get the phase difference.

I've still got just a slight conceptual block when considering the multiplication of $2\pi$. In my above understanding, we're equating $2\p$ radians to one wavelength, hence the multiplication. How do we know that one wavelength is equivalent to $2\pi$ radians?

Thanks for your help!
-Uniqueuy

You can use any angle units you want.

Cycles per second * $2\pi$ = radians per second.
Cycles per second * 360 = degrees per second.