Angular frequency versus normal frequency?

[EnchantedEggs]

Hi all,

I'm struggling with the concept of angular frequency in the context of sinusoidal waves. We describe sinusoidal waves with equations like [itex] y(x,t) = Asin(kx-\omega t) [/itex], where [itex] \omega [/itex] is the angular frequency, yes? But what does this quantity physically represent? The rate at which points on the curve rotate about their positions?

And how is angular frequency related to regular frequency ([itex]f = \frac{1}{T} [/itex]), physically? As in, in intuitive physical terms?? Am I right in saying the angular frequency is the rate of oscillations of points on the curve whereas regular frequency is the rate at which the peaks of the curve ... pass through a given point in a given time period...?

I think I'm confusing myself even more as I type here :(

 

[SteamKing]

You are confused because you don't know all of the definitions.

The period, T, is the time it takes for one cycle to complete. For a sinusoidal wave, this period is the amount of time it take the wave to start at zero amplitude, for example, run thru all of the amplitudes between +1 and -1 and return to zero amplitude. Since the angular displacement required for this to occur is 2pi radians, then the angular frequency, omega, equals the angular displacement divided by the period, or omega = 2 pi/T

 

[Khashishi]

The "angular" part of angular frequency is essentially a metaphor for a general oscillation, since nothing might be physically rotating. For a mass on a spring bouncing back and forth, there is no angle, but we can express the state of the system (the position and velocity of the mass) in terms of an "angle" which represents the phase of the oscillation. It's not a real angle, but the math is the same.

 

[EnchantedEggs]

You are confused because you don't know all of the definitions.

The period, T, is the time it takes for one cycle to complete. For a sinusoidal wave, this period is the amount of time it take the wave to start at zero amplitude, for example, run thru all of the amplitudes between +1 and -1 and return to zero amplitude. Since the angular displacement required for this to occur is 2pi radians, then the angular frequency, omega, equals the angular displacement divided by the period, or omega = 2 pi/T

Well goodness me, that makes a whole lot more sense. Thanks! It's been so long since I studied physics, I really can't remember the details any more. It's pretty sad. I'll be up to speed soon, I hope!

 

[EnchantedEggs]

The "angular" part of angular frequency is essentially a metaphor for a general oscillation, since nothing might be physically rotating. For a mass on a spring bouncing back and forth, there is no angle, but we can express the state of the system (the position and velocity of the mass) in terms of an "angle" which represents the phase of the oscillation. It's not a real angle, but the math is the same.

Huh. Well, that makes more sense than the jumbled mess of stuff going on in my head. Thanks!

 

[sophiecentaur]

One practical advantage of using angular frequency rather than 1/T is that, when you are doing the sums - calculus, in particular - if you use f, you keep getting a factor of 2π coming in, every time you integrate or differentiate. If you use ω, you don't.

 

[KL7AJ]

Hi all,

I'm struggling with the concept of angular frequency in the context of sinusoidal waves. We describe sinusoidal waves with equations like [itex] y(x,t) = Asin(kx-\omega t) [/itex], where [itex] \omega [/itex] is the angular frequency, yes? But what does this quantity physically represent? The rate at which points on the curve rotate about their positions?

And how is angular frequency related to regular frequency ([itex]f = \frac{1}{T} [/itex]), physically? As in, in intuitive physical terms?? Am I right in saying the angular frequency is the rate of oscillations of points on the curve whereas regular frequency is the rate at which the peaks of the curve ... pass through a given point in a given time period...?

I think I'm confusing myself even more as I type here :(

There's no fundamental difference, as you can always convert from angular frequency to "normal" frequency by a simple multiplication or division. Actually, most of us electronics techs use regular frequency, which is why 2pi sprouts up in all our formulas, while the physics folks like angular frequency for its conciseness. We do occasionally refer to the "unit frequency" which is the resonant frequency of a one farad capacitor and a one henry inductor in series. :) Not very useful, but mathematically elegant. :)
Eric

 

[Minhtran1092]

I apologize in advance for digging up this post but I appreciate this question since most textbooks that I've encountered takes this angular-frequency and time relationship for granted (i.e., state it without much explanation). The following is how I make sense of angular frequency:Any cyclic event has a one-to-one correspondence with a continuous interval of length 2π (with range [0,2π]).

The progress of the completion of an event cycle is represented as a ratio of the angular position to 2π. The angular position (denoted by Φ) can be thought of as the absolute measure from 0 (no completion of an event) towards 2π (a completion of event).
> If the ratio falls in the interval (0,1), i.e., Φ/2π with 0<Φ<2π, then the event is partially complete.
> If the ratio is 0, i.e., Φ=0, then there is no progress. If the ratio is 1, i.e., Φ=2π, then the event is complete. If the ratio is greater than 1, then Φ>2π, and it is interpreted to mean that the event has occurred more than once (the event has precisely occurred Φ/2π, Φ>2π, times).
> We can take the sign of the ratio Φ/2π to indicate clockwise/counterclockwise direction of progress. Conventionally, progress is measured by a positive ratio though one can assign meaning to a negative ratio.

Frequency is a measure of progress towards the completion of an event, and is represented by a ratio of quantities. Typically, it is expressed as a ratio of angular displacement from Φ=0 and time-progress. It assumes that the event will occur in a unit time (conventionally taken to be 1 second). The unit of frequency is the Hertz (Hz, cycles/second -- see below) but formally, it is unitless (it can be expressed as a ratio of angular position (rad/rad) or a ratio of time (seconds/seconds).

Frequency, defined to the be a ratio of angular position: Φ/2π, is a measure of progress towards the completion of a cycle (corresponding to 2π, as discussed above).

The definition of frequency can be extended to events that occur in a duration that is more (or less) than 1 second. If it takes T seconds for an event to occur, then Frequency is taken to be the ratio: 1/T, the ratio of 1 second to T (note that T has units of second), the time it takes for a cycle to be completed.

Notice that the two definitions of Frequency (in terms of angular displacement and in terms of time-duration) allows us to relate the progress of completion of an event in terms of two quantities: Φ and T. Specifically: Φ/2π = 1/T or Φ=2π/T.
Remember that Frequency is a ratio. Let F denote that ratio. We can thus write:

F = Φ/2π = 1/T

Φ = 2π/T

In physics and digital processing, Φ is referred to as the "Angular Frequency", a quantity that relates angular displacement to the time that is needed for an event to complete. That is:

F = 1/T
Φ = 2π/T = 2πF

 

[Khashishi]

Looks good.

 

[sophiecentaur]

Frequency is a measure of progress towards the completion of an event,

How can that statement make sense? Frequency is defined as the reciprocal of the period of oscillation (with the factor 2π included or not, depending).
But I can't understand what the bottom line of your argument / 'explanation' is. The quantities involved in oscillations and their relationships are well established. What you have written is ok for your own private thought processes but how is it supposed actually to help somebody who's new to the topic? It just seems to inject further confusion.

 

[Minhtran1092]

How can that statement make sense? Frequency is defined as the reciprocal of the period of oscillation (with the factor 2π included or not, depending).

But I can't understand what the bottom line of your argument / 'explanation' is. The quantities involved in oscillations and their relationships are well established. What you have written is ok for your own private thought processes but how is it supposed actually to help somebody who's new to the topic? It just seems to inject further confusion.

I offered an interpretation of the definition of frequency. This is not an argument about definition. What do you mean by "well estabished quantities" -- do you mean the deliberate choice to use 2π? Why was 2π chosen? Why do you think frequency was given the definition 1/T? Why the choice of ONE second and not FIVE seconds? The interpretation is equivalent if we define F=5/T as well -- the choice of whatever is considered to be "unit" time is arbitrary, as is the choice to choose 2π. There's no reason why I can't call the duration you perceive to be "one second", "five seconds". All that changes is the relationship between what "ten seconds" means by your definition and by mine.

If interpretation were meant to be private, then there'd be no purpose to mathematical conferences. Math is partially about interpretation (as I'm sure you're aware, it's also about deduction and induction).

 

[sophiecentaur]

We define frequency as the number of cycles of oscillation in the basic unit of time. That unit is arbitrary and could be 1.317 of our seconds on planet Zog. Their values of frequency for the same passing radio wave would be 1.317 times ours.
We use angular frequency in many calculations because the radian is not arbitrary like our degree is. Zoggians use 380 degrees in a circle but their radians would still be the (non arbitrary) same as ours.

 

[David Lewis]

Frequency is... represented by a ratio of quantities. Typically, it is expressed as a ratio of angular displacement from Φ=0 and time-progress. It assumes that the event will occur in a unit time (conventionally taken to be 1 second).

The first two sentences are OK because you're manipulating physical quantities (displacement and time). However, the restriction that the cycle* occurs during a unit time mixes a unit of measure into the formula. Secondly, you are allowed to put any amount of time (or any number of seconds) over which the cycles occur into the denominator of the ratio:

For example, 20 cycles divided by 5 seconds = 4 Hz.

* I assume you meant some number of cycles.

 

[David Lewis]

The unit of frequency is the Hertz (Hz, cycles/second -- see below) but formally, it is unitless

The derived quantity of frequency is time to the negative one power.
Hz can also be written s^-1.

 

[sophiecentaur]

Why was 2π chosen?

2π chose itself. It's the constant that emerges from the basic trigonometric functions involved with circular and oscillatory motion. The relationship between frequency and angular frequency is always the same.
There really is no better way of coming to terms than to go through the Maths from basics. "Mathematical conferences" don't waste time discussing such things. They are given.

 

[godbolerr]

In order to clarify confusion over angular and regular frequency, I read many threads on this forum as well as others. I have documented my understanding in the form of video. Not sure if I can post link to that video here as a link as I am new to PF.

 

[berkeman]

In order to clarify confusion over angular and regular frequency, I read many threads on this forum as well as others. I have documented my understanding in the form of video. Not sure if I can post link to that video here as a link as I am new to PF.

Please don't post your videos yet. We've already had this conversation. Thank you.