The reciprocal relationship between frequency and period

[Frank Castle]

I was asked by a friend to explain why the frequency, $f$ and period, $T$ of a wave. The initial explanation I gave to them was as follows:

Heuristically, the period of a wave is defined as $T=\frac{\text{number of units time}}{\text{cycle}}$, and its frequency as $f=\frac{\text{number of cycles}}{\text{unit time}}$ hence we see that the two quantities are reciprocal to one another, such that $f=\frac{1}{T}$.

However, they weren't happy with this explanation, it didn't fully explain it for them; I think they're finding it hard to reason why the equation should be $f=\frac{1}{T}$. I think the "change of plurals", i.e. in the sense that one is number of units time per cycle, and the other is number of cycles per unit time is also "throwing" them.
As such, I have come up with an alternative explanation, but I'd like to check first that I'm not saying anything incorrect or misleading:

Suppose that a wave repeats itself every $n$ units of time such that $T=n\frac{\text{ units time}}{\text{cycle}}$. Then in one unit of time the wave will have completed $\frac{1}{n}\text{ cycles}$, such that it completes full cycles at a rate of $\frac{1}{n}\frac{\text{cycles}}{\text{unit time}}=\frac{1}{T}$. This rate is referred to as frequency , $f$, that is "how frequently the wave completes a cycle". As such, it follows that $f=\frac{1}{T}$.

I would really appreciate any feedback on this description, and if it can be improved upon at all?!

 

[Dale]

However, they weren't happy with this explanation

I wouldn't be happy with it either, it seems to get things backwards. It is a definition. Frequency is defined such that $f=1/T$. The fact that the units are reciprocal is a consequence of the definition, not the other way around.

 

[pixel]

Not sure what kind of understanding your friend is looking for, but maybe some simple examples would make the relationship clear to him:

If the period is 1/2 sec., then in one second there will be two cycles, hence f = 2 cycles/sec.

If the period is 2 sec., then in one second there will be a half a cycle, hence f = 1/2 cycles/sec.

 

[pixel]

Frequency is defined such that $f=1/T$.

Really? Could one say period is defined as the reciprocal of frequency? It seems to me that frequency and period are independently defined quantities.

 

[Frank Castle]

I wouldn't be happy with it either, it seems to get things backwards. It is a definition. Frequency is defined such that $f=1/T$. The fact that the units are reciprocal is a consequence of the definition, not the other way around.

Fair enough. I appreciate the feedback. I was taking this approach since they were struggling to see why the relation is defined as $f=\frac{1}{T}$; They're seeking a justification for why it is defined this way.
A lot of introductory physics texts seem to take this approach.

I offered one reason for why it is defined this way... Since $T$ is the amount of time taken for one full cycle, the fraction of the cycle completed per unit time would be $\frac{\frac{\text{unit time}}{T}}{\text{unit time}}=\frac{1}{T}$.

 

[andrewkirk]

For some reason I find it more intuitive to start with the frequency and derive the period than the other way around.

I would say something like: Imagine a wave that completes $n$ cycles per second. We can that number $n$ the frequency of the wave. We use another word 'period' for the time it takes to complete a single cycle. Since $n$ cycles are completed in a second, that means the type taken to complete one cycle must be $1/n$. Note that that number is the reciprocal of the frequency.

 

[Frank Castle]

Really? Could one say period is defined as the reciprocal of frequency? It seems to me that frequency and period are independently defined quantities.

To be honest, that's what I thought as well. I think this is why they are struggling somewhat to understand why the relationship between them is $f=\frac{1}{T}$. I think they can see somewhat that the two quantities are reciprocal in nature, but not why they are related exactly in this way.

 

[PeroK]

I was asked by a friend to explain why the frequency, $f$ and period, $T$ of a wave. The initial explanation I gave to them was as follows:

Heuristically, the period of a wave is defined as $T=\frac{\text{number of units time}}{\text{cycle}}$, and its frequency as $f=\frac{\text{number of cycles}}{\text{unit time}}$ hence we see that the two quantities are reciprocal to one another, such that $f=\frac{1}{T}$.

However, they weren't happy with this explanation, it didn't fully explain it for them; I think they're finding it hard to reason why the equation should be $f=\frac{1}{T}$. I think the "change of plurals", i.e. in the sense that one is number of units time per cycle, and the other is number of cycles per unit time is also "throwing" them.

I would really appreciate any feedback on this description, and if it can be improved upon at all?!

I'm beginning to suspect that this "friend" that you're often trying to explain physics to is you yourself!

Anyway, if there are $n$ buses per hour, then the time between buses is $60/n$ minutes. And if there is a bus every $t$ minutes then there are $60/t$ buses per hour.

Is that not all there is to it?

 

[Frank Castle]

For some reason I find it more intuitive to start with the frequency and derive the period than the other way around.

I would say something like: Imagine a wave that completes $n$ cycles per second. We can that number $n$ the frequency of the wave. We use another word 'period' for the time it takes to complete a single cycle. Since $n$ cycles are completed in a second, that means the type taken to complete one cycle must be $1/n$. Note that that number is the reciprocal of the frequency.

Would my reasoning be OK though? I mean, it seems to make sense to me that if a periodic motion takes $n$ units of time to complete one cycle then, in one unit of time it with complete $\frac{1}{n}$ cycles. Hence, the frequency at which it completes full cycles will be $\frac{1}{n}$ cycles per unit time.

I'm beginning to suspect that this "friend" that you're often trying to explain physics to is you yourself!

In this case it's genuinely not (although I have been guilty of using that cover in the past out of embarrassment ). Sometimes I've asked by a peer to explain something and have realized that I perhaps don't know the reason why something is defined as such well enough to give a reasonable explanation.

Anyway, if there are nnn buses per hour, then the time between buses is 60/n60/n60/n minutes. And if there is a bus every ttt minutes then there are 60/t60/t60/t buses per hour.

Is that not all there is to it?

This is my understanding, yes.

In essence one is asking "what is the fraction of the period can one fit into a single unit of time, per unit time", right?! In the sense that, if one cycle takes $t$ units of time, then the fraction of cycles completed per unit time is $\frac{1}{t}$.

I just wanted to clarify before I pass on the information.

 

[andrewkirk]

Would my reasoning be OK though?

Absolutely. The two approaches are entirely equivalent and just as logically valid as one another. It's just that personally I find it easier to grasp when I start with the number of cycles per second rather than the number of seconds per cycle. That is also the approach most non-physicists are familiar with, since record turntable speeds, engine speeds and sound frequencies are all quoted as cycles per second rather than seconds per cycle. It tends to be only very slow things, often astronomical things, like rotations of the Earth about its axis or revolutions around the Sun (and an odd one - the 17-year life-cycle of certain cicadas), that are quoted as a period rather than a frequency.

 

[Dale]

They're seeking a justification for why it is defined this way.

Hmm, how can a definition be justified? It is automatically and tautologically true by definition.

I mean I can define "flubnubitz" to be $f \equiv m q v$. It is as justified as any other definition.

 

[DrClaude]

Could the problem be do to "waves"? You could use @PeroK bus example, or a more physical example like a pendulum.

 

[Frank Castle]

Absolutely. The two approaches are entirely equivalent and just as logically valid as one another. It's just that personally I find it easier to grasp when I start with the number of cycles per second rather than the number of seconds per cycle. That is also the approach most non-physicists are familiar with, since record turntable speeds, engine speeds and sound frequencies are all quoted as cycles per second rather than seconds per cycle. It tends to be only very slow things, often astronomical things, like rotations of the Earth about its axis or revolutions around the Sun (and an odd one - the 17-year life-cycle of certain cicadas), that are quoted as a period rather than a frequency.

To be honest, I find it easier to grasp starting from cycles per second rather than seconds per cycle too. Perhaps that is a better approach to take.

Could the problem be do to "waves"? You could use @PeroK bus example, or a more physical example like a pendulum.

Yes, I think that's probably a good idea. Something like: A pendulum takes $t$ seconds to swing from left to right and back again (which we shall refer to as a "full swing"). This is what is referred to as the period, $T$ of the swing. One can then ask, if the period is $t$ seconds, how many full swings of the pendulum will there be in $1$ second. Clearly, the answer is there will be $\frac{1}{t}$ full swings per second. This is what is referred to as the frequency, $f$ of the swing, i.e. the rate at which the pendulum swings. From this we see that period and frequency are inversely related, in particular, from the above it can be seen that $f=\frac{1}{T}$.

 

[A.T.]

They're seeking a justification for why it is defined this way.

The justification for common definitions is that they are common, so many people seem to find them useful.