Frequency of Beat in Transverse Wave

[terryds]

By addition of transverse wave, I can get a beat.
$y_1 = A\ \sin (\omega_1 t + kx)\\ y_2 = A\ \sin(\omega_2 t + kx)\\ --------------------------------- + \\ y_1 + y_2 = 2A\ \cos(\frac{\omega_1-\omega_2}{2} \ t) \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)$

So, I get new amplitude as a function of t (A' = 2A cos (((ω_1 - ω_2)/2)t)

$y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)$

But, I don't know how to determine the frequency of the beat

If I relate $y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)$ to the form y = A sin (ωt + kx), I get
$f=\frac{f_1+f_2}{2}$

But, if I relate the amplitude to the form y = A sin (ωt), I get
$f=\frac{f_1-f_2}{2}$Which one is the beat frequency?
Does those two things have relation to the group velocity and phase velocity? Which one is for group, which one is for phase?

And, I remember the correct formula is $f_{beat} = |f_1 - f_2|$ which I don't know where it come from.
Please help. I really don't get the idea.

 

[Simon Bridge]

Both are correct - so one is the beat frequency and the other is the sound frequency ... to tell the difference, you have to use physics.
What, physically, is the beat frequency? What physical phenomenon do the words refer to?
Which of those frequencies fits the physics?

 

[terryds]

Both are correct - so one is the beat frequency and the other is the sound frequency ... to tell the difference, you have to use physics.
What, physically, is the beat frequency? What physical phenomenon do the words refer to?
Which of those frequencies fits the physics?

I think the sound frequency is $f=\frac{f_1+f_2}{2}$, and the beat frequency is $f=\frac{f_1-f_2}{2}$ (since beat relates with amplitude)

The beat frequency is number of beat in one second, right? But, really I still can't tell it from the sound frequency.. Please help
But, I guess beat frequency is number of waves with the same amplitude in one second, right? (so others with different amplitude doesn't count)
And, I guess sound frequency is just number of waves in one second no matter the amplitude is.
And, by assuming so, beat frequency must be less than the sound frequency which means beat frequency is $f=\frac{f_1-f_2}{2}$, and the sound frequency is $f=\frac{f_1+f_2}{2}$

Is my assumption (guess) correct?

But, what about the formula $f_{beat} = |f_1-f_2|$? I really don't get it.

 

[Nugatory]

But, what about the formula $f_{beat} = |f_1-f_2|$? I really don't get it.

This is one of those situations where a picture helps a lot. You might want to tryplotting a graph of $y_1 + y_2 = 2A\ \cos(\frac{\omega_1-\omega_2}{2} \ t) \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)$. There's plenty of free graphing software out there and you can take $x=0$ so that you only need two axes: $t$ along the horizontal and $y$ along the vertical. Or you can google for "adding waves beat" which will bring up some nice animations.

But one question about your original post: For a transverse wave $y=A\sin(\omega{t}+kx)$, $k$ and $\omega$ are not independent; a slightly different value of $\omega$ implies a slightly different value of $k$. Might it be that your original equations were supposed to be $y_1=A\sin(\omega_1{t}+k_1x)$ and $y_2=A\sin(\omega_2{t}+k_2x)$?

 

[terryds]

This is one of those situations where a picture helps a lot. You might want to tryplotting a graph of $y_1 + y_2 = 2A\ \cos(\frac{\omega_1-\omega_2}{2} \ t) \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)$. There's plenty of free graphing software out there and you can take $x=0$ so that you only need two axes: $t$ along the horizontal and $y$ along the vertical. Or you can google for "adding waves beat" which will bring up some nice animations.

But one question about your original post: For a transverse wave $y=A\sin(\omega{t}+kx)$, $k$ and $\omega$ are not independent; a slightly different value of $\omega$ implies a slightly different value of $k$. Might it be that your original equations were supposed to be $y_1=A\sin(\omega_1{t}+k_1x)$ and $y_2=A\sin(\omega_2{t}+k_2x)$?

I've seen the animation : http://surendranath.tripod.com/GPA/Waves/Beats/Beats.html

Alright, it's supposed to be $y_1=A\sin(\omega_1{t}+k_1x)$ and $y_2=A\sin(\omega_2{t}+k_2x)$
Please answer my question... I'm really curious about beat frequency.

I think the sound frequency is $f=\frac{f_1+f_2}{2}$, and the beat frequency is $f=\frac{f_1-f_2}{2}$ (since beat relates with amplitude)

The beat frequency is number of beat in one second, right? But, really I still can't tell it from the sound frequency.. Please help
But, I guess beat frequency is number of waves with the same amplitude in one second, right? (so others with different amplitude doesn't count)
And, I guess sound frequency is just number of waves in one second no matter the amplitude is.
And, by assuming so, beat frequency must be less than the sound frequency which means beat frequency is $f=\frac{f_1-f_2}{2}$, and the sound frequency is $f=\frac{f_1+f_2}{2}$

Is my assumption (guess) correct?

But, what about the formula $f_{beat} = |f_1-f_2|$? I really don't get it.

 

[Simon Bridge]

I think the sound frequency is $f=\frac{f_1+f_2}{2}$, and the beat frequency is $f=\frac{f_1-f_2}{2}$ (since beat relates with amplitude)

What if you swap the order of the sine terms:
$y_1 + y_2 = 2A\ \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)\cos(\frac{\omega_1-\omega_2}{2} \ t)$
... now which one is associated with the amplitude?

But, what about the formula $f_{beat} = |f_1-f_2|$? I really don't get it.

It goes back to my original question - if you know what beats are, what phenomenon the word labells, then you can answer those questions for yourself. The beat frequency is the number of beats per second, but what is a "beat"?
If there are beats in a sound of only one note, what do you hear?

 

[terryds]

What if you swap the order of the sine terms:
$y_1 + y_2 = 2A\ \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)\cos(\frac{\omega_1-\omega_2}{2} \ t)$
... now which one is associated with the amplitude?It goes back to my original question - if you know what beats are, what phenomenon the word labells, then you can answer those questions for yourself. The beat frequency is the number of beats per second, but what is a "beat"?
If there are beats in a sound of only one note, what do you hear?

If I associate 2A sin ((ω_1+ω_2)/2 * t + kx) with A', then the form becomes y' = A' cos ((ω1-ω2)/2 * t) (the form of simple harmonic oscillation (but actually it's not), and the amplitude now is a function of x and t.

Beats are like audio pattern caused by interference of waves. So, it's kinda like quiet - geting loud - getting quiet -loud -and so on...
So, one beat is defined as quiet - loud - quiet or loud-quiet-loud. In other words, it is from one peak/trough to another peak/through with the same amplitude, right?
And, I can say that the frequency of a beat is the frequency of the 'envelope', right?

And, one wave is defined as just peak to peak or trough to trough no matter the amplitude is.
Is it correct?

If there is a beat of only one note, there is no beat, is it right? I mean, by adding two equal waves (one note means one frequency), I just get a wave with double amplitude, and no beat since the amplitude is constant, right?
Still, I don't know how $\frac{f_1-f_2}{2}$ becomes $|f_1-f_2|$.. What makes it is doubled? Does it mean that the frequency of beat is the frequency of two envelopes? (I'm getting really confused now hahaha)

 

[Simon Bridge]

If I associate 2A sin ((ω_1+ω_2)/2 * t + kx) with A', then the form becomes y' = A' cos ((ω1-ω2)/2 * t) (the form of simple harmonic oscillation (but actually it's not), and the amplitude now is a function of x and t.

So there is an issue there isn't there?

Beats are like audio pattern caused by interference of waves. So, it's kinda like quiet - geting loud - getting quiet -loud -and so on...
So, one beat is defined as quiet - loud - quiet or loud-quiet-loud. In other words, it is from one peak/trough to another peak/through with the same amplitude, right?

That's pretty good. The beat is the repeated pattern of volume (sound).
They happen when $\omega_1\approx\omega_2$.

And, I can say that the frequency of a beat is the frequency of the 'envelope', right?

Yes.

And, one wave is defined as just peak to peak or trough to trough no matter the amplitude is.
Is it correct?

No. Strictly, the wave is the whole thing. In common language a wave is a hump of stuff moving around - think: what is the thing that gets surfed on called? This is why it is common to talk about one wavelength being the length of one wave and stuff - but that is not strictly correct.

The wavelength and the period are only valid for strictly repeating functions.
In maths, if it repeats in time, then the time between repeats is the period. There may be a number of crests and troughs in between. If it repeats in space, the wavelength is the distance between repeats. So the wavelength of the wave you have would actually be the wavelength of the beats ...
In physics we deconstruct waves into sums and products of individual sine waves - and we assign a wavelength to each component separately.
So the language gets messy. This is why we are careful to distinguish between the frequency of the sound and the frequency of the beats... the whole wave does not have a meaningful frequency.

The bigger the frequency, the faster the oscillations.
Now think about it ... which is going to be bigger: the beat frequency, or the sound frequency?
Is $|f_1-f_2|$ bigger or smaller than $f_1+f_2$?
Now do you see how to find which is which?

Still, I don't know how $\frac{f_1-f_2}{2}$ becomes $|f_1-f_2|$.. What makes it is doubled? Does it mean that the frequency of beat is the frequency of two envelopes? (I'm getting really confused now hahaha)

https://academo.org/demos/wave-interference-beat-frequency/

 

[olivermsun]

Strictly, the wave is the whole thing. In common language a wave is a hump of stuff moving around - think: what is the thing that gets surfed on called? This is why it is common to talk about one wavelength being the length of one wave and stuff - but that is not strictly correct.

The wavelength and the period are only valid for strictly repeating functions.
...
In physics we deconstruct waves into sums and products of individual sine waves - and we assign a wavelength to each component separately.
So the language gets messy.

It gets even messier because for the waves that get surfed on, the components are not separate.

This is why we are careful to distinguish between the frequency of the sound and the frequency of the beats... the whole wave does not have a meaningful frequency.

This is definitely important to understand.. there are different "frequencies" associated with different parts of the phenomenon.

 

[terryds]

So there is an issue there isn't there?

That's pretty good. The beat is the repeated pattern of volume (sound).
They happen when $\omega_1\approx\omega_2$.

Yes.

No. Strictly, the wave is the whole thing. In common language a wave is a hump of stuff moving around - think: what is the thing that gets surfed on called? This is why it is common to talk about one wavelength being the length of one wave and stuff - but that is not strictly correct.

The wavelength and the period are only valid for strictly repeating functions.
In maths, if it repeats in time, then the time between repeats is the period. There may be a number of crests and troughs in between. If it repeats in space, the wavelength is the distance between repeats. So the wavelength of the wave you have would actually be the wavelength of the beats ...
In physics we deconstruct waves into sums and products of individual sine waves - and we assign a wavelength to each component separately.
So the language gets messy. This is why we are careful to distinguish between the frequency of the sound and the frequency of the beats... the whole wave does not have a meaningful frequency.

The bigger the frequency, the faster the oscillations.
Now think about it ... which is going to be bigger: the beat frequency, or the sound frequency?
Is $|f_1-f_2|$ bigger or smaller than $f_1+f_2$?
Now do you see how to find which is which?https://academo.org/demos/wave-interference-beat-frequency/

I got it! So, the 'sound frequency' here is phase frequency, right?
Of course, the phase frequency will be bigger than the beat frequency (envelope).

So, phase frequency = $\frac{f_1+f_2}{2}$ (we don't multiply it by 2) and the beat frequency = $|f_1-f_2|$ since an envelope contains two beats, right??

 

[Simon Bridge]

Well done.