Answering Sound Wave Questions

[Biker]

1) How good our approximations of thinking about waves as a function like sin?
2) The power of a sound wave, I have looked on the internet for derivations and all of them used calculus which is a quite expected thing so I skipped it because I still didnt take integration. However I found a proof posted by someone it was like this:
$E = 0.5 ~ K ~~A^2$
Where E represents the energy of the wave.
He then used
$f = \frac{1}{2\pi} (\frac{k}{m})^{0.5}$
to get'
$I = 2 \pi^2 v p f^2 A^2$

Which is actually the correct form. But this derivation doesn't make sense at all physically. To me it is just like plug and chug because the system requires integration to be analysed. However, the weird thing that it gave a correct equation which might be coincidence but not really sure if math coincides in this situation. I would see this working if actually every molecule had the same energy and it is equal to E above.

3) Is the speed of waves in medium only constant because we modeled the medium as springs and it is as always a good approximation to the real world? and that the speed would change if the medium behaved differently?

4 ) How come air molecules can behave so nicely and harmonically while the wave depend on molecules bumping into each other? Isn't there is some motion due to temperature and what so ever? effects are negligible? Is there a graphical reasoning how air molecules spread a wave in 3d because the forces between gases are weak and again it will depend on bumping. It is just seems more of probabilistic than organised. A model with a springs between air molecules would make sense but not really sure about real world. The problem is I care too much about graphical reasoning, The mathematics isn't hard but the Imagining part is.

Are the forces between gases strong enough to actually to force a gas into a particular shape?Thank you. :D

 

[Student100]

1) How good our approximations of thinking about waves as a function like sin?
2) The power of a sound wave, I have looked on the internet for derivations and all of them used calculus which is a quite expected thing so I skipped it because I still didnt take integration. However I found a proof posted by someone it was like this:
$E = 0.5 ~ K ~~A^2$
Where E represents the energy of the wave.
He then used
$f = \frac{1}{2\pi} (\frac{k}{m})^{0.5}$
to get'
$I = 2 \pi^2 v p f^2 A^2$

Which is actually the correct form. But this derivation doesn't make sense at all physically. To me it is just like plug and chug because the system requires integration to be analysed. However, the weird thing that it gave a correct equation which might be coincidence but not really sure if math coincides in this situation. I would see this working if actually every molecule had the same energy and it is equal to E above.

Bit confused by this, what do mean by it needs integration to be analyzed? Once you integrate and obtain a mathematical model for whatever, it's just as valid as before within it's domain of applicability. It shouldn't make any less sense because certain steps were baked in already. This is also the intensity of a sound wave, not the power, and is defined as the power divided by the area.

As to whether his derivation is correct, I don't know. I'd have to see what exactly he's doing before judging.

3) Is the speed of waves in medium only constant because we modeled the medium as springs and it is as always a good approximation to the real world? and that the speed would change if the medium behaved differently?

Waves travel at different "speeds" depending on the medium. I can't really decipher what you wrote in a coherent way. Sound

4 ) How come air molecules can behave so nicely and harmonically while the wave depend on molecules bumping into each other? Isn't there is some motion due to temperature and what so ever? effects are negligible? Is there a graphical reasoning how air molecules spread a wave in 3d because the forces between gases are weak and again it will depend on bumping. It is just seems more of probabilistic than organised. A model with a springs between air molecules would make sense but not really sure about real world. The problem is I care too much about graphical reasoning, The mathematics isn't hard but the Imagining part is.

Air behaves anything but nicely. But here again, and maybe it's just me, but I don't understand what you're trying to say.

Are the forces between gases strong enough to actually to force a gas into a particular shape?Thank you. :D

What do you mean? Gasses assume the shape of their container. Atmospheric gasses are gravitational bound, for the most part.

 

[Biker]

Bit confused by this, what do mean by it needs integration to be analyzed? Once you integrate and obtain a mathematical model for whatever, it's just as valid as before within it's domain of applicability. It shouldn't make any less sense because certain steps were baked in already. This is also the intensity of a sound wave, not the power, and is defined as the power divided by the area.

I meant intensity sorry. What I mean is if you look at this derivation: http://physics.info/intensity/ They model the air as molecules or balls connected with springs and they use integration to calculate the whole energy of a wave then continue on to find the intensity. This is a good approach, the mathematics makes sense.

But up there he just said E = 0.5 k A^2, Now is that the energy of the whole wave? Did he assume that you can treat the whole air as a single object (Which not sure if it is true) connected to a spring? Why does this even work?
Then he got the period and substituted for K. Period. He assumed the mass m to be the mass of the air

Waves travel at different "speeds" depending on the medium. I can't really decipher what you wrote in a coherent way. Sound

Don't we model the forces between air molecules as springs? and just because of this it turns out that the speed of wave is not related to frequency, Amplitude .. whatsoever. But only depend on the medium charateristics..

Now does this model work for all substances?

Air behaves anything but nicely. But here again, and maybe it's just me, but I don't understand what you're trying to say.
What do you mean? Gasses assume the shape of their container. Atmospheric gasses are gravitational bound, for the most part.

What I mean is if the inter-molecular forces are strong enough to force air molecules into simple harmonic motion (For example connected by springs, strings if one molecules moves the other will move too ..etc). Otherwise it would depend on molecules approaching other molecules and "bumping" into them to make a wave but this would be more probabilistic than organized ( Random). If we assume the forces are weak then what is a bit hard for me to grasp is why all these thermal vibration, random bumping due to different velocity ..etc cause this organized harmonic motion.Thank you for spending time on reading this.

What do you mean? Gasses assume the shape of their container. Atmospheric gasses are gravitational bound, for the most part.[/QUOTE]

 

[Drakkith]

What I mean is if the inter-molecular forces are strong enough to force air molecules into simple harmonic motion (For example connected by springs, strings if one molecules moves the other will move too ..etc). Otherwise it would depend on molecules approaching other molecules and "bumping" into them to make a wave but this would be more probabilistic than organized ( Random). If we assume the forces are weak then what is a bit hard for me to grasp is why all these thermal vibration, random bumping due to different velocity ..etc cause this organized harmonic motion.

Air is mostly empty space, but there are still something like 10²⁵ molecules of gas in even a single liter of air. That's a lot of molecules, and they do in fact bump into each other quite often even though their mean free path is much larger than their individual sizes. While the path of each air molecule is "random" in the sense that it is chaotic, the collective behavior of the gas is definitely not random and a sound wave is the result of this collective behavior.

 

[Cutter Ketch]

1) Some sounds are nearly sine waves (flutes, bells), but most sounds are much more complicated than a simple sine wave. However, believe it or not any wave form can be written as a sum of sine waves with different amplitudes, frequencies, and phases. What's more, as long as the response of the medium is linear, the propagation of the constituent sine waves can be treated independently. So at normal amplitudes where the response is linear the propagation of a sine wave is everything you need to know to describe any sound.

2) Well I suppose it would be nice to have a derivation, but I can't help. At least it's good to know the correct form.

3) Harmonic response is often a good approximation for all sorts of things. Here's why. If an object is bound to a location it is near a minimum of a potential well. At the minimum of a potential the slope is zero. The first non-zero derivative is the quadratic term. Higher order terms are necessarily smaller. In other words a parabola is always a good fit to the potential close enough to a minimum, so all bound objects behave like harmonic oscillators if you don't drive them to too high an amplitude. Normal sound amplitudes are low enough to avoid nonlinear effects.

4) this has been answered, but I'll reiterate. When you are averaging over Avagadros number of molecules the average behavior becomes very smooth and predictable.

 

[Biker]

1) Some sounds are nearly sine waves (flutes, bells), but most sounds are much more complicated than a simple sine wave. However, believe it or not any wave form can be written as a sum of sine waves with different amplitudes, frequencies, and phases. What's more, as long as the response of the medium is linear, the propagation of the constituent sine waves can be treated independently. So at normal amplitudes where the response is linear the propagation of a sine wave is everything you need to know to describe any sound.3) Harmonic response is often a good approximation for all sorts of things. Here's why. If an object is bound to a location it is near a minimum of a potential well. At the minimum of a potential the slope is zero. The first non-zero derivative is the quadratic term. Higher order terms are necessarily smaller. In other words a parabola is always a good fit to the potential close enough to a minimum, so all bound objects behave like harmonic oscillators if you don't drive them to too high an amplitude. Normal sound amplitudes are low enough to avoid nonlinear effects.

Could you elaborate more on 3 and what do you mean about linear response?

 

[sophiecentaur]

Some sounds are nearly sine waves (flutes, bells),

A Bell doesn't produce anything like a sinusoidal wave. There are many different modes at work, all over the surface of a bell. A flute can be quite a pure sine wave (plus the hiss of the air). If you want a good sine wave, you are best to use an electronically generated one.
But there are two separate issues here. Whatever the waveform that's introduced into the air (say a sinusoidal tone from a good quality loudspeaker or any audible waveform you choose) the fact that is reaches a good quality microphone and will give a good version of the input wave at its electrical output involves the response of the 'channel', which is the air it passes through. Air will produce very little distortion for normal air pressure levels and a limited range of sound intensity. If you drive it very hard, the interaction of the molecules will come into play at the peaks of pressure in the cycle. This is a source of non linearity but is really not relevant to normal sound propagation.