Sound Power Challenge

[chad kincham]

The amount of energy contained in two different sinusoids, of the same amplitude, of different frequencies, is directly proportional to their frequencies.

For those who aren't familiar with the scientific language; I'll provide this example:
we have two different continuous waves. One peaks and falls at twice the rate of the other; but both waves are the same height. The wave that peaks and falls faster contains twice the energy of the wave that is rising and falling at half the rate of the other.

Let's take two electronic sinusoids and pass them through a coil. Lower frequencies meet less impedance in a coil rather than do higher frequencies. This means that it would take a greater amplitude, at a higher frequency, to pass the same amount of energy as that of a lower frequency, through a coil of the same value; as at a lower frequency, more current would pass through the coil, all amplitudes equal.

Let's convert electronic power into sound power.

Audio drivers (speakers) Are driven by passing current through a coil which is in proximity to a fixed magnetic field. The efficiency of drivers at any particular frequency across their ranges gets very complicated.

Let's simplify it for the sake of the question. The difference that I will speak of is dramatic enough; that we won't need to analyze this to the Nth degree. Drivers are most efficient at their resonant frequency. Let's leave it there4.

We have two drivers with different resonant frequencies, but their efficiencies are equal at their respective resonant frequencies. Both are of the same impedance. Yes, it's a perfect theoretical world.

We run a sinusoid through each of the drivers which matches the resonant frequency of each driver.

So far, because of the higher impedance met by the higher frequency we have less work being done by the high frequency driver. More work is being done by the low frequency driver.

At this point one might expect that we would have more sound power from the low frequency driver. This is not the case in reality. Even at a lower impedance. it takes far more electrical power to deliver the same amount of sound power at a lower frequency.

Why is this true?

Back in the late '80s, early '90s I made some phone calls and visits to try to get an answer to this question. I spoke with numerous engineers and scientists who had no idea. After years of reading and asking others, I finally found a Physicist who worked for JHU, who answered my question. I'm satisfied with his answer; but I thought it would be fun to present this challenge; and if any answers would match the answer that I finally accepted. BTW, of all of the people I spoke with only, I received only two different answers. I accepted both of them; but the answer of "I don't know." didn't help me to move onto the next step toward my invention idea.

1. Our hearing is more insensitive to low frequency sound.

2. Loudspeaker efficiency is much lower below the resonance frequency of a loudspeaker box.

 

[HARK!]

1. Our hearing is more insensitive to low frequency sound.

Nope, someone with excellent hearing can hear for 20 Hz the 20 KHz. However, the ear is most sensitive between about 600 Hz to about 5 KHz. In other words the ear is most sensitive at about the middle of it's range, so like eyes, so like scales, so like other measuring instruments so like electronic circuits, so like everything that I can think of.

2. Loudspeaker efficiency is much lower below the resonance frequency of a loudspeaker box.

I thought that I was talking about drivers suspended in free air. I didn't bother to go back to the OP and check. It doesn't matter. The original model, before I oversimplified it for you, has two drivers. Both are of equal efficiency for the frequency that they were reproducing. Both have a Fs that is equal to the frequency that they are converting to sound power.

All things ideal in a perfect world.

This isn't a trick question.

 

[Mark Quayle]

The sound comes from its' origin no matter where the listener is in reference to that origin.

Of course, and I said as much.

At high frequencies, sound is highly directive. The higher the frequency, the more it behaves like light. In the C region, the sound will a narrow dispersion pattern. Outside of that pattern, the amplitude of that sound pattern will quickly roll off. That dispersion pattern is emitted from the front of the driver. Now let's put that driver in a lead cabinet, with walls 3 feet thick. We might not expect to hear much coming off of the cabinet; as 3 foot thick lead walls don't vibrate very much.Let's make it a perfect world and say that we will hear nothing. Then we suspend that cabinet in free air; so that we don't hear any reflected sound. Now we will suspend ourselves behind the cabinet, in the opposite direction of where the sound is being directed. We will hear where the sound is coming from; but we won't hear it from where it is being directed.

We take the same cabinet in the same conditions, and put a frequency in the X region through it; and the amplitude of the frequency will be virtually the same in all directions; as in the X region sound behaves like heat.

Of course, and I said as much.

Directivity is how much the amplitude rolls off, off axis to the driver. The faster it rolls off, off axis, the higher the directivity.

That is your definition of directivity, I suppose, but you have once again described a result of the cause, and not the cause. You have described a phenomenon and given it a name based on its appearance (I suppose) in a diagram. Maybe you found it in a book and assumed to book to be authoritative in and of itself. I don't know. What I hear you to describe, is what the books call the 'directivity FACTOR', which is not the same thing as 'directivity'. TO ME, 'directivity' in this context has to do with the subjective ability to know from what direction the sound is traveling, and the sound's wavelength relative to the distance of one ear compared to the other off 90dg from the sound source. But, I'm glad to at least have this understood, so that there's no point in attempting to compare the virtues of the ear vs the spl meter. Also humorous to me is the fact that this whole question is, according to you, unrelated to the question you wished answered in your OP.

I did edit one of my posts to ask questions concerning phenomena that I am not altogether sure of or would like explained, that I find interesting and even funny/curious. Read the last two paragraphs in #37

 

[Mark Quayle]

1. Our hearing is more insensitive to low frequency sound.

2. Loudspeaker efficiency is much lower below the resonance frequency of a loudspeaker box.

#2 -- or, that is, below the resonant frequency of the box-speaker combination. The resonant frequency of a speaker is sometimes a huge effect, specially as the frequency response rolls off below that resonant frequency. When I design a bass reflex cabinet, I port it to allow a bandwidth below its transducer's resonant frequency to add the waveform from the back of the speaker cone out the port to the waveform from the front of the speaker cone, instead of canceling each other (such as from an unbaffled speaker cone). But you are correct for a box designed that way --the rolloff of response below that bandwidth, the driver long since having lost efficiency, is generally steep.

 

[HARK!]

Another thing, do low frequencies of sound in a gas medium maintain integrity over distances better than high frequencies,

Do they? If so; why?

 

[HARK!]

But you are correct for a box designed that way --the rolloff of response below that bandwidth, the driver long since having lost efficiency, is generally steep.

The OP mentions nothing of boxes. It mentions drivers. Drivers are capable of producing frequencies which have a wavelength that is equal to or shorter than the diameter of the cone. If the wave is any longer, the amplitude rolls of sharply.

The speed of sound in air changes at different temperatures and different altitudes; but for the sake of demonstration, let's round it off to 1000 ft per second. Let's use a 12" speaker for demonstration.


Vw = fλ,

Vw is the speed of sound that we are using 1000
λ is 1 foot for a 12"cone
f is the frequency

1000 = f X 1

This means that the lowest frequency that a 12" driver can effectively reproduce in free air, is ~1 KHz.

This is an ear piercing frequency.

 

[Mark Quayle]

Do they? If so; why?

Concerning the question do low frequencies maintain their integrity (in a gas medium) over distance more than high frequencies do?", I'm not sure. That was my question. I have heard it asserted so, and was hoping you could lend some light to the question.

I have noticed that high frequency waves on surface water tend to fragment more quickly than low frequencies,and while I don't know how to describe a reason why, I can intuitively see that it is so, but I don't know if that holds true for under the surface. I would like to know.

 

[Mark Quayle]

The OP mentions nothing of boxes. It mentions drivers. Drivers are capable of producing frequencies which have a wavelength that is equal to or shorter than the diameter of the cone. If the wave is any longer, the amplitude rolls of sharply.

The speed of sound in air changes at different temperatures and different altitudes; but for the sake of demonstration, let's round it off to 1000 ft per second. Let's use a 12" speaker for demonstration.


Vw = fλ,

Vw is the speed of sound that we are using 1000
λ is 1 foot for a 12"cone
f is the frequency

1000 = f X 1

This means that the lowest frequency that a 12" driver can effectively reproduce in free air, is ~1 KHz.

This is an ear piercing frequency.

So, as you have aptly demonstrated, not a perfect world.

 

[HARK!]

Concerning the question do low frequencies maintain their integrity (in a gas medium) over distance more than high frequencies do?", I'm not sure. That was my question. I have heard it asserted so, and was hoping you could lend some light to the question.

Integrity? I'm not sure what you mean by that. High frequencies don't travel as far through air as low frequencies. This can be observed by listening to an oncoming train. I'm not talking about the doppler effect; because the same hold true after the train passes. When the rain is distant we hear more low frequency noise; but as it grows closer; we hear more high frequency noise disproportionately to the low frequency noise. Some of this can be explained by the Fletcher-Munson effect; but not all of it this is a scientific phenomenon that I believe is near the heart of the question.

I mentioned that as I was working through this challenge, I had begun to question the JHU Physicists answer.

I feel that this portion of this post has brought us close enough to his answer to reveal his answer.

I know that you were leading onto this before; but the timing of the next portion of your post is perfect.

His answer complies with the law of conservation of energy. He said that in the longer wave, that there was more compression in the wave, to reach the same amplitude. All of these molecules being pushed together in this wide compression wave was generating more heat; and my electrical power was being dissipated as heat in its' conversion to sound power in the air.

This guy was a brilliant Physicist and there were three other JHU Physicists sitting there listening to his answer. We al considered the matter settled at that time. However, I'm seriously questioning this explanation at this time.

High frequencies don't travel as far through air, as do low frequencies at the same amplitude. I suspect that is because they are going trough more compression and rarefaction cycles. This is generating heat. I suspect that is where the energy is being lost. So why would more energy be lost in a longer wave which can travel a greater distance? The only reason that I can think of is that it started out with more energy. Now the velocity at which the wave is being produced is something that might be considered, regarding how quickly its' amplitude was being ramped up; so I considered it. I used a hydraulic analogy. After I did that; I didn't give much consideration to the velocity at which the wave was being generated.

I have noticed that high frequency waves on surface water tend to fragment more quickly than low frequencies,and while I don't know how to describe a reason why, I can intuitively see that it is so, but I don't know if that holds true for under the surface. I would like to know.

I wasn't aware that surface waves had a significant effect on the motion of water below the surface; but here we are in the water!

Let's make it a perfect world. We are in the middle of an ocean, with no wind, no moon, with a smooth mirror surface.

I considered this. I could cast relatively small stone in the water and generate a high frequency wave of about a half inch in height. As the wave spread; that height would quickly diminish, in comparison to a wave that had the same height, but 100,000 times its' wavelength. How could a generate a such a long wave, at a half inch in height? I'd have to raise a large portion of the ocean floor. That would take far more energy than casting a stone; and that wave would travel a much longer distance before it started losing height.

 

[HARK!]

So, as you have aptly demonstrated, not a perfect world.

That's right. There are endless factors; but complicated problems are nothing more than a string of simple problems. KISS Keep It Stupid Simple.

 

[chad kincham]

#2 -- or, that is, below the resonant frequency of the box-speaker combination. The resonant frequency of a speaker is sometimes a huge effect, specially as the frequency response rolls off below that resonant frequency. When I design a bass reflex cabinet, I port it to allow a bandwidth below its transducer's resonant frequency to add the waveform from the back of the speaker cone out the port to the waveform from the front of the speaker cone, instead of canceling each other (such as from an unbaffled speaker cone). But you are correct for a box designed that way --the rolloff of response below that bandwidth, the driver long since having lost efficiency, is generally steep.

In this area I am far from an expert, or even a highly knowledgeable amateur - I just took a shot at answering the OP based on pretty limited knowledge, to see how close it is to right.