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Noise Pollution
The Question:
Are there traffic noise pollution equations?
I am studying Noise Pollution, and am highly interested in any equations that may relate the type of vehicle, acceleration, velocity, and/or distance from source to sink to the external noise produced by the vehicle.  The quantity must be measured in dB(A).

The answer:

Definitely, this is a very good question. It was even a good question for me because I was forced to open the textbooks about sounds I used to read before, and remember these equations.

First of all, sound is a mechanical wave, that propagates through a medium; this medium may be air, where the speed of sound reaches 340 m/s, or may be water (V=1480 m/s), and solids (V=5200 m/s). when waves propagate, they cause disturbance in air pressure, this is termed as compression and rarefaction. The main equations that govern this part are:
V=(Elasticity  modulus of the medium / Density )^(1/2) à (1)
V= frequency X wave length.           à (2)

Thus as you see, the frequency, which is the number of complete wave lengths that happen in one second, is a key factor in determining the speed of the sound in the medium.

Letís shift to the other side, with some definitions. The first definition is Intensity of the wave (I), which is the power per unit area.
I= Power / Area à (3)
Thus it is the rate at which sound flows through a unit area A perpendicular to the direction of the wave propagation. There is another equation that says:
P = (pi)(density X velocity X (frequency X Amplitude )^2) à (3)
(pi = 22 / 7)
Then the power is equal to that above mentioned equation; from which we can get an equation for the intensity.
The second part is to link the previous equation with the Decibels level.
The equation says:

dB = 10 log (I/I0) à (4)
where I0 is called the threshold of hearing, which is the minimum sound intensity that that human ear can distinguish. This value I0 = 10 E-12  Watt/m^2.

Finally, let us transfer to the final point; after which we will link all these equations together. The final part is a famous concept called: Doppler effect. Its definition is:
"Doppler effect is experienced whenever there is a relative motion between the source and the observer. When the source and observer are moving towards each other, the frequency heard by the observer is higher than the frequency of the source. When the source and the observer move away from each other, the observer hears a frequency which is lower than the source frequency". (Serway: 464)

you can feel Doppler effect with ambulances and police cars, when they approach towards you, or go away. You will note the change in their frequency and hence their noise level.
Dopplerís equation says:
1-If we have a stationary source and observer moving away from it:
relative frequency = original frequency X ( 1 - (relative speed between the observer and the source / sound velocity ) )
The negative sign turns to be positive when the observer is moving towards the source .
2-If the observer is stationary, while the source is in motion:
relative frequency = original frequency/ (1 -  (relative speed between the observer and the source / sound velocity ))
The negative sign turns to be positive when the source is moving towards the observer.
3-Both are moving toward each other:

fí = f ( (V +Vo)/(V-Vs))
fí is the relative velocity
V= speed of sound
Vo= Observer speed
Vs= Source speed

Signs will be reversed in case they are moving in opposite directions.

From all this dilemma, what do we get?
First know the speed of the source, and second determine the frequency. Then calculate the power and followed by the intensity. Usually the area is taken to be your ear drum area !!! Then calculate the dB level due to the moving object.

There more points I want to add; however they are not closely relevant to your question but to sound in general. Anyway, I am ready to receive your questions in the future.

Looking forward to your reply,

Moataz Attallah
Undergraduate, Mechanical Engineering Junior
The American University in Cairo


Abbott, A.F. Ordinary Level Physics. Heinemann Educational Books, London. Fourth edition 1984.

Serway, Raymond. Physics for Scientists and Engineers. Saunders Collage publishing, USA. Third updated edition 1990.

Reply from The person who asked the question

        Thanks for your input about the noise pollution physics questions that I
fielded awhile ago. I understand all that you sent, and it is very helpful. I was looking through my physics text and found some more useful stuff.  My question now deals with specifics of a few of the equations.

In regards to the equation (dB = Power/Area) I am looking for the dB level of a sound source (moving automobile) I don't exactly understand the Power in watts.  What would I use for a car traveling at say 60 mph?  Is there a way of determining this?  And also, when cross sectional area is used to determine the Intensity of a sound source, do you use the cross sectional area of the human ear drum, or the area that the receiver takes up in space.  I assume the ear drum Area is needed.  If so what might this be?
        My overall objective is to come up with rough estimates based on equations of the sound level (dBA) produced by a moving vehicle.  The speeds will be the input as well as the distance from the source to the receiver.  Any assistance with this will be most appreciated.


-Nate (go AIR FORCE!)

And my reply:

Dear Nate, (if you allow me to call you dear)!!

First of all thanks for your comment about the answers. I need to know this. Concerning your research, I would like to inform you who is the source of the answers you got. It is simply me; an undergraduate junior mechanical engineering student. However, I am deeply interested in physics; so I am working as an undergraduate teaching assistant in physics, specifically in Electricity and Magnetism courses. So please consider this fact; because I am afraid you may be doing some serious research. If you are an undergraduate like me, and your questions are just for your own interest, or some sort of curiosity, here are my answers.

Imagine, Nate that you came one day to Egypt where I am, and you decided to walk in one of our roads. You will find different sorts of noise pollution; horns, shouting of car drivers, noise coming from motors, and ect. Thus if you are conducting a research entitled noise pollution according to the type of the vehicle you have to specify which part you are actually studying.

Letís focus a little bit. Perhaps it is the motor or the horn or even the shouting. All these words meet in one focus called "FREQUENCY." Frequency is the key word. Physical-wise the definition means the number of complete wave lengths (cycles) produced in one second.
 Every object has its own specific frequency. Even you Nate, you voice has a specific frequency. And by the way, today voice frequency has become a method of identification like the finger print exactly. So you can find electronic lock works with the voice print. The care motor has a frequency, and this frequency changes with different stages of the car motion. You know definitely that the motor of the car moving at 60 mph differs from the one moving at 10 mph. So to speak, the motor voice print is not constant.

Second it is the horn. This one, luckily, has a constant value.

Letís shift. Now we have a car, moving with 60 mph. We want to determine the decibel level at that speed for that specific car. You know that there are sound oscillators that can determine the sound frequency, similar to the charts you see on a sound-mixer, that shows the variation of the voice frequency when a song is being played in your CD-drive. Use this device to measure the frequency of the car at that speed, and assume that that value of frequency is specific for all the similar cars, say in its cc standard. Then start your calculation.

Based on Dopplerís equations, the frequency may differ according to the car motion whether against or with your motion. Second, plug the obtained value of the frequency into the other equation about the watt thing, that you do not like 8). Then divide the power by the ear drum. I know that it is getting so disgusting, concerning the ear drum ; ). But you know there are somethings in life that we have to assume. What is the ear drum area???!!!
Simply, use a microcope to know this !!! Just kidding. Assume it say 0.25 square centimeters. Then get the sound intensity. Put its value to the other equation of the decibels level dB= 10 log (I / I0) . thus we finally got the decibel level.

Frankly Nate, this is the maximum I can say. However, I still find the thing you are  talking about extremely interesting. I would like to know more, how you are thinking, and what are the questions you are attempting to answer. Excuse my curiosity, but may be by this way I can find more helpful stuff for you.

Pleased to help you, Nate. Keep in touch, and reply if you need further data. I can ask our professors at the American University in Cairo. I know some of them are doing some research about noise and I can ask them for any avalilable data.

Moataz Attallah
Undergraduate teaching assistant-Physics Unit.
The American University in Cairo
113 Kasr Al-Aini street,
Cairo, Egypt.
Science Building, Physics Unit.