Amplitude based on number of equal sources

Question

A homework problem states the following:

In a urban road with a lot of traffic the sound level produced by a flow of a 100 cars per minute during the day is $B_1 = 80$ dB. Calculate the sound level $B_2$ at night when the flow slows down to 5 cars per minute.

My line of thought was to suppose that, for any given position, the average amplitude of a sound wave coming from a vehicle is $A$. Then, due to the superposition of waves, if there are $n$ cars the amplitude of the resulting wave must be $A_n = nA$. This means that the total intensity with $n$ cars is $$ I_{100} = \left( \frac{1}{2} \omega^2 v \rho \right) A_{n}^2 = k (nA)^2 $$ Therefore I deduced that the intensity must scale with the square of the number of cars, since all of the first terms are constant as they only depend on the wave's nature and on the medium it is traveling through. The intensities in the two cases must then be $$ I_5 = k (5A)^2 = 25A^2 k \qquad I_{100} = k (100A)^2 = 10^4 A^2 k $$ Such that $$ I_{100} = k \frac{10^4}{25} 25 A^2 = \left(25 k A^2 \right) \frac{10^4}{25} = 400 I_{5} $$ Therefore, the sound level of 5 cars should be $$ B_2 = 10 \log_{10} \left(\frac{I_5}{I_0} \right) = 10 \log_{10} \left(\frac{I_{100}}{400I_0} \right) = 10 \log_{10} \left(\frac{I_{100}}{I_0} \right) - 10 \log_{10} \left(400 \right) = B_1 - 10 \log_{10}(400) = 53.97 \text{ dB} $$ However the solutions at the end of the book say the following:

$\Delta B = B_1 - B_2 = 10 \log_{10} \left(\frac{5}{100} \right) = - 13 \text{ dB, } B_2 = 67 \text{ dB}$

Why would the intensity scale linearly with the number of cars if it is dependent on the square of the amplitude of the and on the square of the maximum pressure change?

Answer 1

There is an implicit assumption here that the cars travelling on the road emit sound independently of each other, so that the phase of waves emitted by car A is randomly offset from the phase of waves emitted by car B. This means that the sound from car A won't destructively or constructively interfere with the sound from car B. Of course, many components of the sound (call it a wave) from car A will interfere with many components (another wave) of the sound from car B, but there will be destructive and constructive interference in a complicated mess that will average out to no significant effect. You could contrive examples where that is not true, but it will be true most of the time. We say that the sources (the cars) are incoherent with respect to each other. Your assumption was that they are coherent, meaning there is a constant phase relationship between waves arriving at the receiver from each pair of cars. Indeed, your assumption was stronger than that: you assumed that the phase difference was $0$.

For incoherent sources, it is fine to simply sum the intensities, which is what the author of your book did.

Even if your cars were capable of emitting sound coherently from fixed positions at zero velocity, if they were travelling at different speeds (cars typically do that), or were located at different average distances from the receiver (e.g. more traffic density in one lane than the other), then this would create phase differences that change over time, rendering the situation incoherent.