Infinite Point Sources of Sound & The Zeta Function

[DivisionByZro]

Homework Statement

An infinite number of incoherent sources of sound are located on the x-axis at positions given by n² (in meters) with n= 1,2,3,4,5... If all the sources emit with a power of 10.0W, calculate the sound level of the total sound wave at the origin. Prove your answer using the Riemann Zeta Function extended to the whole complex plane using analytic continuation.

(This is an April fool's extra problem, it's not going to be marked)

Homework Equations

Riemann Zeta Function
I_Total=I₁+I₂+2sqrt(I₁I₂)
Where I is the intensity

The Attempt at a Solution

I can see that their distances increase exponentially, and so I'm trying to make the inverse square law for sound fit in somewhere. Also, I can see that this could create a plane source, given by an integral of all my point sources (Infinitely many).

But honestly I have little experience with the Zeta Function and I don't see the relation with this problem (Wouldn't be surprised if there wasn't a connection)

It's obviously a joke, but I'm still interested in the solution, so any help is appreciated!

 

[mark726]

I would approach this problem by first considering the basic principles of sound propagation. The intensity of a sound wave decreases with distance according to the inverse square law, meaning that the intensity at a distance r from a point source is proportional to 1/r^2.

In this case, we have an infinite number of point sources, each with a power of 10.0W, located at positions given by n^2 (in meters). This means that the intensity at the origin (where we are trying to calculate the total sound level) will be given by the sum of the intensities from each source at that point.

To calculate this sum, we can use the formula for the intensity of a point source at a distance r, and sum it up for all the sources at the origin:

I_total = Σ(10.0W / r^2)

Since the sources are located at positions given by n^2, the distance from each source to the origin will be given by √n^2 = n. So we can rewrite the sum as:

I_total = Σ(10.0W / n^2)

We can now use the Riemann Zeta Function, which is defined as:

ζ(s) = Σ(1/n^s)

Where s is a complex number. This function is typically defined for real values of s greater than 1, but it can be extended to the whole complex plane using analytic continuation.

If we set s = 2 in the above formula, we get:

ζ(2) = Σ(1/n^2)

Comparing this to our formula for the total intensity, we can see that:

I_total = 10.0W * ζ(2)

This means that the sound level at the origin will be given by:

L_total = 10log(I_total / I_ref) = 10log(10.0W * ζ(2) / I_ref)

Where I_ref is the reference intensity (usually taken to be 10^-12 W/m^2). So the sound level at the origin will be:

L_total = 10log(10^-12 * ζ(2)) = 10log(ζ(2)) - 120 dB

This is the final answer, and it can be proven using the Riemann Zeta Function extended to the whole complex plane using analytic continuation.