In a choir, several people sing the same note at the same time. The sound each person makes consists of a tonic and certain overtones, but much of what we hear is the tonic frequency, so I'd like to concentrate on that.
A very crude approximation is that the tonic is a sine-wave of some frequency,
$$
H(t) = A \sin (kt + d)
$$
where the constant $k$ determines the frequency, and the phase, $d$, determines the instant at which the signal reaches its peak: if you and I begin singing at slightly different moments, we'll each have our own "d" value. To simplify things, let's assume that we adjust our units of time to make $k = 1$, and adjust our units of sound intensity to make $A = 1$, so that
$$
H(t) = \sin(t + d).
$$
By the principle of superposition, the sound made by several singers looks like
$$
C(t) = \sum_{i = 1}^n \sin (t + d_i)
$$
where the $d_i$ are (I would guess) uniformly distributed random variables in the interval, say, $0 \le d_i \le 2\pi$.
It seems to me that for any given time $t_0$, the values $A\sin(t_0 + d_i)$ are distributed between $-1$ to $1$, with a distribution that's symmetric about $0$: any given singer's sine-wave is just as likely to be in the "negative" half-cycle as in the positive one, etc.
That is to say: it appears that the expected value of the sound produced is zero.
I recognize that this isn't exactly the right question to ask, for this looks at the expectation over all sets of phases rather than for a specific choir. One might form a choir (i.e., a set of phases) which was nice and loud, and then offset everyone by a half-cycle and get another nice loud choir, but the sum of the two choirs would be zero, which is no problem: the individual choirs were plenty loud.
I guess the question I have is then this:
What's the expected maximum of $|C(t)|$, on the interval $0 \le t \le 2\pi$, with the expectation taken with respect to iid uniform choices of the phases $d_i$? Experiments in matlab suggest to me that it might be something around $0.9 \sqrt{n}$ (where $0.9$ must surely come from some weird combination of constants involving $\pi$, etc.)
The peculiar thing is that a 100-person choir seems to me to be much louder than just 10 times the loudness of a single singer. Given the logarithmic nature of perception for most senses, this seems to wildly contradict the estimate I gave above.
Can someone suggest some insight into this?
[Let's assume, for the sake of argument, that the choir is arranged in a circle around me, the conductor, so that if everyone sings at exactly the same moment, the sounds all reach my (single) ear at exactly the same moment, OK?]
I realize that the larger question of why choirs work is a combination of perception, physics, math, and probably some other things, but the math question here is about the expected amplitude of a sum of random-phase sine-waves, and that's what I'm hoping to have answered here on MSE.
Best Answer
A better approach than the one I was taking in the comments is to see the whole process as a question about random variables in the plane. Let us define $$ X_j := \begin{bmatrix}\cos d_j \\ \sin d_j\end{bmatrix}\quad \text{and} \quad S_n := \sum_{j=1}^{n} X_j,$$ where $d_j$ is a family of iid uniform random variables on $[0, 2 \pi]$. Then, $S_n$ is a random vector in $\mathbb{R}^2$ with first and second coordinates given respectively by $$ A_n = \sum_{i=1}^n \cos d_j \quad \text{and} \quad B_n = \sum_{i=1}^n \sin d_j. $$ As mentioned in the comments, we have that $$ C(t) = \sum_{i = 1}^n \sin (t + d_i) = \sin t \Big(\sum_{j=1}^n \cos d_j\Big) + \cos t \Big( \sum_{j=1}^n \sin d_j \Big) = \langle (\cos t, \sin t), (B_n, A_n) \rangle, $$ implying that $$ \sup_{t \in [0,1]} |C(t)| = \sqrt{A_n^2 + B_n^2} = \lVert S_n \rVert $$ Thus, you just want to understand well the behavior of $S_n$, a sum of iid random vectors. Notice that the marginals of $X_j$ have the same distribution, and that $$ \mathbb{E}[\cos d_j] = \mathbb{E}[\sin d_j] = 0 \quad \text{and} \quad \mathbb{E}[\cos^2 d_j] = \mathbb{E}[\sin^2 d_j] = \frac12, $$ by applying the expected value operator to $\sin^2 d_j + \cos^2 d_j = 1$. Also, we have $$ \mathbb{E}[\cos d_j \sin d_j] = \frac12 \mathbb{E}[\sin (2d_j)] = 0. $$ Thus we can apply the multidimensional Central Limit Theorem for the sum $S_n$, and see that $$ \frac{S_n}{\sqrt{n}} \to Z $$ in distribution, where $Z$ is a normal of mean $0$ and covariance matrix $\Sigma = \begin{bmatrix}\frac12 & 0 \\ 0 & \frac12\end{bmatrix}$. I am not very familiar with these multidimensional results but I am quite confident that from here you should be able to derive precise estimates on the distribution of $\lVert S_n \rVert$ and its moments.