When do waves experience convolution? Excuse the elementary nature of the question, I am new to Physics S.E. If I can be redirected somewhere more appropriate, that'd be appreciated too.
I am curious when waves$-$particularly sound waves$-$experience convolution. I've been reading that when creating a musical chord from sound waves, we simply add their sinusoidal functions together:
http://www-users.math.umn.edu/~rogness/math1155/soundwaves/
However, in differential equations, I was often assured that convolution was the method of choice when "adding" sound waves. Since I am not an applied mathematician, I don't know what to make of this.
Under what real-world circumstances would you use convolution to describe the result of two distinct functions describing sound waves?
 A: Addition of sound waves results from the linearity of the wave equation that describes them. 
I'm going to hazard a guess at the meaning of the cryptic statement:

However, in differential equations, I was often assured that convolution was the method of choice when "adding" sound waves. Since I am not an applied mathematician, I don't know what to make of this.

I'm guessing what this means is this: one can wholly characterize system's acoustic response by its impulse response if that system is (1) linear and (2) time shift invariant, meaning that that if the system is subject to an input at some time $t$, its response will be the same as if it were subject to the same input at time $0$, aside from a delay $t$ in the output. The impulse response $h(t)$ is the image of the Dirac delta under the transformation wrought by the system. Therefore, since a general pulse can be made by summing up Dirac deltas:
$$f(t) = \int_\mathbb{R} f(t-u)\,\delta(u)\,\mathrm{d} u$$
and the conditions (1) and (2) above mean that the response of a sum of inputs is the sum of the responses to the separate inputs, then the ouptut in response to the above as input must be:
$$\int_\mathbb{R} f(t-u)\,h(u)\,\mathrm{d} u$$
since each $\delta$ at the input leads to an $h$ at the output. Thus we calculate the output in response to any input as the convolution of the input with the system's impulse response.
This line of argument applies to systems described by linear, time shift invariant differential equations, hence, I believe, the comment.
Since, as commented by Alfred Centauri:

Convolution and multiplication are duals, i.e., multiplication in the time (frequency) domain is convolution in the frequency (time) domain. Addition, on the other hand, in the time domain is addition in the frequency domain too so it isn't clear to me that you might be thinking of.

this convolution in the time domain maps to multiplication of the Fourier transforms of the input pulse and the Fourier transform of the impulse response. This is why we can simply multiply system transfer functions in the frequency domain.
