# Derivation of the relation between pressure perturbation and its source in Acoustics

I have a doubt about the derivation of the relation between pressure perturbation and its source in Acoustics:

$$p(\vec{x},t)=\int_V\frac{q\left(\vec{y}, t-|\vec{x}-\vec{y}|/c\right)}{4\pi|\vec{x}-\vec{y}|}d^3\vec{y}\hspace{30pt}(1)$$

where:

$$\vec{x}:=$$ position of a given point

$$\vec{y}:=$$ position of a point in the source

$$p:=$$ pressure perturbation

$$q:=$$ source of sound

$$c:=$$ speed of sound

My professor wrote the derivation of expression (1) in his notes (which are in portuguese). My doubt is about the following step that he made :

So, summarizing, my professor just got this:

$$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\right)\left[\int_V\frac{q\left(\vec{y}, t-|\vec{x}-\vec{y}|/c\right)}{4\pi|\vec{x}-\vec{y}|}d^3\vec{y}-p\right]=0\hspace{30pt}(2)$$

and picked (without any explanation) the trivial solution, which is (1) .

My doubt is:

Why should (1) be the chosen solution, when (2) has also other solutions (i.e. the other solutions of the differential equation)?

(I didn't find any explanations in books either.)

The solution of equation (2) is (1) plus a function $f$ with the property of being null under the action of left operator, i.e.:
$$p(\vec{x},t)=\int_V\frac{q\left(\vec{y}, t-|\vec{x}-\vec{y}|/c\right)}{4\pi|\vec{x}-\vec{y}|}d^3\vec{y}+f$$
such that $\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}- \nabla^2\right)f=0.$
But if the sound source is null ($q=0$), then there's no sound ($p=0$), and to satisfy this condition we would need $f=0$.