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

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


$\vec{x}:=$ 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 :

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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.)


I got an idea:

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 $f=0$.

The solution of (2) is then (1).

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