So, there is a sound at $S$, whose intensity $I$ obeys the inverse square law ($I \sim \frac{1}{x^2}$). At point $P$, at a distance $r$ from $S$, the air molecules oscillate with an amplitude of $8μm$. Point $Q$ is at a distance of $2r$ from $S$. What is the amplitude of the air molecules at $Q$? What is the relationship between amplitude and distance?
[Physics] Amplitude at distance from source
classical-mechanicswaves
Related Solutions
Your last equation;
$$ I= \tfrac{1}{2} \rho v \omega^2 s_m^2 $$
is derived by taking the equation for a plane wave:
$$ s(x, t) = s_m \sin(\omega t - kx) $$
and calculating the kinetic energy associated with the wave. The calculation is described in this article from the Physics Hypertextbook.
For a spherical wave the prefactor $s_m$ is not a constant but decreases as $r^{-1}$. This means $s_m^2$ decreases as $r^{-2}$, and therefore both equations give an inverse square dependance of intensity on distance.
In the equation:
$$y(x,t)=Asin ~k(x-vt)$$
$A$ can be varied independently of $k$ and $v$ and hence of $f$. That is what is meant by saying that the amplitude doesn't depend on the frequency. Now, when you write the equation as:
$$A = \frac{y(x,t)}{sin ~k(x-vt)}$$
it means that the ratio of the height of the string from the mean position at some point to a function($sin ~k(x-vt)$) is always constant and equal to the amplitude. So it is basically saying that the height of the string at such $(x,t)$ where $k(x-vt) = (2n+\frac{1}{2})\pi$ is equal to the amplitude. But does this mean that you can change the ratio by changing $k$ and $v$? No you can't. And why is that? Because the numerator $y(x,t)$ also depends on $k$ and $v$.
In other words, when writing your equation for $A$ you are wrongly assuming that you can change the $k$ and $v$ in the denominator without changing the numerator. Whatever changes you make in the denominator will be reflected in the numerator and hence the ratio will remain fixed. That indeed is what the equation is saying- no matter what your $k$ and $v$ are, take this ratio and you get the amplitude.
Attempting to give more physical intuition. Imagine that you have a vibrating string and you need to find its amplitude. At what point and time will you measure its height to ensure that you got the amplitude? The crest of the wave will keep moving, making it hard to take readings.
So imagine you get another string also tied at both ends and exactly identical to the first, except that its amplitude is always 1 (which I am not saying is trivial to set up, but well this is a thought experiment). So now you just go ahead and measure the heights of both the strings at the same point simultaneously and take a ratio of the heights. You get the amplitude of the string, just as your equation says.
Now, how will you go about changing the $k$ or $v$ or $\omega$ in one string (the unit amplitude one) without changing them in the other? You have to use the same driving source so that the strings are phase matched so you cant vary the frequency of one without changing the other as well. Since they are identical you cannot change the material of one without the other. So you cannot change the denominator of your equation without also changing the numerator simultaneously and canceling out the change.
And finally a short technical answer that will make perfect sense if you are mathematically oriented.
Notice that $y(x,t)=Asin(kx-wt)$ has the $k$ and $w$ in the phase part. So no matter what you do to them, you cannot affect the amplitude part.
Specifically, the ratio of $y$ and $sin(kx-wt)$ is meant to cancel out the phase factor in the $y$ and leave the amplitude. By changing the frequency you only change the phase part that is canceled out anyways.
Best Answer
Yes, Electro, the intensity $I$ - and energy density $T_{00}$ and similar quantities - is proportional to the squared amplitude, $$ I \sim A^2 $$ Because the intensity must go as $$ I \sim \frac{1}{r^2} $$ as the energy gets spread over the sphere of area $4\pi r^2$, it follows that $$ A \sim \frac{1}{r}.$$ See e.g. the $1/r$ factor in this formula: