Describe the wave equation and the pressure wave given that $T=263 \text{ K}$, $\rho=1.34 \text{ kg/m^3}$, $f=200 \text{ Hz}$. It is known that $500 \text{ m}$ away, the sound intensity level $\beta=43 \text{ dB}$. The source is a point source and the waves can be seen as spherical. The medium is air.

My attempt:

The waves are spherical. This means that the wave equation can be written as $s(r,t)=\frac{A_s}{r}cos(kr-\omega t)$. Using the given data and that the medium is air, $v=\sqrt{\frac{\gamma R T}{M}}=106 *10^3 \text{ m/s}$, $k=\frac{2\pi f}{v}=0.011 \text{1/m}$, $\omega=2\pi f=1256 \text{1/s}$. Using the sound intensity formula $\beta=10 db*log(\frac{I}{I_0})$ we get that $I=1.996*10^{-8} \text{W/m^2}$ and using $\beta=20 db*log(\frac{p}{p_0})$ we get $p=0.0028 \text{ Pa}$. My question is how to proceed from here.

My initial thought was to use the relationships $I=\frac{p_{max}^2}{2\rho v}$, $p_max=BkA$, $v=\sqrt\frac{B}{\rho}$ where this $A$ would equal $\frac{A_s}{r}$ at $r=500$. But I don't know whether those formulas are valid for spherical waves or not.

My other thought was to use the pressure wave equation, which I think can be written as $p(r,t)=\frac{A_p}{r}cos(kr-\omega t)$. Then use that $p=0.0028 \text{ Pa} = \frac{A_p}{r}$ at $r=500$. But I don't know whether that is the correct form of the pressure wave equation or whether $p=0.0028 \text{ Pa} = \frac{A_p}{r}$ is correct.

Any help on how to move forward?



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