# Spherical Waves: Dependence of Intensity and Amplitude on Radius

This explanation is meant to clarify the reason why, in a spherical wave, intensity varies as $\frac{1}{r^2},$ (intensity decreases in proportion to the square of the distance from the source), and displacement amplitude varies as $\frac{1}{r}$ (Serway and Beichner 528-529). The latter fact is the main difference between equation 17.2 (Serway and Beichner 523) and equation 17.9 (Serway and Beichner 529): amplitude is not constant in the latter case, since area is changing. To show the relation for intensity, we have: \begin{align} I =\frac{\mathscr{P}_{\mathrm{av}}}{4\pi r^2} &\Rightarrow I_1 =\frac{\mathscr{P}_{\mathrm{av}}}{4\pi r_1^2}\\ I_2 =\frac{\mathscr{P}_{\mathrm{av}}}{4\pi r_2^2} &\Rightarrow\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\\ &\Rightarrow I_2=\frac{r_1^2}{r_2^2} I_1 = k\frac{1}{r_2^2} \end{align} The last equation $I_2=k\frac{1}{r_2^2}$ states that intensity is proportional to $\frac{1}{r^2},$ the desired result.

The key to seeing the validity of this derivation is the statement immediately following equation 17.8 (Serway and Beichner 528), which states that $\mathscr{P}_{\mathrm{av}}$ is the same for any spherical surface centered at the source. This is important because normally, power depends on area as seen by its formula $\mathscr{P} =\frac{1}{2}\rho Av \left(\omega s_{\mathrm{max}}\right)^2$ (Serway and Beichner 526), and hence on $r,$ so the implications above would not then follow.

To show the result for amplitude, we have: \begin{align} \frac{1}{2} \rho v(\omega s_\mathrm{max})^2 =\frac{\mathscr{P}_{av}}{4\pi r^2} \Rightarrow s_\mathrm{max} =\sqrt{\frac{2\mathscr{P}_{av}}{4\pi r^2\rho v\omega^2}} =\frac{1}{r}\sqrt{\frac{2P_{av}}{4\pi \rho v \omega^2}} =k\frac{1}{r} \end{align} Note that if we used $\mathscr{P}$ instead of $\mathscr{P}_{\mathrm{av}},$ then this implication would not hold, because the last equation reduces to an identity since power in the numerator also varies with the area $A=4\pi r^2$ as does the denominator. \begin{align} s_{\mathrm{max}} =\sqrt{\frac{2\mathscr{P}}{4\pi r^2\rho v\omega^2}} =\sqrt{\frac{2 \left( \frac{1}{2}4\pi r^2\rho v\omega^2 s_{\mathrm{max}}^2 \right)} {4\pi r^2\rho v\omega^2}} =s_{\mathrm{max}} \end{align} Obviously, this equation no longer shows any dependence on $r$ because the areas cancelled out. Another way to show the dependence of amplitude on radius is by using the result $\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}.$ Then we obtain \begin{align} \left. \begin{matrix} I_1&=\frac{1}{2}\rho v\left(\omega s_1\right)^2\\ I_2&=\frac{1}{2}\rho v\left(\omega s_2\right)^2\\ \end{matrix} \right\} &\Rightarrow\frac{I_1}{I_2}=\frac{\omega s_1^2}{\omega s_2^2}\\ &\Rightarrow\frac{s_1^2}{s_2^2}=\frac{r_2^2}{r_1^2}\\ &\Rightarrow s_2 =\sqrt{s_1^2\frac{r_1^2}{r_2^2}} =\frac{s_1 r_1}{r_2} =\frac{s_0}{r_2} =k\frac{1}{r_2} \end{align} The last equation, $s_2=k\frac{1}{r_2},$ is the desired result. We also see that indeed, as stated after equation 17.9 (Serway and Beichner 529), that $s_0=s_1r_1$ is the displacement amplitude at unit distance from the source, assuming we take $s_1$ to be such amplitude.

To summarize, the results above for a spherical sound wave combined with the results on p. 526 for any period sound wave, give the following results for a spherical sound wave: \begin{align} I &\propto\omega^2\\ I &\propto s_{\mathrm{max}}^2\\ I &\propto\frac{1}{r^2}\\ s_{\mathrm{max}} &\propto\frac{1}{r} \end{align}

An open question still remains, how were we justified to assume $\omega$ did not depend on $r$? This assumption is implicit in our use of a single angular frequency $\omega$ but two different amplitudes $s_1$ and $s_2.$ If we assumed different angular frequencies, they would not have cancelled, and worse, if they depend on $r,$ then this would change the relation between amplitude and radius.

## Works Cited

Serway, Raymond A. and Robert J. Beichner. Physics for Scientists and Engineers. 5th Edition. Brooks/Cole, 2000. Print.