Subjective Type

Two atmospheric sound sources A and B emit isotropically at constant power. The sound levels $$\beta$$ of their emissions are plotted in Fig. 17-40 versus the radial distance $$r$$ from the sources. The vertical axis scale is set by $$\beta_1 = 85.0 \space dB$$ and $$\beta_2 = 65.0 \space dB$$. What are (a) the ratio of the larger power to the smaller and (b) the sound level difference at $$r = 10 \space m$$?

Solution

We know that, $$\beta = 10 \log \left (\frac{P} {I_0 4 \pi r^{2}} \right )$$.
Taking difference (for sounds A and B) we find
$$\Delta \beta = 10 \log \left (\frac{P_A} {I_0 4 \pi r^{2}} \right ) - 10 \log \left (\frac{P_B} {I_0 4 \pi r^{2}} \right ) = 10 \log \left (\frac{P_A} {P_B} \right )$$
using well-known properties of logarithms.

Thus, we see that $$\Delta \beta$$ is independent of $$r$$ and can be evaluated anywhere.

(a) We can solve the above relation (once we know $$\Delta \beta = 0.5$$) for the ratio of powers : we find $$P_A/P_B \approx 3.2$$.

(b) At $$ r = 1000 \space m$$ it is easily seen (in the graph) that $$\Delta \beta = 5.0 \space dB$$. This is the same $$\Delta \beta$$ we expect to find, then, at $$r = 10\space m$$.

SIMILAR QUESTIONS

Physical World

A point source emits sound waves isotropically. The intensity of the waves 2.50 m from the source is $$1.91 \times 10^{-4} W/m^{2}$$. Assuming that the energy of the waves is conserved , find the power of the source.

Physical World

The sound intensity is $$ 0.0080 \mathrm{W} / \mathrm{m}^{2} $$ at a distance of $$ 10 \mathrm{m} $$ from an isotropic point source of sound. What is the power of the source?

Physical World

A loudspeaker at a rock concert generates $$10^{-2}\ W/m^2$$ at $$20\ m$$ at a frequency of $$1\ kHz$$. Assume that the speaker spreads its energy uniformly in all directions.What is the total acoustic power output of the speaker?

Contact Details