Question

To understand the concept of intensity; the relationship between the power of the source and the intensity of the wave; and the dependence of intensity on distance.Since waves transfer energy from one point to another, one can define the power of a wave as the rate at which the wave transports energy. The intensity of a wave, in contrast, is the power relative to a certain surface. Consider a wave traveling across a surface perpendicular to the direction of propagation. The intensity Iof the wave is defined as the ratio of the power P of the wave to the area A of that surface:I=PA.Note that the surface may be real (physical, like an eardrum or a windowpane) or mathematical. Quite frequently, we will be interested in the intensity produced by a relatively small source at a relatively large distance. If the source emits waves uniformly in all possible directions (produces spherical waves), the formula given here makes it possible to find the intensity at a distance r from the source:I=P/4?r^2.Note that, in all parts of this problem, assume that the source generates spherical waves, so that this intensity formula is applicable.Intensity is measured in watts per square meter (W/m^2). All the information presented here is pertinent to any kind of wave. In this problem, we will be focusing on sound waves.A popular car stereo has four speakers, each rated at 60 W. In answering the following questions, assume that the speakers produce sound at their maximum power.A:Find the intensity I of the sound waves produced by one 60-Wspeaker at a distance of 1.0 m.Express your answer numerically in watts per square meter. Use two significant figures.B:Find the intensity I of the sound waves produced by one 60-Wspeaker at a distance of 1.5 m.Express your answer numerically in watts per square meter. Use two significant figures.C:Find the intensity I of the sound waves produced by four 60-Wspeakers as heard by the driver. Assume that the driver is located 1.0 m from each of the two front speakers and 1.5 m from each of the two rear speakers.Express your answer numerically in watts per meter squared.The threshold of hearing is defined as the minimum discernible intensity of the sound. It is approximately 10^?12W/m^2. Find the distance d from the car at which the sound from the stereo can still be discerned. Assume that the windows are rolled down and that each speaker actually produces 0.06 W of sound, as suggested in the last follow-up comment.

Answer 1

a. 4.77 W/
`m^(2)`

b. 2.122 W/
`m^(2)`

c. 13.78 W/
`m^(2)`

d. 69 km

Step-by-step explanation:

a.

power P of speaker = 60 W

distance d of speaker = 1.0 m

intensity I of speaker = P/4π
`d^(2)`

I = 60/(4 x 3.142 x
`1^(2)`) = 60/12.568

Intensity = 4.77 W/
`m^(2)`

b.

at a distance of 1.5 m, intensity will be

I = P/4π
`d^(2)`

I = 60/(4 x 3.142 x
`1.5^(2)`) = 60/28.278

Intensity = 2.122 W/
`m^(2)`

c.

combined intensity of the two front speakers will be 2 x 4.77 = 9.54 W/
`m^(2)`

combined intensity of the two back speaker will be 2 x 2.122 = 4.244 W/
`m^(2)`

total sound intensity perceived by the driver will be the superimposition of these four speakers.

I = 9.54 + 4.244 = 13.78 W/
`m^(2)`

d.

Minimum discernible intensity of sound =
`10^(-12)` W/
`m^(2)`

each speaker produces 0.06 W of power.

We assume that each speaker spreads outward evenly.

from I = P/4π
`d^(2)`,

d =
`\sqrt{(P)/(4*\pi*I ) }`

d =
`\sqrt{\frac{0.06} {4*\pi*10^(-12 ) }` = 69094.35 m ≅ 69 km