By measurement you determine that sound waves are spreading...

By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m$^2$ at a distance of 4.3 m from the source. (a) What is the intensity at a distance of 3.1 m from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?

Key Concepts

Inverse Square Law

The inverse square law states that the intensity of a physical quantity, like sound or light, emitted by a point source decreases in proportion to the square of the distance from the source. This law is vital for determining how the intensity of a wave changes when the distance from its source is altered.

Sound Intensity

Sound intensity refers to the power per unit area carried by a sound wave, typically measured in watts per square meter. It describes how much energy passes through a specified area and is fundamental in assessing sound levels at various distances from the source.

Power and Energy

Power is the rate at which energy is emitted or transferred, measured in watts, while energy represents the total amount of work done over time. Understanding the relationship between power and energy, including the necessary unit conversions (such as converting hours to seconds), is essential for calculating total energy output from a constant power source.

Transcript

00:01 So there are sound spreading out equally in all directions from a point source.

00:06 So we have intensity of the sound waves, 0 .026 watts per meter squared.

00:14 And we know that the distance r is 4 .3 meters from the source.

00:20 And i'm just going to go ahead and call this i1 and r1, since we're going to be talking about different intensities and different distances later.

00:28 And recall the main equation for power intensity, which is this.

00:39 I is equal to p over 4 pi r squared.

00:43 And i really hope to you this is more than just an equation and that this is intuitively understandable to you.

00:49 What this is, it means we have a power that is spreading out over a surface area because remember 4 pi r squared is the surface area of a sphere.

00:57 So it describes is some power spreading out, over a sphere equally in all directions.

01:06 And so if you take any distance, it's going to be smeared out over an area.

01:11 So it's going to, you take the whole power and divide by the surface area to get the intensity.

01:16 So we want to know basically what the intensity is at 3 .1 meters away.

01:24 So i2 is equal to what, given that r2 is equal to 3 .1 meters.

01:32 So power is is like energy per unit time, and that is conserved.

01:37 So wherever you are on the sphere, whether you're this far away or this far away or this far away, the power is going to be equal the same.

01:44 So we can say that p is equal to 4 pi r squared times i.

01:52 And so everywhere the power is the same.

01:55 So that means 4 pi r1 squared is equal to 4 pi r2 squared.

02:02 Oops, forgot the i...

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