(II) At a painfully loud concert, a 120 -dB sound wave...

(II) At a painfully loud concert, a 120 -dB sound wave travels away from a loudspeaker at 343 $\mathrm{m} / \mathrm{s}$ . How much sound wave energy is contained in each $1.0-\mathrm{cm}^{3}$ volume of air in the region near this loudspeaker?

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Key Concepts

Sound Intensity and the Decibel Scale

Sound intensity quantifies the power carried by sound waves per unit area and is often expressed in decibels (dB), a logarithmic scale that compares the measured intensity with a standard reference intensity. The decibel scale compresses a wide range of sound levels into a manageable range and is used to represent the relative loudness of sounds in a way that approximates human hearing perception.

Relationship Between Intensity and Energy Density

In a propagating sound wave, the energy density—which represents the energy stored per unit volume—is directly related to the intensity of the wave and the speed at which it travels. Specifically, the average energy density can be calculated by dividing the intensity of the sound by its speed, linking how energetic the wave is to its ability to do work over both space and time.

Volume Conversion in Energy Calculations

When calculating the total energy contained in a small volume, it is necessary to convert between standard volume units so that the energy density (expressed in energy per unit volume) can be multiplied by the volume of interest. Understanding unit conversion, such as transforming cubic centimeters to cubic meters, is essential for accurately determining the energy contained within that volume.

Transcript

00:01 We're asked to finger out how much sound energy is in one cubic centimeter of air in a region near the loudspeaker.

00:12 When the sound, when the sound level is at 120 decibels.

00:22 So we have a sound wave.

00:25 And so we need to figure out the intensity.

00:28 Now, we know that the indecibles, the intensity, the intensity, divided by the reference, the natural log of that times 10 is the sound intensity in decibels.

00:41 So we can invert that and get the sound intensity level is the reference times 10 to the beta in the sound intensity in decibels divided by 10.

00:57 This is 10 to the minus 12.

01:00 Beta was 120.

01:03 So this is 10 to the 12.

01:05 So that means we have.

01:05 Have one watt per meter squared.

01:08 So that means we have one watt of power.

01:12 We have this surface area is a meter squared.

01:15 We have one watt of sound power going through that area.

01:21 So that's kind of like a sound flux.

01:24 Now we want to figure out what the energy is.

01:28 Well, we know that this i is the power over going through the surface area.