Two small loudspeakers emit sound waves of different...

Name: Problem 72, Chapter 14: College Physics
Uploaded: 2020-06-16T19:43:45-08:00
Duration: 5 min 7 s
Channel: Patrick Connors
Description: Two small loudspeakers emit sound waves of different frequencies equally in all directions. Speaker A has an output of 1.00 mW, and speaker B has an output of 1.50 mW. Determine the sound level (in decibels) at point C in Figure P14.72 assuming (a) only speaker A emits sound, (b) only speaker B emits sound, and (c) both speakers emit sound.

Two small loudspeakers emit sound waves of different frequencies equally in all directions. Speaker A has an output of 1.00 mW, and speaker B has an output of 1.50 mW. Determine the sound level (in decibels) at point C in Figure P14.72 assuming (a) only speaker A emits sound, (b) only speaker B emits sound, and (c) both speakers emit sound.

Key Concepts

Decibel Scale

The decibel (dB) is a logarithmic unit used to express the ratio of a given sound intensity to a reference intensity (commonly the threshold of hearing). It is calculated using the formula L = 10 log??(I/I?), where I is the sound intensity and I? is the reference intensity. This scale conveniently allows very large ranges of sound intensity to be represented by a manageable range of decibel values.

Sound Power and Intensity

Sound power is the total amount of energy emitted by a sound source per unit time, while sound intensity is the power per unit area that is received by a surface. In problems involving isotropic radiators (sources that emit sound equally in all directions), the intensity at a given point can be determined by dividing the power by the surface area of a sphere centered on the source, thereby linking the two concepts.

Incoherent Sound Addition

When multiple independent sound sources are present, their contributions to the overall sound field are added at the level of intensities because they typically do not maintain a fixed phase relationship. This means that the total sound intensity is the sum of the individual intensities, even though the corresponding sound levels in decibels must be calculated using the logarithmic formula after the intensities have been summed.

Transcript

00:01 All right, so in this problem, we have two speakers, a and b, that are both creating sound waves.

00:09 A is creating sound waves with a power of one milliwatt, which is 0 .001 watts, and b is producing a power of 15 milliwats or 0 .0015 watts.

00:27 Now we have an observer who is at point c.

00:32 And this is where we get into some geometry here because we know the dimensions of how these things are set up.

00:40 This distance here is four meters.

00:43 Distance from here is two meters from b to that middle point and from a to that middle point is three meters.

00:50 Now if you weren't sleeping in geometry class, you'll see that this distance from a to c is a right triangle and specifically it's a three, four, five triangle.

01:00 So we can get that without doing any work for it.

01:04 From b to c here, we have two, then four.

01:08 For this one, i'm going to use pythagorean theorem.

01:11 If you do the square root of two squared plus four squared, we end up with that side being 4 .5 meters long.

01:19 Now, those values matter, because when we're talking about sound, what matters is how far away is the observer from the source of the sound, which i now have as 5 meters and 4 .5 meters.

01:32 So we're here where i'm collecting my information.

01:34 The radius from a is 5 meters away.

01:37 The radius from b is 4 .5 meters away.

01:42 Now, our goal here is to get the decibels for how many decibels are there from a from b and in total when they put together.

01:49 Our decibel equation is that the intensity in decibels is equal to 10 times the logarithm of our intensity in watts per meter squared divided by the intensity at the threshold of hearing, which that is a constant value.

02:06 Remember back to earlier, this is 10 to the negative 12 watts per meter squared.

02:10 So we need to figure out this intensity.

02:13 And the intensity is from our equation that the intensity of a sound wave is equal to the power of that wave divided by the surface area is being spread out over.

02:24 Now we know the power for both these speakers...

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