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Two 100.0-W speakers, A and B, are separated by a distanceD = 3.6 m. The speakers emit in-...

Two 100.0-W speakers, A and B, are separated by a distanceD = 3.6 m. The speakers emit in-phase sound

waves at a frequency f  = 10,000.0 Hz. PointPl is located at x1 = 4.50 m andy1 = 0 m; pointP2 is located at  x2 = 4.50 m andy2 = -Δy. Neglecting speaker B, what is the intensity, IA1(in W/m2), of the sound at pointP1 due to speaker A? Assume that the sound from the speaker is emitted uniformly in all directions. What is this intensity in terms of decibels (sound level,βA1 )? When both speakers are turned on, there is a maximum in their combined intensities atP1 As one moves towardP2, this intensity reaches a single minimum and then becomes maximized again atP2. How far isP2from  P1, that is, what is Δy? You may assume that L ≫Δy and thatD≫ Δy, which will allow you to simplify

the algebra by using  whena≫b.

Step-by-Step Solution

Solution 1
Step #1 of 8

THINK:

Intensity of the sound wave at a given point,

Distance between the first maxima to central maxima of sound intensity,

Here, power delivered by the speakers,

Distance between speakers to point,

Distance between the speakers,

Co-ordinates of this point are: and

Frequency of the wave,

Step #2 of 8

SKETCH:

Below sketch shows the interference of the two sound waves emitting by a speakers formed first maxima at point

Step #3 of 8

RESEARCH:

Sound intensity at point is,

…… (1)

The level of the sound at this point,

……. (2)

Further, the calculation of:

Let, and and

Step #4 of 8

From the geometry of the figure,

Similarly,

Step #5 of 8
Assume, and,

The value of will be larger than the value of, so

So we can use the Malaren series to estimate the value of, and

By the three-term Malaren approximations is,

Hence, the difference between the distances

Step #6 of 8

But, difference between the distances is,

……. (3)

Step #7 of 8

CALCULATE:

Using equation (1),

Intensity of the sound at point is,

Using equation (2),

level of the sound at the point is,

Step #8 of 8

Distance between the first maxima from the central maxima is,

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