2. A compact sound source radiates 300 W of sound energy uniformly in all direct

Question

2. A compact sound source radiates 300 W of sound energy uniformly in all directions. Assuming no energy is lost, what is the ratio of the sound intensity at a distance of 1.0 m to that at 5.0 m in a. Our th ree dimensional universe? b. A hypothetical two dimensional universe? i. What is an example of a real wave that this would apply to? Explain why. A hypothetical single dimensional universe? c. i. What is an example of a real wave that this would apply to? Explain why. A hypothetical four dimensional universe? d. e. A hvpothetical th dimensional universe? i. Describe in words how an experiment would detect an nth wave, using what we have discussed above.

Explanation / Answer

Sound Intensity: I = P/ A Where, P= Power & A= Area

Given:

P= 300 W

Now area would differ for all cases.

Also with different dimensions (2D, 3D & SO on) , the intensity of sound waves decreases with the increase in distance. In case of 2-D energy is conserved as compared to 3-D as area of 2-D is lesser than that of 3-D & so on.

a) For 3D case, lets consider our Universe as a sphere.

So Area of Sphere with Radius 1m & 5m would be:

A=4/3 r2

For, r= 1m, A=12.57 m2

r= 5m, A=314.16 m2

Thus, I1= 300/12.57 = 23.867 W/m2

I5= 300/314.16 = 0.9549 W/m2, Ratio: I1/ I5 = 24.994

b) For 2-D case, let's consider a circle.

So area of Circle with r = 1m & r = 5m would be,

A= r2

For r=1m, A= 3.14 m2

r=5m, A=78.54 m2

Thus: I1 = 300/3.14 = 95.54 W/ m2 & I5 = 300/78.54 = 3.819 W/ m2

Thus, Ratio: I1/ I5 = 25.02

i) Example: Water wave: When we throw a stone in water it creates circular waves,

c) For 1-D Case, Area can't be calculated as there is only 1 dimension. So taking length for the intensity we have,

I1= 300/1 = 300 W/m & I5 = 300/5 = 60 W/m

Thus, Ratio: I1/I5 = 300/60 = 5

i) Example: Wave created by Oscillating Spring ( Longitudinal Wave). It moves in one Dimension.

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