An underwater microphone is used to record sounds emitted by porpoises. The minimum intensity level the instrument can record is 10 dB. Assuming a porpoise emits sound at a rate of 0.05 J/s, what will be the max distance at which the animal will still be recorded? Neglect sound absorption in water, and treat the porpoise as a point sound source.

Question

An underwater microphone is used to record sounds emitted by porpoises. The minimum intensity level the instrument can record is 10 dB. Assuming a porpoise emits sound at a rate of 0.05 J/s, what will be the max distance at which the animal will still be recorded? Neglect sound absorption in water, and treat the porpoise as a point sound source.

..i am very unsure of how to go about answering this question, any help would be great.

Answer 1

To determine the maximum distance at which the porpoise's sound will still be recorded by the underwater microphone, we need to consider the relationship between intensity level, distance, and power.

The intensity level (IL) is measured in decibels (dB) and is given by the formula:

IL = 10 * log10(I / I₀)

where I is the intensity of the sound wave and I₀ is the reference intensity (usually the threshold of human hearing, which is approximately 1 x 10^(-12) W/m^2).

In this case, the minimum intensity level the microphone can record is 10 dB. We need to find the corresponding intensity (I) that corresponds to this intensity level.

Therefore, we can rearrange the formula to solve for I:

I = I₀ * 10^(IL / 10)

Substituting the given values:

I = (1 x 10^(-12) W/m^2) * 10^(10 dB / 10)
I = 1 x 10^(-12) W/m^2 * 10^1

I = 1 x 10^(-12) W/m^2 * 10

I = 1 x 10^(-11) W/m^2

Now, let's consider the power (P) of the sound emitted by the porpoise, which is given as 0.05 J/s. This power is spread over a spherical surface area as the sound waves propagate outward.

Therefore, we can use the following equation to relate the intensity (I), power (P), and distance (r):

I = P / (4πr^2)

Where:
- I is the intensity,
- P is the power, and
- r is the distance from the source.

Rearranging the equation to solve for the distance (r):

r = √(P / (4πI))

Substituting the given values:

r = √(0.05 J/s / (4π * 1 x 10^(-11) W/m^2))

Calculating the value:

r ≈ √(0.05 / (12.57 x 1 x 10^(-11)))

r ≈ √(3.98 x 10^9)

r ≈ 63,090 meters

Therefore, the maximum distance at which the animal's sound will still be recorded by the underwater microphone is approximately 63,090 meters.