Physics Bloopers: Sound Level Problem Gone Wrong

[keydetpiper]

In my high school physics class the other day the teacher botched an example problem involving sound level (decibels).

Homework Statement

A single sound source produces 105 dB at a distance of 5.0 m. How far away must the observer be to not hear this sound at all?

Homework Equations

The Attempt at a Solution

The reference intensity I₀ is the lowest intensity a human can hear, 1 x 10^-12 W/m², so when the source is far enough away its intensity will be equal to this. The intensity of a sound falls off as 1/d², so after substituting the information in the problem the equation above can be rewritten as follows:

In this equation, d₀ represents the distance to the sound source that produces the reference intensity and d is the distance to the original source, 5.0 m for this problem.
When this is solved for d₀, the result is 177828d, or about 890 km. This can't be right! What happened? Thanks in advance for your help.

 

[Steve4Physics]

EDIT: Whoops. This is a really old question. I answered it by mistake thinking it was recent.

In my high school physics class the other day the teacher botched an example problem involving sound level (decibels).

Homework Statement

A single sound source produces 105 dB at a distance of 5.0 m. How far away must the observer be to not hear this sound at all?

Homework Equations

View attachment 126674

The Attempt at a Solution

The reference intensity I₀ is the lowest intensity a human can hear, 1 x 10^-12 W/m², so when the source is far enough away its intensity will be equal to this. The intensity of a sound falls off as 1/d², so after substituting the information in the problem the equation above can be rewritten as follows:
View attachment 126675
In this equation, d₀ represents the distance to the sound source that produces the reference intensity and d is the distance to the original source, 5.0 m for this problem.
When this is solved for d₀, the result is 177828d, or about 890 km. This can't be right! What happened? Thanks in advance for your help.

$$\text{β = 10log(}\frac{d₀²}{d²})$$ is unhelpful in this question.

The formula only applies to relative (not absolute) dB values. E.g. if d₀ = 10d, then β = 10log(10²) = 20dB.

That means there is a 20dB *difference* between levels at the two distances. The absolute levels (referenced to 10⁻¹²W/m²) could be 100dB and 80dB, or 20dB and 0dB for example, depending on ther source's power output.

The formula contains no information about the absolute threshold level (10⁻¹²W/m²). For example if the threshold level were changed to, say, 10⁻¹¹W/m², the formula would give the same result, which makes no sense.

I’d take this approach:

Use 105 = 10log(I/10⁻¹²) to work out I/10⁻¹².

Apply the inverse square law:$$\frac{d₀²}{5²} = \frac{I}{10⁻¹²}$$to find d₀.