Intensity Level of 76 trombones

[Coushander]

Homework Statement

If the intensity level at distance d of one trombone is 70 dB, what is the intensity level of 76 identical trombones, all at distance d?

Homework Equations

β = (10 dB) log₁₀(I/I₀)

I₀= 1.0 x 10^-12 W/m²

I = power/area

The Attempt at a Solution

70db = 10db log₁₀(I/I₀)

7db = log₁₀(I/I₀)

10⁷ = 10^{log₁₀(I/I₀)}

10⁷ = (I/I₀)

I = 0.00001 W/m²

Then I multiplied that by 76 and put it back in the general equation to generate β = 98db, which is wrong. β = 89db.

 

[PeterO]

Homework Statement

If the intensity level at distance d of one trombone is 70 dB, what is the intensity level of 76 identical trombones, all at distance d?

Homework Equations

β = (10 dB) log₁₀(I/I₀)

I₀= 1.0 x 10^-12 W/m²

I = power/area

The Attempt at a Solution

70db = 10db log₁₀(I/I₀)

7db = log₁₀(I/I₀)

10⁷ = 10^{log₁₀(I/I₀)}

10⁷ = (I/I₀)

I = 0.00001 W/m²

Then I multiplied that by 76 and put it back in the general equation to generate β = 98db, which is wrong. β = 89db.

Your argument seems sound, but you arithmetic must have gone astray.

For each doubling of intensity, we increase 3 dB [10*log₁₀2 = 3]

1 trombone = 70dB
2 -> 73 dB
4 -> 76 dB
8 -> 79 dB
16 -> 82 dB
32 -> 85 dB
64 -> 88 dB
128 -> 91 dB

so 89 dB seems quite possible

indeed 10*log₁₀76 = 18.8,
so final intensity is 70 + 18.8 = 88.8, or perhaps 89 dB

[must be annoying to see it is that simple to calculate]

 

[Coushander]

You're right, it was an arithmetic error.

Working it through again gave me 88.57dB this time (did it twice to make sure).

 

[PeterO]

You're right, it was an arithmetic error.

Working it through again gave me 88.57dB this time (did it twice to make sure).

You actually should be getting 88.808 dB. Not sure why the discrepecy? your figure is the sound of 72 trombones!