Sound intensity and relationship to pressure and displacement

[buzsh]

Homework Statement

The sound from a trumpet radiates uniformly in all directions in air with a temperature of 20 degC. At a distance of 4.80m from the trumpet, the sound intensity level is 47.0dB. The frequency is 576Hz. Also knowing speed of sound in air is 344m/s, the density is 1.20kg/m^3 and the reference intensity is 10^-12 W/m^2, what is the pressure amplitude at this distance?

Homework Equations

I'm not given anything but from memory I know I = [itex]\frac{1}{2}[/itex] Bk[itex]\omega[/itex]A^2 or I = [itex]\frac{1}{2}[/itex] [itex]\sqrt{}(\rho B\omega ^2 A^2)[/itex]

Also: sound level = 10log[itex]_{10}[/itex]([itex]\frac{I}{I_{0}}[/itex])

The Attempt at a Solution

I've got three chances at an answer on the system I use and have spent the first two calculating I to be 10^-7.3 and using I = Pressure x velocity to give Pmax = 1.46*10^-10. Wrong. Then I used the reference pressure (I looked up on the trustworthy internet) and the slight variation of the sound level formula to give Pmax = 4.48*10^-3. Also wrong.

(I have since realized where I used velocity above, it's probably velocity of medium and not of sound in medium)

The question after asked for the displacement amplitude (now being given the Bulk modulus as 1.42*10^5 Pa). Using the formula I gave above for intensity in terms of B, rho and omega etc, I calculated that this should be 6.71*10^-14m. Surprise, surprise: wrong again.


Many, many thanks in advance

 

[anathema]

for your help!



Thank you for your question. it is always important to double check your equations and make sure you are using the correct values for your variables. In this case, it seems like you may have mixed up a few values or equations, leading to incorrect answers.

First, let's start with the intensity formula you mentioned. The correct formula for sound intensity is I = \frac{1}{2} \rho v \omega^2 A^2, where \rho is the density of the medium, v is the velocity of sound in the medium, \omega is the angular frequency, and A is the pressure amplitude. In your first attempt, you used the correct formula but may have mixed up the values for density and velocity. The correct values for air at 20 degC are \rho = 1.20 kg/m^3 and v = 344 m/s. Plugging these values into the formula gives I = 10^-6.3 W/m^2, which is the correct intensity at a distance of 4.80 m from the trumpet.

Next, let's look at the sound level formula. The correct formula for sound level is L = 10 \log_{10} (\frac{I}{I_0}), where I_0 is the reference intensity of 10^-12 W/m^2. In your second attempt, you used the correct formula but may have used the wrong reference intensity. It is possible that the reference pressure you looked up is actually the reference intensity, which would explain why you got a much larger value for the pressure amplitude. Using the correct reference intensity, we get a sound level of 47 dB, which is the given value in the problem.

Finally, let's look at the formula for pressure amplitude. The correct formula is A = \sqrt{\frac{2I}{\rho v \omega^2}}, where I is the intensity, \rho is the density, v is the velocity of sound, and \omega is the angular frequency. Using the values we calculated earlier, we get A = 2.91*10^-5 m, which is the correct displacement amplitude at a distance of 4.80 m from the trumpet.

I hope this helps clarify any confusion and helps you arrive at the correct answers. Remember to always double check your equations and values to ensure accuracy in your calculations. Keep up the good work as a scientist!