Calculating the Itensity of a band

[roldy]

Homework Statement

An acoustic signal consists of three tones, each of different frequency and different effective pressure amplitude: $$P_1=5\times10^{-2}$$ Pa,$$P_2=7\times10^{-2}$$ Pa,$$P_3=.1$$ Pa, and $$f_1=104$$ Hz,$$f_2=190$$ Hz,$$f_3=237$$ Hz. Find the intensity in the 100 to 110 Hz band.

Homework Equations

Sound Pressure Level=SPL=$$20log\frac{P_e}{P_0}, P_0=2\times10^{-5}Pa$$

Sound Intensity Level=SIL$$10log\frac{I}{I_0}, I_0=10^{-12) Watts/m^2$$

Intensity Spectrum=S=[tex]\frac{\DeltaI}{\Deltaf}

Bandwidth=w=$$f_{upper}-f_{lower}$$

The Attempt at a Solution

I think I could figure this out if I knew how the frequencies came into play. To find the intensity in the band, I would multiply the Intensity Spectrum (S) by the bandwidth. How do I obtain the Intensity Spectrum from the sound pressures and frequencies given?

 

[mark726]

To find the intensity in the 100 to 110 Hz band, we first need to calculate the sound intensity level (SIL) for each of the three tones using the given equations. Then, we can use the bandwidth formula to determine the intensity in the specific frequency range.

Using the given sound pressure levels and frequencies, we can calculate the SIL for each tone as follows:

SIL_1=10log\frac{P_1^2}{P_0^2}=10log\frac{(5\times10^{-2})^2}{(2\times10^{-5})^2}=70 dB

SIL_2=10log\frac{P_2^2}{P_0^2}=10log\frac{(7\times10^{-2})^2}{(2\times10^{-5})^2}=74 dB

SIL_3=10log\frac{P_3^2}{P_0^2}=10log\frac{(0.1)^2}{(2\times10^{-5})^2}=80 dB

Next, we can calculate the intensity spectrum (S) for each tone using the given equation:

S_1=\frac{\DeltaI_1}{\Deltaf_1}=\frac{10^{-12}}{104}=9.62\times10^{-15} W/m^2

S_2=\frac{\DeltaI_2}{\Deltaf_2}=\frac{10^{-12}}{190}=5.26\times10^{-15} W/m^2

S_3=\frac{\DeltaI_3}{\Deltaf_3}=\frac{10^{-12}}{237}=4.22\times10^{-15} W/m^2

Now, to find the intensity in the 100 to 110 Hz band, we need to calculate the bandwidth (w) for this frequency range:

w=110-100=10 Hz

Finally, we can use the formula for bandwidth to calculate the intensity in the 100 to 110 Hz band as follows:

Intensity=\sum_iS_iw=\sum_i\frac{\DeltaI_i}{\Deltaf_i}w=\frac{\DeltaI_1}{\Deltaf_1}w+\frac{\DeltaI_2}{\Deltaf_2}w+\frac{\DeltaI_3}{\Deltaf