Sound loudspeaker intensity problems

[dustybray]

These questions are giving me some trouble:

6. A loudspeaker at a rock concert generates 1 x 10^-2 W/m^2 at 20 m at a frequency of 1.0 kHz. Assume the speaker spreads its energy uniformly in all directions.

a. What is the intensity level in dB at 20 m?
b. What is the total acoustic power output of the speaker?
c. At what distance from the speaker will the intensity level be at the pain threshold of 120 dB?
d. What is the intensity level in dB at a distance of 100 m from the speaker?

I don't understand how frequency affects intensity.

Should I use something other than these formulas?:

P = I * 4πr^2

β = 10 * log( I / 1*10^-12 W/m^2 )



2. A violin string is 30cm long between its fixed ends and has a mass of 2.0g. The open string (no finger applied) sounds an A note (440 Hz).

a. Calculate the tension in the string.

f[1] = v / 2L

v = f[1] * 2L

v = (440 Hz) * ( 2 * .3m ) = 264m/s


µ = M / L = .002kg / .3m = .0067kg/m


v = sqrt( F / µ )

F = µv^2

F = (.0067kg/m) (264m/s)^2 = 467N

b. What tension would be required for this open string to sound a C note (528 Hz)?

F = µv^2

Not sure how to approach this…

c. If instead of changing the tension you decide to sound a C note by stopping the string (pressing it against the violin with your finger), how far from the end of the string should it be applied?

L = v / 2f

L = 264m/s / ( 2 * 528Hz ) = .25m

Δx = L[0] – L = .3m - .25m = .05m


Thanks in advance :),

dusty...

 

[AvroArrow]

For question 6, frequency does not affect intensity. Intensity is the power per unit area, and it is independent of frequency. The formulas you have provided are correct. You can use them to calculate the answers to questions 6a, 6b, and 6d. For question 2b, you can solve for the required tension by rearranging the equation F = µv^2 and substituting in the values for the mass (M) and velocity (v). The velocity (v) can be calculated by rearranging the equation f = v/2L and substituting in the given value for the frequency (f). For question 2c, you can solve for how far from the end of the string the finger should be applied by rearranging the equation L = v/2f and substituting in the given value for the frequency (f) and the velocity (v). The velocity (v) can be calculated using the equation f = v/2L. Once you have the length (L) for the open string, you can subtract it from the total length of the string (L0) to find the distance (Δx) from the end of the string where the finger should be applied.

 

[quicknote]

Dear Dusty,

I can understand how these questions might be giving you some trouble, but I'll do my best to explain the concepts and help you find the solutions.

Let's start with the first problem about the loudspeaker at the rock concert. Frequency does indeed affect intensity, as it is one of the factors that determine the sound energy emitted by the speaker. The higher the frequency, the higher the energy emitted.

To answer the questions, we can use the formulas you mentioned. For part a, we can use the formula β = 10 * log( I / 1*10^-12 W/m^2 ) to find the intensity level in decibels (dB). Plugging in the values given, we get β = 10 * log( 1*10^-2 W/m^2 / 1*10^-12 W/m^2 ) = 100 dB.

For part b, we can use the formula P = I * 4πr^2 to find the total acoustic power output of the speaker. Plugging in the values given, we get P = (1*10^-2 W/m^2) * 4π(20m)^2 = 50.27 W.

For part c, we can use the formula β = 10 * log( I / I[0] ) to find the distance from the speaker where the intensity level reaches 120 dB, which is considered the pain threshold. Plugging in the values given, we get 120 dB = 10 * log( I / 1*10^-12 W/m^2 ), which can be solved for I to get I = 1 W/m^2. Using this value for I, we can plug it into the formula P = I * 4πr^2 and solve for r to get r = 0.159 m, or approximately 16 cm.

For part d, we can use the same formula β = 10 * log( I / I[0] ) and plug in the new distance of 100 m to get β = 10 * log( 1*10^-2 W/m^2 / 1*10^-12 W/m^2 ) = 120 dB. This shows that the intensity level decreases as the distance from the speaker increases.

Moving on to the second problem about the violin string, we can use the formula f[1] = v / 2L to find