How Does a Moving Speaker Affect Sound Frequency Perception?

[songoku]

Homework Statement

A block with a speaker bolted to right side of it is put on the table. The left side of the block is connected to a spring having spring constant k and the block is free to oscillate in horizontal direction. The total mass of the block and speaker is m, and the amplitude of this unit's motion is A. The speaker emits sound waves of frequency f and speed of sound is v

a. Determine the highest frequency heard by the person
b. Determine the lowest frequency heard by the person
c. If the maximum sound level heard by the person is β when the speaker is at its closest distance d from him, what is the minimum sound level heard by the person?

Homework Equations

maybe:
Doppler
T = 2π √(m/k)

The Attempt at a Solution

a. The highest frequency is when the speaker is the closest to the person. I am thinking using Doppler to find the frequency:
$$f_2=\frac{v±v_o}{v±v_s}f_1$$

But the speed of the speaker is not constant so I don't think Doppler can be used.

The maximum speed is Aω = A√(k/m), but I am stuck...

b. Don't know

 

[frogjg2003]

Well, it's asking for the maximum and minimum frequencies, so you only need the maximum and minimum speeds, which are ±Aω. That's parts a and b.
I'm assuming that sound level refers to the value measured in dB. That means that the sound level is given by $L=20 \log_{10}\left(\frac{p}{p_0}\right)$ which can be rearrange to be L=20log₁₀(p)-β. Here p and p₀ are the pressures at the two points. The pressure is inversely proportional to distance, so you can use p₀=λ/d and p=λ/(d+Δd) get L as a function of β, d, and Δd. What is Δd?

 

[songoku]

Well, it's asking for the maximum and minimum frequencies, so you only need the maximum and minimum speeds, which are ±Aω. That's parts a and b.
I'm assuming that sound level refers to the value measured in dB. That means that the sound level is given by $L=20 \log_{10}\left(\frac{p}{p_0}\right)$ which can be rearrange to be L=20log₁₀(p)-β. Here p and p₀ are the pressures at the two points. The pressure is inversely proportional to distance, so you can use p₀=λ/d and p=λ/(d+Δd) get L as a function of β, d, and Δd. What is Δd?

I have never encountered the formula p₀=λ/d. λ is the wavelength of the sound wave, and what is d?

 

[frogjg2003]

Oh, i used λ as an arbitrary constant. When you manipulate the math, it will be remove from the equations. d is the distance you were given in the diagram.

 

[songoku]

Oh, i used λ as an arbitrary constant. When you manipulate the math, it will be remove from the equations. d is the distance you were given in the diagram.

Actually I still don't really understand the formula and the idea but let me try:
p₀=λ/d and p=λ/(d+Δd), where Δd equals to 2A + d

L = 20 log (P/P₀)
= 20 log P - 20 log p₀
= 20 log λ - 20 log d - 20 log λ + 20 log (2d + 2A)
= 20 log 2 [(d + A)/d]

There is no β term

or

L = 20 log (P/P₀)
= 20 log P - 20 log p₀
= 20 log λ - 20 log d - β

The term λ does not cancel out

Please help

 

[songoku]

I did another work and got answer like this:

$$minimum~sound~level = β + 20 log (\frac{d}{2A+d})$$

Is this the correct answer?

 

[frogjg2003]

That looks right.

 

[songoku]

OK thanks a lot for your help