What Are the Highest and Lowest Frequencies Heard Due to the Doppler Effect?

[ramin86]

A block with a speaker bolted to it is connected to a spring having spring constant k = 21.0 N/m, as in Figure P17.40. The total mass of the block and speaker is 46.00 kg, and the amplitude of this unit's motion is 0.475 m. Assume that the speed of sound is 343 m/s.

Figure P17.40 (http://www.webassign.net/pse/p17-38.gif)

(a) If the speaker emits sound waves of frequency 470 Hz, determine the highest and lowest frequencies heard by the person to the right of the speaker to the nearest 0.01 Hz.
Hz (highest)
Hz (lowest)
(b) If the maximum sound level heard by the person is 66.0 dB when he is closest to the speaker, 1.00 m away, what is the minimum sound level heard by the observer?
dB

Was thinking you would have to use f'=(v -v0 / v) but that won't work. Not sure how to fit in the given values such as the spring constant and the amplitude. Please help!

 

[Gamma]

In the doppler shift equation, you need the velocity of the source (speaker). The source under goes SHM and you need the information given such as k, m to find velocity.

The displacement of the speaker is given by x = A sin(wt) where A is the amplidtude (given) and w = sqrt (k/m).

Find the velocity by differentiating the above.

V= Aw cos(wt). Listner will hear minimum frequency when the speaker moves away from him and maximum frequency when it moves towards him. Speaker's maximum speed is Aw at the equillibrium point O.

A....O.....B * (Listner)

Min frequency f- = f c/ (c + v)
Max frequency f+ = f c / (c-v)

 

[thepatientmental]

To solve this problem, we first need to understand the concept of the Doppler effect. The Doppler effect is the change in frequency of a wave as the source or observer moves. In this case, the source of the sound waves is the speaker attached to the block, and the observer is the person to the right of the speaker. As the block and speaker move back and forth due to the spring's oscillations, the frequency of the sound waves emitted by the speaker will change, resulting in a different frequency heard by the observer.

To determine the highest and lowest frequencies heard by the observer, we can use the formula for the Doppler effect: f’ = f (v± vo)/(v± vs), where f is the original frequency, v is the speed of sound, vo is the velocity of the observer, and vs is the velocity of the source.

In this problem, the observer is stationary, so vo = 0. The block and speaker are moving back and forth with an amplitude of 0.475 m, so the maximum velocity of the source (vs) will be equal to the speed of sound multiplied by the ratio of the amplitude to the period of oscillation (T = 2π√(m/k)). Therefore, vs = 343 * (0.475 / (2π√(46.00/21.0))) = 3.37 m/s.

Now, we can plug in the given values to the formula to find the highest and lowest frequencies heard by the observer:
f’ = 470 (343 + 3.37) / (343 + 0) = 470.49 Hz (highest frequency)
f’ = 470 (343 - 3.37) / (343 + 0) = 469.51 Hz (lowest frequency)

Therefore, the highest frequency heard by the observer will be 470.49 Hz and the lowest frequency will be 469.51 Hz.

For part (b), we can use the formula for sound level in decibels (dB): L = 20 log (P/P0), where P is the sound pressure level and P0 is the reference sound pressure level (usually taken as 20 micropascals). We know that the maximum sound level heard by the observer is 66.0 dB when he is 1.00 m away from the speaker. So, we can use this value to find the sound pressure level at