A speaker which produces a frequency of 312 Hz is moving away from an observer and towards a wall at 3 m/s when it is 30 degrees Celsius. What is the frequency of the sound that travels directly to the observer? What is the frequency of the sound that bounces off the wall and then gets to the observer? How many beats per second are heard by the observer?

FIRST QUESTION: f2=312(350/350-3)
SECOND QUESTION: f2=312(350/350+3)
THIRD QUESTION: SECOND QUESTION - FIRST QUESTION

Is that right?

all correct.

Solution

To answer these questions, we need to apply the Doppler effect equation, which describes how the frequency of a wave changes when the source or observer is in motion relative to each other. The equation for the Doppler effect is as follows:

f2 = f1 * (v + vo) / (v + vs)

Where:
f2 = frequency heard by the observer
f1 = frequency emitted by the source
v = speed of sound in air (approximately 343 m/s at 30 degrees Celsius)
vo = speed of the observer
vs = speed of the source

Now let's calculate the answers:

First question - frequency of the sound that travels directly to the observer:
Given: f1 = 312 Hz, vo = 0 (since the observer is not moving), v = 343 m/s
Plugging the values into the equation:
f2 = 312 * (343 + 0) / (343 + 0) = 312 Hz

So, the frequency of the sound that travels directly to the observer is 312 Hz.

Second question - frequency of the sound that bounces off the wall and then gets to the observer:
Given: f1 = 312 Hz, vs = 3 m/s, v = 343 m/s
Plugging the values into the equation:
f2 = 312 * (343 + 0) / (343 + 3) = 305.75 Hz

So, the frequency of the sound that bounces off the wall and then gets to the observer is approximately 305.75 Hz.

Third question - the number of beats per second heard by the observer:
To calculate beats per second, we subtract the frequency of the first question from the frequency of the second question:

Beats per second = f2 (Second question) - f2 (First question)
= 305.75 Hz - 312 Hz
= -6.25 Hz (Note that the negative sign indicates that the frequencies are slightly lower than each other, resulting in beats.)

So, the observer would hear approximately 6.25 beats per second.

In conclusion:
- The frequency of the sound that travels directly to the observer is 312 Hz.
- The frequency of the sound that bounces off the wall and then gets to the observer is approximately 305.75 Hz.
- The observer would hear approximately 6.25 beats per second.