Final answer:
To find the wavelength of a sound wave received by an observer moving towards a stationary siren, we first determine the observed frequency using the Doppler Effect formula, and then calculate the wavelength using the speed of sound and this frequency. The correct answer, when rounded to three significant figures, is 0.872 m.
Step-by-step explanation:
The question involves the concept of the Doppler Effect, which is a change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In this case, we are looking for the wavelength of a sound wave received by the observer who is moving towards the source.
To find the wavelength received by the observer, we first need to calculate the observed frequency using the Doppler Effect formula: f' = f(v + vo) / v, where:
f' is the observed frequency
f is the source frequency
v is the speed of sound in the medium
vo is the speed of the observer towards the source
For the given values:
f = 390 Hz (source frequency)
v = 345 m/s (speed of sound in air)
vo = 5.0 m/s (speed of the observer)
Plugging in the values:
f' = 390 Hz (345 m/s + 5.0 m/s) / 345 m/s
f' = 390 Hz * 350 m/s / 345 m/s
f' = 390 Hz * (350/345)
f' = 392.75 Hz (observed frequency)
Now, we use the relationship between the speed of sound, frequency, and wavelength: v = f' × λ, where λ is the wavelength. We can rearrange this to solve for wavelength: λ = v / f'.
λ = 345 m/s / 392.75 Hz λ = 0.8784 m
The wavelength of the sound wave received by the observer is approximately 0.878 meters. Therefore, the closest option and the correct answer is:
3) 0.872 m