To determine the frequency registered by the observer when a source is moving towards them while wind is affecting the sound waves, we first need to apply the Doppler effect formula, taking into account the influence of the wind. Let's break this down step-by-step.
Understanding the Doppler Effect
The Doppler effect describes how the frequency of sound changes for an observer moving relative to a sound source. When the source moves towards the observer, the frequency increases, and when it moves away, the frequency decreases. The formula for the observed frequency (f') when the source is moving towards a stationary observer is:
f' = f (v + vo) / (v - vs)
Where:
- f = frequency of the source (500 Hz)
- v = speed of sound (300 m/s)
- vo = speed of the observer (0 m/s, since the observer is stationary)
- vs = speed of the source (20 m/s)
Calculating the Frequency without Wind
Plugging in the values without considering the wind:
f' = 500 Hz (300 m/s + 0 m/s) / (300 m/s - 20 m/s)
f' = 500 Hz (300 m/s) / (280 m/s)
f' = 500 Hz * 1.0714
f' ≈ 535.7 Hz
Incorporating Wind Effects
The wind affects how sound travels. Since the wind is blowing at 20 m/s at an angle of 60° north of east, we need to calculate the effective speed of sound relative to the observer. The wind has both a horizontal (east-west) and a vertical (north-south) component.
Calculating the Wind Components
Using trigonometry:
- Wind's eastward component: 20 m/s * cos(60°) = 20 m/s * 0.5 = 10 m/s
- Wind's northward component: 20 m/s * sin(60°) = 20 m/s * √3/2 ≈ 17.32 m/s
Since the observer is located east of the source, we only consider the eastward component of the wind for adjusting the effective speed of sound.
Effective Speed of Sound
The effective speed of sound (v_eff) toward the observer is the speed of sound plus the wind's eastward component:
v_eff = v + wind's eastward component
v_eff = 300 m/s + 10 m/s = 310 m/s
Calculating the Final Observed Frequency
Now, substituting the effective speed of sound back into the Doppler effect formula:
f' = 500 Hz (310 m/s) / (300 m/s - 20 m/s)
f' = 500 Hz (310 m/s) / (280 m/s)
f' ≈ 553.57 Hz
Final Result
So, the approximate frequency registered by the observer, taking into account the wind, would be around 554 Hz. This shows how both the motion of the source and the wind can significantly affect the perceived frequency of sound.
