Observed Frequency Calculations

Introduction

• The observed frequency (fo for observer) of a wave can be calculated using the Doppler effect formula when certain parameters are known. These parameters include the frequency of the source (fs for source), the speed of the source (vs), and the speed of sound (v).

Doppler Effect Formula

• The Doppler effect formula is used to calculate the observed frequency, and it is given by:
• fo = fs (v + vo) / (v ± vs)
• Where:
  • fo is the observed frequency.
  • fs is the frequency of the source.
  • v is the speed of sound in the medium.
  • vo is the velocity of the observer relative to the medium.
  • vs is the velocity of the source relative to the medium.
  • The 'plus or minus' sign depends on whether the source is moving towards or away from the observer.

Calculating Observed Frequency - Example

• Let's consider an example to calculate the observed frequency when a fire engine is moving away from a stationary observer while the source has a siren with a frequency of 512 Hz.
• The speed of sound (v) is given as 340 m/s.

Step 1: Set up the Doppler Effect Formula

• Insert the known values into the formula and consider the sign based on the direction of motion:
• fo = 512 Hz x (340 m/s + 0 m/s) / (340 m/s + 18 m/s)

Step 2: Simplify the Calculation

• Calculate the terms in the brackets, which will become a factor to multiply the source frequency:
• fo = 512 Hz x (340 m/s) / (358 m/s)

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Observed Frequency Calculations

Step 3: Perform the Calculation

• Compute the final result:
• fo = 512 Hz x 0.949
• fo ≈ 485.19 Hz

Summary

• The observed frequency of a wave can be calculated using the Doppler effect formula when you know the frequency of the source, the speed of the source, and the speed of sound.
• The formula accounts for whether the source is moving towards or away from the observer, and it helps determine how the perceived frequency changes due to relative motion. In the example given, the fire engine's siren frequency of 512 Hz appears to drop to approximately 485.19 Hz as it moves away from the stationary observer.