The Doppler effect is a phenomenon observed when there is relative motion between a wave source and an observer. It causes a change in the perceived frequency of the wave. In the case of an approaching receiver, the observed frequency is higher than the emitted frequency. While commonly associated with sound waves, the Doppler effect applies to all types of waves, including electromagnetic waves like light.

The formula for the Doppler effect with an approaching receiver is:

\[ f_o = f_s \left(\frac{v + v_r}{v - v_s}\right) \]

Where:

- \( f_o \) = observed frequency (Hz)
- \( f_s \) = source frequency (Hz)
- \( v \) = speed of wave in the medium (m/s)
- \( v_r \) = velocity of the receiver (m/s, positive when approaching)
- \( v_s \) = velocity of the source (m/s, positive when receding)

Let's work through an example to calculate the observed frequency:

- Given:
- Source frequency (\( f_s \)) = 500 Hz
- Speed of sound (\( v \)) = 343 m/s
- Receiver velocity (\( v_r \)) = 25 m/s (approaching)
- Source velocity (\( v_s \)) = 5 m/s (receding)

- Substitute the values into the Doppler effect formula: \[ f_o = 500 \text{ Hz} \left(\frac{343 \text{ m/s} + 25 \text{ m/s}}{343 \text{ m/s} - 5 \text{ m/s}}\right) \]
- Simplify: \[ f_o = 500 \text{ Hz} \left(\frac{368 \text{ m/s}}{338 \text{ m/s}}\right) \]
- Calculate: \[ f_o = 500 \text{ Hz} \times 1.0888 \approx 544.4 \text{ Hz} \]

The observed frequency is approximately 544.4 Hz, which is higher than the source frequency due to the approaching receiver.

The following diagram illustrates the Doppler effect with an approaching receiver:

This diagram shows:

- The source (blue circle) moving away from the receiver
- The receiver (red circle) moving towards the source
- The sound waves (green curves) being compressed as the receiver approaches
- The source velocity (\( v_s \)) and receiver velocity (\( v_r \)) represented by arrows

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