Find an expression for apparent frequency for source of sound when source is coming towards stationary observer .?

To determine the apparent frequency of a sound when the source is moving towards a stationary observer, we can use the Doppler effect. This phenomenon describes how the frequency of a wave changes for an observer moving relative to the source of the wave. Let's break this down step by step.

The Doppler Effect Explained

The Doppler effect occurs when there is relative motion between a sound source and an observer. When the source moves towards the observer, the sound waves are compressed, leading to a higher frequency perceived by the observer. Conversely, if the source were moving away, the frequency would decrease.

Mathematical Representation

The formula for the apparent frequency (\( f' \)) when the source is moving towards a stationary observer can be expressed as:

f' = f \times \frac{v}{v - v_s}

• f' = apparent frequency
• f = actual frequency of the sound emitted by the source
• v = speed of sound in the medium (approximately 343 m/s in air at room temperature)
• v_s = speed of the sound source towards the observer

Understanding the Components

In this equation:

• The term \( v \) represents the speed of sound, which is constant in a given medium under standard conditions.
• The term \( v_s \) is the speed of the source. When the source moves towards the observer, this value is positive, which reduces the denominator and increases the apparent frequency.

Example Scenario

Imagine a car with a siren that emits a sound frequency of 500 Hz. If the car is moving towards you at a speed of 30 m/s, we can calculate the apparent frequency you would hear.

Using the formula:

f' = 500 Hz × (343 m/s) / (343 m/s - 30 m/s)

Calculating the denominator:

343 m/s - 30 m/s = 313 m/s

Now substituting back into the equation:

f' = 500 Hz × (343 m/s) / (313 m/s)

Calculating this gives:

f' ≈ 500 Hz × 1.095

Thus, the apparent frequency you would hear is approximately 547.5 Hz. This increase in frequency is what you perceive as the car approaches you.

Real-World Applications

The Doppler effect is not just a theoretical concept; it has practical applications in various fields:

• Radar and Sonar: Used to determine the speed of moving objects, such as cars or submarines.
• Astronomy: Helps in measuring the speed of stars and galaxies moving towards or away from Earth.
• Medical Imaging: Utilized in Doppler ultrasound to assess blood flow in vessels.

Understanding the apparent frequency due to the Doppler effect is crucial in these applications, as it allows for accurate measurements and interpretations of data. By grasping the relationship between motion and frequency, we can better understand how sound behaves in our environment.