Define Doppler effect. Derive the expression for apparent frequency when the source is in motion and observer is at rest.

The Doppler effect is a fascinating phenomenon that occurs when there is a relative motion between a sound source and an observer. It explains how the frequency of a wave changes for an observer moving relative to the source of the wave. You might have noticed this effect when an ambulance passes by; the siren sounds higher in pitch as it approaches and lower as it moves away. This change in frequency is what we refer to as the Doppler effect.

Understanding the Basics of the Doppler Effect

To grasp the concept, let’s break it down. The Doppler effect can be observed with any type of wave, including sound and light. When the source of the wave moves towards the observer, the waves get compressed, leading to a higher frequency (or pitch, in the case of sound). Conversely, when the source moves away, the waves are stretched, resulting in a lower frequency.

Deriving the Expression for Apparent Frequency

Let’s derive the expression for the apparent frequency when the source is in motion and the observer is at rest. We will consider the following variables:

• f: Actual frequency of the source
• v: Speed of sound in the medium
• v_s: Speed of the source (positive if moving towards the observer, negative if moving away)
• f': Apparent frequency observed

When the source is moving towards a stationary observer, the wavelength of the sound waves decreases because the source is catching up with the waves it has already emitted. The relationship between the speed of sound, frequency, and wavelength is given by:

v = f × λ

Where λ is the wavelength. When the source moves, the effective wavelength (λ') can be expressed as:

λ' = λ - (v_s / f)

Substituting this into the wave equation gives:

v = f' × λ'

Now, since λ' is the new wavelength when the source is moving, we can express it as:

v = f' × (λ - (v_s / f))

Rearranging this equation to solve for the apparent frequency (f') gives us:

f' = f × (v / (v - v_s))

When the Source Moves Away

If the source is moving away from the observer, the wavelength increases, and the expression modifies slightly. The apparent frequency can be derived similarly:

f' = f × (v / (v + v_s))

Real-World Applications

The Doppler effect has numerous applications in various fields. For instance, in astronomy, it helps determine the speed at which stars and galaxies are moving towards or away from Earth. In medical imaging, Doppler ultrasound is used to measure blood flow. Understanding this effect not only enhances our grasp of wave mechanics but also enriches our comprehension of the universe around us.

In summary, the Doppler effect illustrates how motion influences the perception of sound frequency, and the derived formulas allow us to calculate the apparent frequency based on the motion of the source relative to the observer. This concept is not just theoretical; it has practical implications that impact various scientific and technological fields.