(II) Starting from Eq. 36–16a, show that the Doppler shift in wavelength is, if v ≪ c ,


∆λ / λ = v/c .

Verified step by step guidance

1. Identify the original equation (Eq. 36-16a) for the Doppler shift, which typically is expressed as \( f' = f \left(\frac{c + v}{c - v}\right) \) where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v \) is the velocity of the observer relative to the source, and \( c \) is the speed of light.
2. Express the Doppler shift in terms of wavelength. Since frequency \( f \) and wavelength \( \lambda \) are related by \( f = \frac{c}{\lambda} \), substitute this into the Doppler shift equation to get the relationship in terms of wavelength.
3. Simplify the equation under the assumption that \( v \ll c \). This allows for the approximation \( \frac{v}{c} \) to be considered very small, leading to simplifications in the mathematical expression.
4. Expand the denominator using the binomial approximation, \( (1 - x)^{-1} \approx 1 + x \) for \( x \ll 1 \), to simplify the expression further.
5. Isolate the change in wavelength \( \Delta \lambda \) and express it as a fraction of the original wavelength \( \lambda \), leading to the final form \( \Delta \lambda / \lambda = v/c \).

Key Concepts

Doppler Effect

The Doppler Effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In the context of sound and light, it explains how the observed frequency increases as the source approaches and decreases as it recedes. This effect is crucial for understanding how motion affects wave properties, particularly in astrophysics and acoustics.

Wavelength Shift

Wavelength shift is the change in the wavelength of a wave due to relative motion between the source and the observer. In the case of the Doppler Effect, this shift can be quantified as an increase or decrease in wavelength depending on whether the source is moving towards or away from the observer. This concept is essential for deriving the relationship between velocity and wavelength change in the given equation.

Relativity of Motion

The relativity of motion is a fundamental principle in physics that states that the motion of an object is always measured relative to another object. In the context of the Doppler shift, it emphasizes that the observed changes in wavelength depend on the relative velocities of the source and observer. This principle is particularly relevant when considering speeds that are much less than the speed of light (v ≪ c), simplifying the mathematical relationships involved.