Question

doppler effect
u : source speed
v : sound speed
source
moving
observer
rest
λ = v/f
δx = u·t
δx = u/f
v - u ⇒ u < v
v > v - u
source moving towards the obse.
f₀ > f

Explanation:

Step1: Recall Doppler - effect principles

The Doppler - effect formula for a moving source approaching a stationary observer is derived from the change in wavelength. The original wavelength $\lambda=\frac{v}{f}$, where $v$ is the speed of sound and $f$ is the original frequency of the source.

Step2: Analyze the change in wavelength

When the source is moving with speed $u$ towards the observer, the distance it moves in one time - period $T=\frac{1}{f}$ is $\Delta x = uT=\frac{u}{f}$. The observed wavelength $\lambda_0$ is $\lambda_0=\lambda-\Delta x=\frac{v}{f}-\frac{u}{f}=\frac{v - u}{f}$.

Step3: Calculate the observed frequency

Since $f_0=\frac{v}{\lambda_0}$, substituting $\lambda_0=\frac{v - u}{f}$ into the formula for $f_0$, we get $f_0 = f\frac{v}{v - u}$. This shows that when the source is moving towards the observer ($u>0$), the observed frequency $f_0$ is greater than the original frequency $f$.

Answer:

The formula for the observed frequency $f_0$ when a source moving with speed $u$ approaches a stationary observer is $f_0 = f\frac{v}{v - u}$, where $v$ is the speed of sound and $f$ is the original frequency of the source.