Question

A friend of yours is loudly singing a single note at 400 Hz while racing toward you at 25.0 m/s on a day when the speed of sound is 340 m/s. a. What frequency do you hear? b. What frequency does your friend hear if you suddenly start singing at 400 Hz?

Short answer

Part a

The frequency of your friend's note as heard by you is $432Hz$.

Part b

The frequency heard by your friend by your note is$429Hz$.

Step-by-step solution

Given information

The frequency of the source is $f_0=400Hz$

The speed of the source as it is approaching you is $v_s=25.0m/s$.

The speed of sound is $v=340m/s$.

Part a

The Doppler effect for an approaching source is given by $f_{+}=\frac{f_0}{1-{\displaystyle \frac{v_s}{v}}}$.

Substitute the given values

role="math" localid="1649779767311" $$\begin{gathered} f_{+}=\frac{400}{1-{\displaystyle \frac{25}{340}}} \\ \approx 432Hz \end{gathered}$$

Therefore, you will hear the frequency of$432Hz$.

Part b

The frequency heard by your friend by your note is given by $f_{+}=f_0\left(1+\frac{v_0}{v}\right)$.

Substitute the given values

$$\begin{gathered} f_{+}=400\left(1+\frac{25}{340}\right) \\ \approx 429Hz \end{gathered}$$

Therefore, the frequency heard by your friend by your note is $429Hz$.