A trumpet player hears 5 beats per second when she plays a note and simultaneously sounds a 440 Hz tuning fork. After pulling her tuning valve out to slightly increase the length of her trumpet, she hears 3 beats per second against the tuning fork. Was her initial frequency 435 Hz or 445 Hz? Explain.

A trumpet is played alongside a tuning fork of 440Hz and beat frequency is 5Hz

Then length of trumpet is increased and so beat frequency is decreased to 3Hz.

$\left|f_s-f_t\right|=f_b$

Where, fs is frequency of source (in this case trumpet)

ft is frequency of tuning fork, (in this case 440Hz)

fb is frequency of beats (in this case 5Hz)

$$\begin{gathered} \left|f_s-440Hz\right|=5Hz \\ \Rightarrow f_s=445Hz/435Hz \end{gathered}$$

So there are two possible values of frequency of trumpet, i.e. 445Hz or 435Hz.

Now it is given that when we slightly increased the length of the pipe, i.e. we increased the wavelength (decreased the frequency of trumpet) beat frequency is dropped to 3Hz. This means that initial frequency is higher that frequency of tuning fork. Therefore we get that initial frequency of trumpet was 445Hz.