Beat (acoustics)/Just fifth displace stretch

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A wild-goose chase

First calculate $f_{B}$ using the beat period measured directly from the graph (then converting to frequency.)

$T_{B}=8T_{0}+3T_{x}=\left(6\times 8+3=51\right)T_{x}$$\implies f_{B}={\frac {1}{51}}f_{x}$ CHECK WITH BELOW

Now, check against the formula: $T_{B}=\left|p\Delta f_{q}-q\Delta f_{p}\right|:$

$\Delta f_{p}={\frac {\widetilde {N}}{\widetilde {T}}}-{\frac {N}{T}}$, where ${\frac {N}{T}}={\frac {3}{6}}$ (since there are 3 p-waves in time $T_{0}$.)

${\widetilde {N}}=(8\times 3)+1=25$

${\widetilde {T}}=(8\times 6+3)T_{x}=51T_{x}$$\implies {\widetilde {f_{p}}}={\frac {25}{51}}T_{x}$

$q\Delta f=q\left|\Delta f_{p}\right|=2\left|{\frac {1}{2}}-{\frac {25}{51}}\right|f_{x}$ $=\left(1-{\frac {50}{51}}\right)f_{x}={\frac {1}{51}}f_{x}$ THIS CHECKS WITH ABOVE