Beat (acoustics)/Just fifth displace stretch

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A wild-goose chase

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

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

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

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

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

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

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