91Ó°ÊÓ

$\mathrm{v}=\mathrm{λ´Ú}$where v is the speed of wave, f is the frequency and$\mathrm{λ}$ is the wavelength of the wave.

Path difference from points A and B to point Q is $3.0\mathrm{m}-1.0\mathrm{m}=2.0\mathrm{m}$as shown in Figure. Constructive interference implies path difference$=\mathrm{n}\mathrm{λ}\mathrm{where}\mathrm{n}=1,2,3...$

Given that,$2.00m=n\mathrm{λ}$

So,role="math" localid="1664342520874" $\mathrm{λ}=\frac{2.00\mathrm{m}}{\mathrm{n}}$

Apply in formula of speed,

role="math" localid="1664342538111" $$\begin{gathered} \mathrm{f}=\frac{\mathrm{v}}{\mathrm{λ}} \\ =\frac{\mathrm{vn}}{2.00\mathrm{m}} \\ =\frac{\mathrm{n}\left(344\mathrm{m}/\mathrm{s}\right)}{2.00\mathrm{m}} \\ =\mathrm{n}\left(172\mathrm{Hz}\right) \\ \mathrm{Where}\mathrm{n}=1,2,3,... \end{gathered}$$

Hence, the lowest frequency for which constructive interference occurs is 172 Hz.

Destructive interference implies path difference $=(n/2)\mathrm{λ}$,$\mathrm{where}\mathrm{n}=1,3,5...$

Given that,$2.00m=(n/2)\mathrm{λ}$

So,$\mathrm{λ}=\frac{4.00\mathrm{m}}{\mathrm{n}}$

Apply in formula for speed,

$$\begin{gathered} \mathrm{f}=\frac{\mathrm{v}}{\mathrm{λ}} \\ =\frac{\mathrm{vn}}{4.00\mathrm{m}} \\ =\frac{\mathrm{n}\left(344\mathrm{m}/\mathrm{s}\right)}{4.00\mathrm{m}} \\ =\mathrm{n}\left(86\mathrm{Hz}\right) \\ \mathrm{Where}\mathrm{n}=1,2,3,... \end{gathered}$$

Hence, the lowest frequency for which destructive interference occurs is 86 Hz.