91Ó°ÊÓ

Frequency generated-$120 \mathrm{MHz}$

Distance between antennas-$9.00\mathrm{m}$

The sound is reinforced to its maximum when the path difference between the sound waves is an integral multiple of the wavelength. This is also known as the Constructive interference. The condition is,

${\mathbf{r}}_{\mathbf{2}}\mathbf{-}{\mathbf{r}}_{\mathbf{1}}\mathbf{=}\mathbf{³¾Î»}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{m}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{Â\pm }\mathbf{1}\mathbf{,}\mathbf{Â\pm }\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}$ ...(i)

The sound is cancelled when the path difference between the sound waves is a multiple of the odd number of half wavelengths. This is also known as the destructive interference. The condition is,

${\mathbf{r}}_{\mathbf{2}}\mathbf{-}{\mathbf{r}}_{\mathbf{1}}\mathbf{=}\left(m+\frac{1}{2}\right)\mathbf{λ}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{m}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{Â\pm }\mathbf{1}\mathbf{,}\mathbf{Â\pm }\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}$...(ii)

The wavelength is calculated as follows,

$\begin{array}{l}\mathrm{λ}=\frac{\mathrm{c}}{\mathrm{f}}\\ =\frac{2.998×{10}^8\mathrm{m}/\mathrm{s}}{120×{10}^6\mathrm{Hz}}\\ =2.50\mathrm{m}\end{array}$

Refer to the image below,

The path difference is given as,

Putting the value of the calculated wavelength,

The range in which x must lie is 0 to 9.00 m. Thus, the values of x is,

$$\begin{gathered} for,\mathrm{m}=0 \\ \mathrm{x}=4.50\mathrm{m}. \\ \mathrm{for},\mathrm{m}=+1 \\ \mathrm{x}=+4.50\mathrm{m}-1.25\mathrm{m} \\ =3.25\mathrm{m} \\ \mathrm{for},\mathrm{m}=+2 \\ \mathrm{x}=+4.50\mathrm{m}-2.50\mathrm{m} \\ =2.00\mathrm{m} \\ \mathrm{for},\mathrm{m}=+3 \\ \mathrm{x}=4.50\mathrm{m}-3.75\mathrm{m} \\ =0.75\mathrm{m} \\ \mathrm{for},\mathrm{m}=-1 \\ \mathrm{x}=4.50\mathrm{m}-1.25\mathrm{m} \\ =5.75\mathrm{m} \\ \mathrm{for},\mathrm{m}=-2 \\ \mathrm{x}=4.50\mathrm{m}-2.50\mathrm{m} \\ =7.00\mathrm{m} \\ for,m=-3 \\ \mathrm{x}=4.50\mathrm{m}+3.75\mathrm{m} \\ =8.25\mathrm{m} \end{gathered}$$

For constructive interference to occur, $x$ must have the values x = 0.75 m, 2.00 m, 3.25 m, 4.50 m, 5.75 m, 7.00 m, and 8.25 m.