91Ó°ÊÓ

Distance when the speaker 2 is behind the speaker $1,\mathrm{Δ}x=10\mathrm{cm}$

Amplitude when speaker 2 is in front of speaker $1$ at $30\mathrm{cm},A=2a$

Formula used:

Distance between the maximum and minimum sound intensities is as below:

$$

\begin{aligned}

\Delta x=& 10 \mathrm{~cm}+30 \mathrm{~cm} \\

=& 40 \mathrm{~cm}

\end{aligned}

$$

So, the wavelength of the sound between maximum and minimum intensities is as follows:

$$

\begin{aligned}

&x=\frac{\lambda}{2} \\

&\lambda=2 \Delta x

\end{aligned}

$$

Calculation:

Substitute $\Delta x=40 \mathrm{~cm}$ so, the wavelength will become as below:

$$

\begin{aligned}

& \lambda=2 \Delta x \\

=& 2(40 \mathrm{~cm}) \\

=& 80 \mathrm{~cm}

\end{aligned}

$$

Conclusion:

Hence, the wavelength of the sound is $80 \mathrm{~cm}$

Distance when the speaker 2 is behind the speaker $1, \Delta x=10 \mathrm{~cm}$

Amplitude when speaker 2 is in front of speaker $1$ at $30 \mathrm{~cm}, A=2 a$

Formula Used:

The expression for the phase difference between the two sound waves is as below:

$$

\Delta \phi=2 \pi \frac{\Delta x}{\lambda}

$$

Calculation:

Substitute $\Delta x=40 \mathrm{~cm}$ and $\lambda=80 \mathrm{~cm}$ so, the phase difference is as follows:

$$

\begin{aligned}

& \phi=\frac{2 \pi}{\lambda} \Delta x \\

=& \frac{2 \pi}{80 \mathrm{~cm}} 40 \mathrm{~cm} \\

=& 2.36 \mathrm{rad}

\end{aligned}

$$

Conclusion:

Hence, the phase difference between the two waves is $2.36 \mathrm{rad}$.

Distance when the speaker 2 is behind the speaker $1, \Delta x=10 \mathrm{~cm}$

Amplitude when speaker 2 is in front of speaker $1$ at $30 \mathrm{~cm}, A=2 a$

Formula Used:

When two waves with a phase difference $\phi$ and each having an amplitude $a$ then the resulting wave has an amplitude as below:

$$

A=2 a \cos \left(\frac{\Delta \phi}{2}\right)

$$

Calculation:

Now, substitute $\phi=2.36 \mathrm{rad}$ so, the amplitude is

$$

\begin{aligned}

&A=2 a \cos \left(\frac{\Delta \phi}{2}\right) \\

&=2 a \cos \left(\frac{2.36 \mathrm{rad}}{2}\right) \\

&=0.76 a

\end{aligned}

$$

Conclusion:

Hence, the amplitude of sound if the speakers are placed side by side is $0.76 a$.