Question

Two identical loudspeakers separated by distance $∆x$each

emit sound waves of wavelength $\lambda$ and amplitude a along the

x-axis. What is the minimum value of the ratio $\frac{∆x}{\lambda }$ for which

the amplitude of their superposition is also a?

Short answer

The solution also provides the identical distance of the sound waves of the wavelengths

Step-by-step solution

Discussion on superposition

When two or more waves cross at a point, the displacement at that point is equal to the sum of the displacements of the individual waves.

Discusion on the phase difernces

The phrase difference is calculated below

$\begin{array}{rcl}@\mathrm{acos}\left(\mathrm{\pi }\frac{∆\mathrm{x}}{\mathrm{\lambda }}\right)& =& \mathrm{a}=\cos \left(\mathrm{\pi }\frac{∆\mathrm{x}}{\mathrm{\lambda }}\right)\\ & =& 1/2=\\ A& =& 2aco9s\left(\frac{∆\varnothing }{2}\right)\\ ∆\varnothing & =& 2\mathrm{\pi }\frac{∆\mathrm{x}}{\mathrm{\lambda }}\\ \mathrm{A}& =& 2\mathrm{acos}\left(\mathrm{\pi }\frac{∆\mathrm{x}}{\mathrm{\lambda }}\right)=\mathrm{\pi }/3\\ \frac{∆\mathrm{x}}{\mathrm{\lambda }}& =& 1/3\\ & & \end{array}$