# Two identical loudspeakers separated by distance triangle x each emit sound waves of wavelength lambda and amplitude a along the and amplitude a along the x-axis. What is the minimum value of the ratio triangle x/lambda for which the amplitude of their superposition is also a?

Two identical loudspeakers separated by distance $\mathrm{△}x$ each emit sound waves of wavelength $\lambda$ and amplitude a along the and amplitude a along the x-axis.
What is the minimum value of the ratio $\mathrm{△}x/\lambda$ for which the amplitude of their superposition is also a?
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Jamari Morgan
We know that the amplitude of the superposition A will be given by
$A=2a\mathrm{cos}\left(\frac{\mathrm{△}\varphi }{2}\right)$
On the other hand, the phase difference is given by
$\mathrm{△}\varphi =2\pi \frac{\mathrm{△}x}{\lambda }$
Combining these two results, we have
$A=2a\mathrm{cos}\left(\pi \frac{\mathrm{△}x}{\lambda }\right)$
Therefore, for A=a we will have
$2a\mathrm{cos}\left(\pi \frac{\mathrm{△}x}{\lambda }\right)=a⇒\mathrm{cos}\left(\pi \frac{\mathrm{△}x}{\lambda }\right)=\frac{1}{2}⇒\left(\pi \frac{\mathrm{△}x}{\lambda }{\right)}_{min}=\frac{\pi }{3}$
Finally,
$\frac{\mathrm{△}x}{\lambda }=\frac{1}{3}$
Result:
1/3