Two 100.0-W speakers, A and B, are separated by a distance \(D=3.6 \mathrm{~m}
.\) The speakers emit in-phase sound waves at a frequency \(f=10,000.0
\mathrm{~Hz}\). Point \(P_{1}\) is located at \(x_{1}=4.50 \mathrm{~m}\) and
\(y_{1}=0 \mathrm{~m} ;\) point \(P_{2}\) is located at \(x_{2}=4.50 \mathrm{~m}\)
and \(y_{2}=-\Delta y .\) Neglecting speaker \(\mathrm{B}\), what is the
intensity, \(I_{\mathrm{A} 1}\) (in \(\mathrm{W} / \mathrm{m}^{2}\) ), of the
sound at point \(P_{1}\) due to speaker \(\mathrm{A}\) ? Assume that the sound
from the speaker is emitted uniformly in all directions. What is this
intensity in terms of decibels (sound level, \(\beta_{\mathrm{A} 1}\) )? When
both speakers are turned on, there is a maximum in their combined intensities
at \(P_{1} .\) As one moves toward \(P_{2},\) this intensity reaches a single
minimum and then becomes maximized again at \(P_{2}\). How far is \(P_{2}\) from
\(P_{1},\) that is, what is \(\Delta y ?\) You may assume that \(L \gg \Delta y\)
and that \(D \gg \Delta y\), which will allow you to simplify the algebra by
using \(\sqrt{a \pm b} \approx a^{1 / 2} \pm \frac{b}{2 a^{1 / 2}}\) when \(a \gg
b\).
