Two 100.0-W speakers, A and B, are separated by a distance D=3.6 m . The speakers emit in-phase sound waves at a frequency f=10,000.0 Hz. Point P1 is located at x1=4.50 m and y1=0 m ; point P2 is located at x2=4.50 m and y2=-Δy . Neglecting speaker B, what is the intensity, IA 1 (in W / m^2 ), of the sound at point P1 due to speaker A ? Assume that the sound from the speaker is emitted uniformly in all directions. What is this intensity in terms of decibels (sound level, βA 1 )? When both speakers are turned on, there is a maximum in their combined intensities at P1 . As one moves toward P2, this intensity reaches a single minimum and then becomes maximized again at P2. How far is P2 from P1, that is, what is Δy ? You may assume that L ≫Δy and that D ≫Δy, which will allow you to simplify the algebra by using √(a ±b) ≈a^1 / 2 ±(b)/(2 a^1 / 2) when a ≫b.

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 $a \gg b$.

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 $a \gg b$.

Text: Two identical speakers S1 (0,0), S2 (10.0 m, 0) oscillate with an initial phase difference of G (S1 sends out waves ahead of S2) and emit a sound at a frequency of 1000 Hz. Assume that each speaker's sound intensity does not decay within a short distance range of 50m and is proportional to the square of the amplitude of the wave. (a) When both speakers are on and G is zero, what is the phase difference Î¸ between the two waves arriving at point P (10m, 10m)? Hint: Î¸ = 2Ï€(r1-r2)/Î», where r1 is the distance from S1 to P and r2 is the distance from S2 to P. What are the amplitude and intensity of the superimposed waves at (10m, 10m)? (b) When both speakers are on and G is zero, what are the positions on the X-axis where an observer will hear 1) no sound, 2) the greatest intensity of sound? Hint: No sound occurs at positions Î¸ = Ï€/2 - (2n-1)Ï€/2. The greatest intensity of sound occurs at positions Î¸ = 2nÏ€. (c) What is the phase difference G if an observer at point Q (20m, 0) receives a sound intensity of 1.40 I0 when both speakers are on? I0 is the intensity when one speaker is on. Hint: The total phase difference is Î¸ = G + Î¸8 at Q. Speed of sound is 340 m/s.