How to solve the problem "A large room contains two speakers that are 3 meters apart …"?

Question

I have this task:

A large room contains two speakers that are 3 meters apart. The sound intensity I of one speaker is twice that of the other, as shown in the figure. Suppose the listener is free to move about the room to find those positions that receive equal amounts of sound from both speakers. Such a location satisfies two conditions:

1. the sound intensity at the listener’s position is directly proportional to the sound level of a source, and

2. the sound intensity is inversely proportional to the square of the distance from the source.

(a) Find the points on the x-axis that receive equal amounts of sound from both speakers.

I tried to solve it this way:

1. let's define two intensities as I_lower, whose source is located at the point (0; 0) and I_higher, whose source is located at the point (3; 0);
2. since each of the intensities is inversely proportional to the square of the distance from the respective source, we can express $I_{higher} = \frac{1}{(3 - x)^2 + y^2}$ and $I_{lower} = \frac{1}{x^2 + y^2}$
3. now we need to find points where $I_{higher} = 2 I_{lower}$ hence the equation $\frac{1}{(3 - x)^2 + y^2} = \frac{2}{x^2 + y^2}$
4. since the task requires to find points only on the X axis, y = 0, hence $\frac{1}{(3 - x)^2} = \frac{2}{x^2}$

Now my problem is that this equation gives two roots $x_1 = 6 + 3 \sqrt{2} \approx 10.24$ and $x_2 = 6 - 3 \sqrt{2} \approx 1.76$, which are obviously incorrect: the listener in order to have equal sound intensities must stand closer to the louder source (1.76) or even further to the right from the louder source (10.24).

The correct answer from the textbook (Larson Calculus) is $x_1 \approx 1.2426$ and $x_2 \approx -7.2426$.

Could someone explain to me, how this answer was obtained? I searched the web for this task and found a couple of mentions, where the equation is almost the same, except that the coefficient 2 is placed on the other side of the equation: $\frac{2}{(3 - x)^2} = \frac{1}{x^2}$, but:

• I really cannot comprehend this, why the intensity from the louder source must be twice as large as the one from the quieter source
• even with this equation the roots are -3, 1, which doesn't correspond to the textbook's answer.

I checked and double checked everything several times, and my suspicion is that either I have an error in my initial model $\frac{1}{(3 - x)^2} = \frac{2}{x^2}$ or the textbook has a typo somewhere.

Thank you.

Answer 1

The right form for the two intensities is

$$ I_L = \frac{C_L}{x^2 + y^2}\\ I_R = \frac{C_R}{(3-x)^2 + y^2} $$ where $C_L$ and $C_R$ are constants.

"The sound intensity of one speaker is twice that of the other" is a vague way of saying that if you stand, say, 1 unit away from each of them, the right-hand one will be louder (by a factor of two). That tells you that $C_R = 2 C_L$. By adjusting your units of measurement, you can just pick $C_L = 1$ and $C_R = 2$. So now you have $$ I_L = \frac{1}{x^2 + y^2}\\ I_R = \frac{2}{(3-x)^2 + y^2} $$ And then the problem asks you to think about $xy$-locations where $I_L$ and $I_R$ are equal. Since they only ask about such points on the $x$-axis, you were right to plug in $y = 0$; this gives $$ \frac{1}{x^2 + 0^2}= \frac{2}{(3-x)^2 + 0^2}\\ {(3-x)^2}= {2}{x^2} $$ ...and I'll bet that you can take it from there.

Short summary: it's a not very well-phrased question.