Question

Two very small, identical speakers, each radiating sound uniformly in all directions, are placed at points S1 and S2 as in Figure 8. The speakers are connected to an audio source in such a way that they radiate in phase, at the common wavelength of 2.00 m. Sound propagates in air at 338 m/s.

(a) Calculate the frequency of the sound.
(b) Point M, a nodal point, is 7.0 m from S1 and more than 7.0 m from S2 Find three possible distances M could be from S2.

I got part a already (f=169 Hz), but I'm not sure how to attack part b. Any ideas?

Answer 1

The idea for finding nodal points is that you want to find a point that is x distance from one speaker and x + (wavelength/2) from the other one so that the waves are completely out of phase from each other and they will completely destructively interfere with one another.

Answer 2

Would the formula \[| PnS1 - PnS2 | = (n \left(\begin{matrix}1 \\ 2\end{matrix}\right) \lambda)\] have something to do with it? I tried rearranging the equation, but it didn't work out.

Answer 3

Or is the second half of the formula \[(n+\left(\begin{matrix}1 \\ 2\end{matrix}\right) \lambda )\] ?

Answer 4

What does Pn represent?

Answer 5

Sorry, that's wrong! One sec.

Answer 6

Should be...
\[d_{S2}=d_{S1}+(n+\frac{1}{2})\lambda\]So you'll get...\[d_{S1}+\frac{1}{2}\lambda,\ d_{S1}+\frac{3}{2}\lambda,\ d_{S1}+\frac{5}{2}\lambda,\ ...\]appropriately.

Answer 7

Oh, the way I have it written down was \[(n 1/2 \lambda )\] I just assumed it was multiplication, but I guess they meant n "and a half"

Answer 8

Yeah, the main idea is you want the difference in the distances from S1 to M and S2 to M to be half the wavelength (or some equivalent point on the wave).

Answer 9

Okay, I think I got it. Thanks so much! That was really helpful.