Two identical loudspeakers are driven in phase by a common oscillator at 800 Hz and face each other at a distance of 1.25 m. Locate the points along the line joining the two speakers where relative minima of sound pressure amplitude would be expected.
v = λ*f
The Attempt at a Solution
First up: v = λ*f <=> 343 m/s = λ*800 Hz <=> λ = 0.429 m
I just did what I learned back in high school, and though: x =n*λ/2 (see the PS), where x is the distance from the beginning of say the imaginary axis (let's say the first speaker). From the theory, we know that every particle that is a node, has a distance that is equal to a integer n, times half the length of the wave. So far so good. Therefore, since the maximum distance is 1.25m, we can say:
0 <= x <= 1.25 m <=> 0 <= n*λ/2 <= 1.25 m <=> ... <=> 0 <= n <= 5.8 <=> n = 0, 1, 2, 3, 4, 5
Then, I took those integers, put them into the formula, and got some certains. I went to check, and the manual's answers were completely different! Here's his solution:
The wavelength is λ = v/f = 0.429m
The two waves moving in opposite directions along the line between the two speakers will add
to produce a standing wave with this distance between nodes: distance N to N = λ/2 = 0.214 m
Because the speakers vibrate in phase, air compressions from each will simultaneously reach the point halfway between the speakers, to produce an antinode of pressure here. A node of pressure will be located at this distance on either side of the midpoint: distance N to A = λ/4 = 0.107 m
Therefore nodes of sound pressure will appear at these distances from either speaker:
1/2*(1.25 m) + 0.107 m = 0.732 m and
1/2*(1.25 m) − 0.107 m = 0.518 m
The standing wave contains a chain of equally-spaced nodes at distances from either speaker of
0.732 m + 0.214 m = 0.947 m
0.947 m + 0.214 m = 1.16 m
and also at 0.518 m − 0.214 m = 0.303 m
0.303 m − 0.214 m = 0.089 1 m
The standing wave exists only along the line segment between the speakers. No nodes or antinodes appear at distances greater than 1.25 m or less than 0, because waves add to give a standing wave only if they are traveling in opposite directions and not in the same direction. In order, the distances from either speaker to the nodes of pressure between the speakers are 0.089 1 m, 0.303 m, 0.518 m, 0.732 m, 0.947 m, and 1.16 m.
Now, I understand his logic. He finds the antinode at the midway, and then goes step-by-step, left and right to find the nodes. What I don't get, is why it matters that the waves meet at the same time, and why he's so sure they create an antinode.
I'm probably missing something, but that's the way we did it at High School, so this is a completely new way to me, and I wonder why mine is wrong.
Any help would be appreciated!
PS: Back in HS, we were taught that x = (2*N +1)*λ/4 when it comes to nodes, and x = Ν*λ/2 when it came to antinodes. Now, in my new book (Serway's Physics for Engineers &...), it says that x = n*λ/2 is for nodes, and x = n*λ/4 for antinodes. Could someone explain to me why there's such a vast difference between the two, if they cover the same subject?