Frequency of first intensity maximum for loudspeakers

[vaizard]

Homework Statement

Two loudspeakers are set up 4.0 meters apart and are driven in phase by the same amplifier. The listener is located a sufficient distance away so the angle to the listener is approximately the same at each speaker as indicated in the drawing. The speed of sound is 344 m/s. What frequency would the speakers need to emit if the first intensity maximum from the central maximum occurs at an angle of 30° from the perpendicular to each speaker?
http://omploader.org/vMTBsaw/phys_diagram1.png

Homework Equations

$$v=\lambda \cdot f$$
There is also $$A = 2 A_0 cos(\frac{\phi}{2})$$ in the solution to the problem but I don't understand where this equation even came from. It's not in my book or anything.

The Attempt at a Solution

Well, I know I'm looking for f, the frequency. So I did $$f = \frac{v_{\mbox{sound}}}{\lambda}$$ but that's about it. Now the actual solution uses $$A=2A_0 cos(\frac{\phi}{2})$$ and $$\phi = k \Delta x$$. I would like to understand where these equations came from. Are they universal for this type of problem, or were they somehow derived from the given information?

http://omploader.org/vMTBsbQ/phys_diagram2.png

 

[Carid]

Surely the first max will occur when the path length from one speaker is a single wavelength longer than to the path length to the other.
The path difference shown here is 4*sin30° = 2metres
So 344m/sec = 2m * frequency
Frequency = 172 Hertz.

 

[thomate1]

As a scientist, it is important to understand the underlying principles and equations used to solve a problem. In this case, we are dealing with the concept of interference of sound waves. The equation A = 2A_0 cos(\frac{\phi}{2}) is known as the superposition principle, which states that the total displacement of two or more waves at any point is equal to the algebraic sum of the individual displacements at that point. In simpler terms, it means that when two waves meet, they combine to form a new wave with an amplitude that is the sum of the individual waves.

In this problem, the two loudspeakers are producing sound waves that interfere with each other. The first intensity maximum occurs when the waves from the two speakers are in phase, meaning they have the same frequency and are peaking at the same time. This occurs at an angle of 30° from the perpendicular to each speaker. This means that the distance between the two speakers (4.0 meters) is equal to half the wavelength of the sound wave at this frequency.

Using the equation v=\lambda \cdot f, we can rearrange it to solve for f, which gives us f=\frac{v}{\lambda}. We know the speed of sound (344 m/s) and we have calculated the wavelength to be 8.0 meters. Plugging these values into the equation, we get a frequency of 43 Hz.

Now, let's look at the equation A = 2A_0 cos(\frac{\phi}{2}). This equation relates the amplitude of the sound wave to the angle at which it is being measured. In this case, the angle is 30°, which means that \phi = 30°. Plugging this into the equation, we get A = 2A_0 cos(15°). This equation allows us to find the amplitude of the sound wave at this specific angle, which is important in understanding the intensity of the sound at that point.

In summary, the equations used in this problem were derived from the principles of interference of sound waves and can be applied to any similar problem. It is important to have a solid understanding of these principles in order to solve problems accurately.