Question

Two speakers face each other, and they each emit a sound of wavelength λ. One speaker is 180∘ out of phase with respect to the other. If we separate the speakers by a distance 1.5λ, how far from the left-most speaker should we place a microphone in order to pick up the loudest sound? Ignore reflections from nearby surfaces. Select all that apply.

a) 1/4λ
b) λ
c) 0λ
d) 1/2 λ
e) 3/4 λ

anyone know the answer to this?

Answer 1

Let's look at this in terms of cosine functions. Function one will be centered at x=0 and will be:\[y _{1}=\cos \left( x \right)\]

Answer 2

Fucntion 2 will be centered at x=3π (2π is one wavelength) and will be 180° (π radians out of phase with Function 1. So Function 2 can then be expressed as:\[y _{2}=\cos \left[ \left( x-3 \pi \right) + \pi \right]=\cos \left( x-2 \pi \right)=\cos(x)\]

Answer 3

So both functions are mathematically identical. This means that they will always be in phase, and as such, the loudest sound will be where ever there is a maximum or minimum in the cosine function. The maxima and minima of cosine functions are at 0, π, 2π, 3π, and so on.

Answer 4

This means if you place a microphone right in front of the speaker on the left, i.e. at 0λ, you will pick up the loudest sound.