Sound Wave Interference Problem

[Ghost Repeater]

Homework Statement

This is just a question about a question in Serway & Jewett's "Physics for Scientists and Engineers 3rd Ed". It's Objective Question 3 from Chapter 18, building on Example 18.1 from the text.

Two identical loudspeakers placed 3.00 m apart are driven by the same oscillator. A listener is originally at point O, located 8.00 m from the center of the line connecting the two speakers. The listener then moves to point P, which is a perpendicular distance 0.350 m from O, and she experiences the first minimum in sound intensity. What is the intensity at P?

a) Less than but close to the intensity at O

b) half the intensity at O

c) very low but not zero

d) zero

e) indeterminate [/B]

Homework Equations

The Attempt at a Solution

My answer to this question was d), zero. My reasoning is that a minimum occurs when the waves interfere destructively, i.e. they cancel each other out completely. This means there can be no amplitude, which means there can be no intensity.

But the solution manual says differently. It says 'The two waves must have slightly different amplitudes at P because of their different distances, so they cannot cancel each other exactly.'

I guess I'm not sure what this means or how it fits with my concept of 'minimum'. I thought the first minimum was where the path length difference that each wave travels is exactly equal to 1/2 a wavelength, so the waves are exactly 180 deg out of phase, meaning they cancel each other out perfectly.

What am I missing here? [/B]

 

[andrewkirk]

The amplitude of the wave is inversely proportional to the square of the distance from the source. At P one wave will have traveled n wavelengths and the other will have traveled (n+1/2). The one that has traveled n+1/2 will be slightly weaker than the other and will cancel against n/(n+1/2) of the other, leaving a wave that is 1/(n+1/2) of the wave that has traveled only n, as the residual.

Usually in these problems they don't worry about amplitude, assuming that all amplitudes are approximately equal so that everything cancels perfectly, because amplitude effects are small at these ranges. But the book answer is technically correct.

 

[Tom.G]

The listener then moves to point P, which is a perpendicular distance 0.350 m from O...

The confusing factor here is the 'perpendicular' to what. My original reading was perpendicular to the line between the two speakers. However moving along this line will not display any peaks or nulls in amplitude. To get a null there has to be a phase difference in the sound from the two speakers. This requires that the observer be unequal distances from the speakers. To get that condition the observer must move away from the line "O", "<first observation point>". So the 'perpendicular distance' in the problem statement must be in a direction parallel to the line between the speakers; or perpendicular to the line "O", "<first observation point>".

Sneaky (or at least unclear) wording... or they were trying to point out you have to take account of unstated or ambiguous conditions, just like in the real world!

Cheers,
Tom

 

[haruspex]

The confusing factor here is the 'perpendicular' to what. My original reading was perpendicular to the line between the two speakers. However moving along this line will not display any peaks or nulls in amplitude. To get a null there has to be a phase difference in the sound from the two speakers. This requires that the observer be unequal distances from the speakers. To get that condition the observer must move away from the line "O", "<first observation point>". So the 'perpendicular distance' in the problem statement must be in a direction parallel to the line between the speakers; or perpendicular to the line "O", "<first observation point>".

Sneaky (or at least unclear) wording... or they were trying to point out you have to take account of unstated or ambiguous conditions, just like in the real world!

Cheers,
Tom

Maybe there was a diagram.

 

[haruspex]

The confusing factor here is the 'perpendicular' to what. My original reading was perpendicular to the line between the two speakers. However moving along this line will not display any peaks or nulls in amplitude. To get a null there has to be a phase difference in the sound from the two speakers. This requires that the observer be unequal distances from the speakers. To get that condition the observer must move away from the line "O", "<first observation point>". So the 'perpendicular distance' in the problem statement must be in a direction parallel to the line between the speakers; or perpendicular to the line "O", "<first observation point>".

Sneaky (or at least unclear) wording... or they were trying to point out you have to take account of unstated or ambiguous conditions, just like in the real world!

Cheers,
Tom

Moreover, the listener probably has two ears.

 

[Tom.G]

Moreover, the listener probably has two ears.

I hope so!