What Is the Resulting Frequency When Two Different Sound Frequencies Combine?

[erisedk]

Homework Statement

When two sound sources of the same amplitude but of slightly different frequencies n1 and n2 are sounded simultaneously, the sound one hears has a frequency equal to

Ans: (n1+n2)/2

Homework Equations

The Attempt at a Solution

I have virtually no clue how that's the answer. I thought maybe the problem was related to beats, but it's clearly not. Beyond this, I just don't know at all.

 

[haruspex]

Homework Statement

When two sound sources of the same amplitude but of slightly different frequencies n1 and n2 are sounded simultaneously, the sound one hears has a frequency equal to

Ans: (n1+n2)/2

Homework Equations

The Attempt at a Solution

I have virtually no clue how that's the answer. I thought maybe the problem was related to beats, but it's clearly not. Beyond this, I just don't know at all.

No, it is related to beats. Write the expression for the addition of two sine waves. What trigonometric formulae do you know that look relevant?

 

[erisedk]

Got it! Asin(2πn1t-kx) + Asin(2πn2t-kx) = 2Asin(πn1t+πn2t2-kx)cos(πn1t-πn2t), which will have the frequency (n1+n2)/2
As for cos(πn1t-πn2t), is it something like variable amplitude term in standing wave equations? Cos it doesn't have any traveling component.

 

[haruspex]

Got it! Asin(2πn1t-kx) + Asin(2πn2t-kx) = 2Asin(πn1t+πn2t2-kx)cos(πn1t-πn2t), which will have the frequency (n1+n2)/2
As for cos(πn1t-πn2t), is it something like variable amplitude term in standing wave equations? Cos it doesn't have any traveling component.

Not exactly. Your equations are not quite right. The two waves being added should have the same speed.
But whether the beat factor in the product form travels is not relevant. The listener is presumed stationary. It's a perception question; the low frequency factor is heard as variation in amplitude, not as a tone.

 

[erisedk]

Not exactly. Your equations are not quite right. The two waves being added should have the same speed.

Oh yeah, v=w/k, and so we can adjust the k's accordingly, which makes the frequency term look like this: sin(πn1t+πn2t2-(k1-k2)/2x).