Beats: frequency of resulting wave vs. beats frequency

• Soren4
In summary, the beats frequency heard from the interference of two sound waves with frequencies ##f_1## and ##f_2## is $$\nu=|f_1-f_2|$$ Nevertheless the frequency of the resulting wave is not ##\nu## but the mean value of the two frequencies $$f_{resulting}=\frac{f_1+f_2}{2}$$, and the distinction between the two is clear in theory but may cause confusion in practice. This can be explained by the imperfections of the ear as an instrument for measuring sound waves. An instrument that can measure sound waves within a certain frequency range may be able to detect the beat frequency even if it does not have a response at the difference frequency
Soren4
The beats frequency heard from the interference of two sound waves with frequencies ##f_1## and ##f_2## is $$\nu=|f_1-f_2|$$

Nevertheless the frequency of the resulting wave is not ##\nu## but the mean value of the two frequencies
$$f_{resulting}=\frac{f_1+f_2}{2}$$

As far as I understood ##\nu## is the frequency at which the maxima of intensity are heard, while ##f_{resulting}## is indeed the frequency of the resulting wave.

In particular I can notice that ##f_{resulting} > \nu## always.

The distincion between the two is clear in theory, but in practice I still have doubts.

Suppose to have an instrument that can measure sound waves iff the frequency is, say, bigger than ##f_{min}## and lower than ##f_{max}##.

Now suppose that two waves interefere in a way such that ##f_{resulting} > f_{min}## but ##\nu <f_{min}##, or, in a way such that ##\nu < f_{max}## but ##f_{resulting} > f_{max}##.

What does the instrument measure in these cases?

Which of the two frequencies "determine" the upper of lower limits for the frequency that can be measured by such instrument?

Soren4 said:
The beats frequency heard from the interference of two sound waves with frequencies ##f_1## and ##f_2## is $$\nu=|f_1-f_2|$$

Nevertheless the frequency of the resulting wave is not ##\nu## but the mean value of the two frequencies
$$f_{resulting}=\frac{f_1+f_2}{2}$$

As far as I understood ##\nu## is the frequency at which the maxima of intensity are heard, while ##f_{resulting}## is indeed the frequency of the resulting wave.

In particular I can notice that ##f_{resulting} > \nu## always.

The distincion between the two is clear in theory, but in practice I still have doubts.
This link says it all, I think.
The reason for hearing a beat of twice the frequency that you would expect is that you hear the two peaks in loudness and your ear ignores the phase inversion of the sum at frequency (f1+f2)/2
The ear is an imperfect instrument for measuring what you present it with and that accounts for the apparent paradox, I think.
The instrument that you describe would be like an AM receiver, I think and, to measure the beat, it would need a wide enough bandwidth to include the two input tones. But, as with an AM receiver, it only needs to admit those tones and needs no response at the difference frequency - to take an extreme case, a 1MHz AM receiver doesn't need any response to audio signals in its antenna input in order to receive audio signals carried on the carrier.

$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$
Let $x=\omega_{1}t$ and $y=\omega_{2}t$.

Svein said:
$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$
Let $x=\omega_{1}t$ and $y=\omega_{2}t$.
Did you want to add some more?

1. What are beats in relation to frequency?

Beats refer to the periodic variation in amplitude or loudness of a sound wave caused by the interference of two sound waves with slightly different frequencies. The resulting wave will have a frequency equal to the difference between the two original frequencies.

2. How is the frequency of resulting wave related to the beats frequency?

The frequency of the resulting wave is equal to the difference between the two original frequencies. This is known as the beats frequency and can be calculated by subtracting the lower frequency from the higher frequency.

3. How do beats impact the perceived pitch of a sound?

Beats can impact the perceived pitch of a sound by creating a pulsing or throbbing effect. This can make the sound appear to have a lower or higher pitch than its actual frequency.

4. Can beats occur with any two sound waves?

Yes, beats can occur with any two sound waves as long as they have slightly different frequencies. This phenomenon can occur in both musical and non-musical sounds.

5. How can beats be used in music?

Beats can be used in music to create a rhythmic effect or to tune instruments. By adjusting the frequencies of two notes, musicians can create beats that add depth and complexity to a musical piece.

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