Two tones have frequencies \(440\,\mathrm{Hz}\) and \(446\,\mathrm{Hz}\). Find the beat frequency.

Two tones have frequencies \(256\,\mathrm{Hz}\) and \(260\,\mathrm{Hz}\). Find the average tone frequency.

A listener hears \(5\,\mathrm{Hz}\) beats when a \(512\,\mathrm{Hz}\) fork is played with another fork. Give two possible frequencies for the other fork.

Two tones have beat frequency \(8\,\mathrm{Hz}\). If one tone is \(300\,\mathrm{Hz}\), what are the two possible frequencies of the other tone?

Two frequencies are \(499\,\mathrm{Hz}\) and \(503\,\mathrm{Hz}\). Find the beat period.

Two tones produce \(12\) loudness maxima in \(3.0\,\mathrm{s}\). Find the beat frequency and the frequency difference.

Two tones have \(f_{\mathrm{avg}}=400\,\mathrm{Hz}\) and \(f_{\mathrm{beat}}=6\,\mathrm{Hz}\). Find the two original frequencies.

A \(440\,\mathrm{Hz}\) tuning fork and a string produce \(3\,\mathrm{Hz}\) beats. After the string is tightened, the beat frequency becomes \(7\,\mathrm{Hz}\). Was the original string frequency above or below \(440\,\mathrm{Hz}\)? Explain.

Two equal-amplitude tones are \(200\,\mathrm{Hz}\) and \(202\,\mathrm{Hz}\). Write the beat frequency and the approximate pitch frequency.

A beat envelope repeats every \(0.25\,\mathrm{s}\). Find the frequency difference between the two tones.

Use \(s(t)=A\cos(\omega_1t)+A\cos(\omega_2t)\) to identify the fast oscillation frequency and beat frequency for \(f_1=600\,\mathrm{Hz}\) and \(f_2=608\,\mathrm{Hz}\).

A student hears beats at \(4\,\mathrm{Hz}\) with a reference fork. Adding a small mass to the unknown fork lowers its frequency and the beat frequency becomes \(1\,\mathrm{Hz}\). Was the unknown initially above or below the reference frequency?

Two tones differ by \(20\,\mathrm{Hz}\). Explain why the result may be heard as roughness or two tones rather than slow beats.

Two tones have angular frequencies \(2800\,\mathrm{rad\,s^{-1}}\) and \(2864\,\mathrm{rad\,s^{-1}}\). Find the beat frequency in hertz.

Derive \(f_{\mathrm{beat}}=|f_1-f_2|\) using the sum of two equal-amplitude cosines.

Given \(f_{\mathrm{avg}}\) and \(f_{\mathrm{beat}}\), derive formulas for \(f_1\) and \(f_2\), assuming \(f_2>f_1\).

A piano string is compared with a \(440\,\mathrm{Hz}\) reference. It gives \(2\,\mathrm{Hz}\) beats. After tension is increased slightly, beats slow to \(1\,\mathrm{Hz}\). Find the original string frequency and explain the inference.

Two equal-amplitude tones have frequencies \(f-\delta\) and \(f+\delta\). Derive the heard carrier frequency and beat frequency in terms of \(f\) and \(\delta\).

A signal is \(s(t)=A\cos(2\pi f_1t)+A\cos(2\pi f_2t)\). Show that the amplitude envelope has maxima twice as often as the cosine factor \(\cos(\pi(f_1-f_2)t)\) completes a signed cycle, and reconcile this with \(f_{\mathrm{beat}}=|f_1-f_2|\).

A listener hears \(b\) beats per second between a reference frequency \(f_0\) and an adjustable oscillator. After the oscillator frequency is deliberately increased by \(\Delta f>b\), the beat frequency becomes \(b'\). Derive how to decide whether the original oscillator was above or below \(f_0\).