17.6 Beats

The study of music provides many examples of the superposition of waves and the constructive and destructive interference that occurs. Very few examples of music being performed consist of a single source playing a single frequency for an extended period of time. You will probably agree that a single frequency of sound for an extended period might be boring to the point of irritation, similar to the unwanted drone of an aircraft engine or a loud fan. Music is pleasant and interesting due to mixing the changing frequencies of various instruments and voices.

An interesting phenomenon that occurs due to the constructive and destructive interference of two or more frequencies of sound is the phenomenon of beats. If two sounds differ in frequencies, the sound waves can be modeled as

$y_{1} = A cos (k_{1} x - 2 π f_{1} t) and y_{2} = A cos (k_{2} x - 2 π f_{2} t) .$

Using the trigonometric identity $cos u + cos v = 2 cos (\frac{u + v}{2}) cos (\frac{u - v}{2})$ and considering the point in space as $x equals 0.0 m,$ we find the resulting sound at a point in space, from the superposition of the two sound waves, is equal to Figure 17.29:

$y (t) = 2 A cos (2 π f_{a v g} t) cos (2 π (\frac{| f_{2} - f_{1} |}{2}) t),$

where the beat frequency is

$f_{beat} = | f_{2} - f_{1} | .$

Graphs plot displacement in centimeters versus time in seconds. Top graph shows two sound waves. Bottom graph shows interference wave with the constructive (double intensity) and destructive (zero intensity) regions indicated. — **Figure 17.29** Beats produced by the constructive and destructive interference of two sound waves that differ in frequency.

These beats can be used by piano tuners to tune a piano. A tuning fork is struck and a note is played on the piano. As the piano tuner tunes the string, the beats have a lower frequency as the frequency of the note played approaches the frequency of the tuning fork.

Function graph showing y = 0.5*cos(2*pi*10*t) + 0.5*cos(2*pi*f2*t) on t in [0, 3], y = cos(pi*(f2 - 10)*t) on t in [0, 3] and y = -cos(pi*(f2 - 10)*t) on t in [0, 3]. Adjustable parameter: Frequency f₂ of the second tone (f₁ is fixed at 10.0 Hz) (f2) = 11 Hz. Viewing window: x from -0.2 to 3.2, y from -1.05 to 1.05. — The sum of two equal-amplitude tones, 0.5cos(2πf₁t) + 0.5cos(2πf₂t), with f₁ held at 10.0 Hz. The dashed pair is the envelope ±cos(π(f₂ − f₁)t) that the product form 2Acos(2π(f₂ − f₁)t/2)·cos(2π(f₁ + f₂)t/2) predicts. At the f₂ = 11.0 Hz drawn here the envelope pinches shut once a second: the beat frequency is |f₂ − f₁| = 1.0 Hz, and three loud-quiet cycles fill this 3-second window. Drag f₂ back toward 10.0 Hz and the beats slow toward silence — the vanishing beat is exactly what the piano tuner listens for — while pushing it to 14.0 Hz gives four beats a second. The frequencies here are deliberately low so that single cycles stay visible; a 256-Hz fork would pack hundreds of cycles between the same envelope pinches.

The study of the superposition of various waves has many interesting applications beyond the study of sound. In later chapters, we will discuss the wave properties of particles. The particles can be modeled as a “wave packet” that results from the superposition of various waves, where the particle moves at the “group velocity” of the wave packet.

Summary

When two sound waves that differ in frequency interfere, beats are created with a beat frequency that is equal to the absolute value of the difference in the frequencies.

Conceptual Questions

Two speakers are attached to variable-frequency signal generator. Speaker A produces a constant-frequency sound wave of 1.00 kHz, and speaker B produces a tone of 1.10 kHz. The beat frequency is 0.10 kHz. If the frequency of each speaker is doubled, what is the beat frequency produced?

The label has been scratched off a tuning fork and you need to know its frequency. From its size, you suspect that it is somewhere around 250 Hz. You find a 250-Hz tuning fork and a 270-Hz tuning fork. When you strike the 250-Hz fork and the fork of unknown frequency, a beat frequency of 5 Hz is produced. When you strike the unknown with the 270-Hz fork, the beat frequency is 15 Hz. What is the unknown frequency? Could you have deduced the frequency using just the 250-Hz fork?

The frequency of the unknown fork is 255 Hz. No, if only the 250 Hz fork is used, listening to the beat frequency could only limit the possible frequencies to 245 Hz or 255 Hz.

Referring to the preceding question, if you had only the 250-Hz fork, could you come up with a solution to the problem of finding the unknown frequency?

A “showy” custom-built car has two brass horns that are supposed to produce the same frequency but actually emit 263.8 and 264.5 Hz. What beat frequency is produced?

The beat frequency is 0.7 Hz.

Problems

What beat frequencies are present: (a) If the musical notes A and C are played together (frequencies of 220 and 264 Hz)? (b) If D and F are played together (frequencies of 297 and 352 Hz)? (c) If all four are played together?

What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz?

$\begin{matrix} f_{B} & = & | f_{1} - f_{2} | \\ | 128.3 Hz - 128.1 Hz | & = & 0.2 Hz; \\ | 128.3 Hz - 127.8 Hz | & = & 0.5 Hz; \\ | 128.1 Hz - 127.8 Hz | & = & 0.3 Hz \end{matrix}$

A piano tuner hears a beat every 2.00 s when listening to a 264.0-Hz tuning fork and a single piano string. What are the two possible frequencies of the string?

Two identical strings, of identical lengths of 2.00 m and linear mass density of $μ equals 0.0065 kg/m,$ are fixed on both ends. String A is under a tension of 120.00 N. String B is under a tension of 130.00 N. They are each plucked and produce sound at the $n equals 10$ mode. What is the beat frequency?

$\begin{matrix} v_{A} = 135.87 m/s, v_{B} = 141.42 m/s, \\ λ_{A} = λ_{B} = 0.40 m \\ Δ f = 13.9 Hz \end{matrix}$

A piano tuner uses a 512-Hz tuning fork to tune a piano. He strikes the fork and hits a key on the piano and hears a beat frequency of 5 Hz. He tightens the string of the piano, and repeats the procedure. Once again he hears a beat frequency of 5 Hz. What happened?

A string with a linear mass density of $μ equals 0.0062 kg/m$ is stretched between two posts 1.30 m apart. The tension in the string is 150.00 N. The string oscillates and produces a sound wave. A 1024-Hz tuning fork is struck and the beat frequency between the two sources is 52.83 Hz. What are the possible frequency and wavelength of the wave on the string?

$\begin{matrix} v = 155.54 m/s, \\ f_{string} = 971.17 Hz, n = 16.23 \\ f_{string} = 1076.83 Hz, n = 18.00 \end{matrix}$
The frequency is 1076.83 Hz and the wavelength is 0.14 m.

A car has two horns, one emitting a frequency of 199 Hz and the other emitting a frequency of 203 Hz. What beat frequency do they produce?

The middle C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?

$\begin{matrix} f_{2} = f_{1} \pm f_{B} = 260.00 Hz \pm 1.50 Hz, \\ so that f_{2} = 261.50 Hz or f_{2} = 258.50 Hz \end{matrix}$

Two tuning forks having frequencies of 460 and 464 Hz are struck simultaneously. What average frequency will you hear, and what will the beat frequency be?

Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?

$\begin{matrix} f_{ace} = \frac{f_{1} + f_{2}}{2}; f_{B} = f_{1} - f_{2} (assume f_{1} > f_{2}) \\ f_{ace} = \frac{(f_{B} + f_{2}) + f_{2}}{2} \Rightarrow \\ f_{2} = 4099.750 Hz \\ f_{1} = 4100.250 Hz \end{matrix}$

Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370, and 392 Hz. What beat frequencies are produced by this discordant combination?