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== 17.6#h(0.6em)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 #strong[beats]. If two sounds differ in frequencies, the sound waves can be modeled as


  #math.equation(block: true, alt: "y sub 1 equals A cos open parenthesis k sub 1 x minus 2 π f sub 1 t close parenthesis and y sub 2 equals A cos open parenthesis k sub 2 x minus 2 π f sub 2 t close parenthesis .")[$y_(1) = A #h(0.2em) "cos" ( k_(1) x − 2 π f_(1) t ) #h(0.2em) "and" #h(0.2em) y_(2) = A #h(0.2em) "cos" ( k_(2) x − 2 π f_(2) t ) .$]
  Using the trigonometric identity #math.equation(block: false, alt: "cos u plus cos v equals 2 cos open parenthesis the fraction u plus v over 2 close parenthesis cos open parenthesis the fraction u minus v over 2 close parenthesis")[$"cos" #h(0.2em) u + "cos" #h(0.2em) v = 2 #h(0.2em) "cos" ( frac(u + v, 2) ) "cos" ( frac(u − v, 2) )$] and considering the point in space as #math.equation(block: false, alt: "x equals 0.0 m,")[$x = 0.0 #h(0.2em) "m,"$] we find the resulting sound at a point in space, from the superposition of the two sound waves, is equal to :


  #math.equation(block: true, alt: "y open parenthesis t close parenthesis equals 2 A cos open parenthesis 2 π f sub a v g t close parenthesis cos open parenthesis 2 π open parenthesis the fraction | f sub 2 minus f sub 1 | over 2 close parenthesis t close parenthesis ,")[$y ( t ) = 2 A #h(0.2em) "cos" ( 2 π f_(a v g) t ) "cos" ( 2 π ( frac(| f_(2) − f_(1) |, 2) ) t ) ,$]
  where the #strong[beat frequency] is


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "f sub beat equals | f sub 2 minus f sub 1 | .")[$f_("beat") = | f_(2) − f_(1) | .$]
]


  #figure(figph[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.], alt: "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.", caption: [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.


  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.{"functions":\[{"color":"\#1f4e79","domain":\[0,3\],"expression":"0.5\*cos(2\*pi\*10\*t) + 0.5\*cos(2\*pi\*f2\*t)","expression2":"","id":"superposition","inequality":"lt","lineDash":\[\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"t","visible":true},{"color":"\#b8442a","domain":\[0,3\],"expression":"cos(pi\*(f2 - 10)\*t)","expression2":"","id":"envelope-upper","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"t","visible":true},{"color":"\#b8442a","domain":\[0,3\],"expression":"-cos(pi\*(f2 - 10)\*t)","expression2":"","id":"envelope-lower","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"t","visible":true}\],"grid":{"axisColor":"\#333333","gridColor":"\#cccccc","majorSpacing":0.5,"minorGridColor":"\#eeeeee","showAxes":true,"showLabels":true},"parameters":\[{"id":"up1-17-7-f2","label":"Frequency f\\u2082 of the second tone (f\\u2081 is fixed at 10.0 Hz)","max":14,"min":10,"name":"f2","step":0.1,"unit":"Hz","value":11}\],"title":"Beats from two tones and their envelope","version":1,"viewport":{"centerX":1.5,"centerY":0,"scale":224}}#examplebox("Example 1")[Find the Beat Frequency Between Two Tuning Forks][
What is the beat frequency produced when a tuning fork of a frequency of 256 Hz and a tuning fork of a frequency of 512 Hz are struck simultaneously?


    Strategy
The beat frequency is the difference of the two frequencies.


    Solution
We use #math.equation(block: false, alt: "f sub beat equals | f sub 2 minus f sub 1 | :")[$f_("beat") = | f_(2) − f_(1) | :$]


    #math.equation(block: true, alt: "| f sub 2 minus f sub 1 | equals open parenthesis 512 minus 256 close parenthesis Hz equals 256 Hz .")[$| f_(2) − f_(1) | = ( 512 − 256 ) #h(0.2em) "Hz" = 256 #h(0.2em) "Hz" "."$]
    Significance
The beat frequency is the absolute value of the difference between the two frequencies. A negative frequency would not make sense.
]


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
What would happen if more than two frequencies interacted? Consider three frequencies.


    
    #solutionbox[
An easy way to understand this event is to use a graph, as shown below. It appears that beats are produced, but with a more complex pattern of interference.


      #figure(figph[Graph plots displacement in centimeters versus time in seconds. Three sound waves and the interference wave are shown in the graph.], alt: "Graph plots displacement in centimeters versus time in seconds. Three sound waves and the interference wave are shown in the graph.", caption: none)
]
]


  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 #emph[A] produces a constant-frequency sound wave of 1.00 kHz, and speaker #emph[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?


    
    #solutionbox[
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?


    
    #solutionbox[
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?


    
    #solutionbox[
#math.equation(block: true, alt: "f sub B, equals, | f sub 1 minus f sub 2 |; | 128.3 Hz minus 128.1 Hz |, equals, 0.2 Hz;; | 128.3 Hz minus 127.8 Hz |, equals, 0.5 Hz;; | 128.1 Hz minus 127.8 Hz |, equals, 0.3 Hz")[$f_("B") & = & | f_(1) − f_(2) | \ | 128.3 #h(0.2em) "Hz" − 128.1 #h(0.2em) "Hz" | & = & 0.2 #h(0.2em) "Hz;" \ | 128.3 #h(0.2em) "Hz" − 127.8 #h(0.2em) "Hz" | & = & 0.5 #h(0.2em) "Hz;" \ | 128.1 #h(0.2em) "Hz" − 127.8 #h(0.2em) "Hz" | & = & 0.3 #h(0.2em) "Hz"$]
]


    
      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 #math.equation(block: false, alt: "μ equals 0.0065 kg/m,")[$μ = 0.0065 #h(0.2em) "kg/m,"$] are fixed on both ends. String #emph[A] is under a tension of 120.00 N. String #emph[B] is under a tension of 130.00 N. They are each plucked and produce sound at the #math.equation(block: false, alt: "n equals 10")[$n = 10$] mode. What is the beat frequency?


    
    #solutionbox[
#math.equation(block: true, alt: "v sub A equals 135.87 m/s, v sub B equals 141.42 m/s,; λ sub A equals λ sub B equals 0.40 m; Δ f equals 13.9 Hz")[$v_(A) = 135.87 #h(0.2em) "m/s," #h(0.5em) v_(B) = 141.42 #h(0.2em) "m/s," \ λ_(A) = λ_(B) = 0.40 #h(0.2em) "m" \ "Δ" f = 13.9 #h(0.2em) "Hz"$]
]


    
      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 #math.equation(block: false, alt: "μ equals 0.0062 kg/m")[$μ = 0.0062 #h(0.2em) "kg/m"$] 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?


    
    #solutionbox[
#math.equation(block: false, alt: "v equals 155.54 m/s,; f sub string equals 971.17 Hz, n equals 16.23; f sub string equals 1076.83 Hz, n equals 18.00")[$v = 155.54 #h(0.2em) "m/s," \ f_("string") = 971.17 #h(0.2em) "Hz," #h(0.5em) n = 16.23 \ f_("string") = 1076.83 #h(0.2em) "Hz," #h(0.5em) n = 18.00$] #linebreak() 
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?


    
    #solutionbox[
#math.equation(block: true, alt: "f sub 2 equals f sub 1 plus or minus f sub B equals 260.00 Hz plus or minus 1.50 Hz,; so that f sub 2 equals 261.50 Hz or f sub 2 equals 258.50 Hz")[$f_(2) = f_(1) ± f_("B") = 260.00 #h(0.2em) "Hz" ± 1.50 #h(0.2em) "Hz," \ "so that" #h(0.2em) f_(2) = 261.50 #h(0.2em) "Hz" #h(0.2em) "or" #h(0.2em) f_(2) = 258.50 #h(0.2em) "Hz"$]
]


    
      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?


    
    #solutionbox[
#math.equation(block: true, alt: "f sub ace equals the fraction f sub 1 plus f sub 2 over 2 ; f sub B equals f sub 1 minus f sub 2 open parenthesis assume f sub 1 greater than f sub 2 close parenthesis; f sub ace equals the fraction open parenthesis f sub B plus f sub 2 close parenthesis plus f sub 2 over 2 ⇒; f sub 2 equals 4099.750 Hz; f sub 1 equals 4100.250 Hz")[$f_("ace") = frac(f_(1) + f_(2), 2) ; f_("B") = f_(1) − f_(2) ( "assume" f_(1) > f_(2) ) \ f_("ace") = frac(( f_("B") + f_(2) ) + f_(2), 2) ⇒ \ f_(2) = 4099.750 #h(0.2em) "Hz" \ f_(1) = 4100.250 #h(0.2em) "Hz"$]
]


    
      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?