LD
Mechanics Physics
Acoustics Leaflets P1.7.1.2
Sound waves
Acoustic beats
Displaying on the oscilloscope
Objects of the experiments
Studying acoustic beats resulting from the superposition of tuning-fork oscillations with slightly different frequencies.
Displaying the beats on the oscilloscope
Determining the beat frequency fb and the frequency f of the superposed oscillation and comparing these frequencies
with the individual frequencies f1 and f2.
Principles
The wave character of sound becomes obvious when the if the two amplitudes A1 and A2 agree exactly. In this case, y(t)
superposition of two sound waves with equal amplitudes A1 can be regarded as an oscillation with the frequency f and a
and A2 and slightly different frequencies f1 und f2 is studied. At time dependent amplitude:
the position of the observer an oscillation comes about with y(t) = a(t) ⋅ cos (2 ⋅ f ⋅ t + ) (IV)
the time dependence
with
y(t) = A1 ⋅ cos (2 ⋅ f1 ⋅ t + 1) + A2 ⋅ cos (2 ⋅ f2 ⋅ t + 2) (I).
− −)
The fact that the phases 1 and 2 of the two individual a(t) = 2 ⋅ A ⋅ cos (2 ⋅ f ⋅ t + (V).
oscillations are completely arbitrary has been taken into ac- The magnitude of the amplitude a varies periodically be-
count. tween 0 and 2A (see Fig. 1), the change occurring twice
In order to calculate the beat signal, the quantities during one period. The number of so-called beats per sec-
ond, the beat frequency fb, is therefore
A1 + A2 − A1 − A2 f1 + f2 − f1 − f2
A= ,A = ,f= ,f = , −
2 2 2 2 fS = 2 ⋅ f = f1 − f2 (VI).
1 + 2 − = 1 − 2
= und When the amplitude a(t) passes zero, the sign change of the
2 2 beat leads to a phase jump in the superposed oscillation.
are introduced. After some transformations, the superposed In this experiment, the superposition of two sound waves
signal is given by generated with tuning-forks that are slightly out of tune is
− −) ⋅ cos (2 ⋅ f ⋅ t + )
studied. The beat signal is received with a microphone and
y(t) = 2 ⋅ A ⋅ cos (2 ⋅ f ⋅ t + then displayed on an oscilloscope. By detuning one tuning-
− − − ) ⋅ sin (2 ⋅ f ⋅ t + ) fork the beat frequency fb is enhanced, or, in other words, the
− 2 ⋅ A ⋅ sin (2 ⋅ f ⋅ t + (II)
beat period
This expression becomes even simpler,
1
− Tb = (VII)
−) ⋅ cos (2 ⋅ f ⋅ t + )
y(t) = 2 ⋅ A ⋅ cos (2 ⋅ f ⋅ t + (III), fb
is made shorter. The frequencies f1, f2, and fb are determined
by measuring the corresponding periods T1, T2, and Tb with
the oscilloscope.
0210-Sel
Fig. 1 Acoustic beats when the individual oscillations have the
equal amplitudes.
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P1.7.1.2 LD Physics Leaflets
Carrying out the experiment
Apparatus – Strike the two tuning-forks with the hammer one immedi-
ately after the other, and compare the audible beats with
1 pair of resonance tuning-forks, 440 Hz . 414 72
the display on the oscilloscope.
1 multi-purpose microphone . . . . . . . . 586 26 – Determine the beat period Tb, and record it.
1 saddle base . . . . . . . . . . . . . . . . 300 11 – Set the time base to 0.5 ms/DIV.
1 two-channel oscilloscope 303 . . . . . . 575 21 – In order to determine the periods T1 und T2, strike each
1 BNC/ 4 mm adapter, 2-pole . . . . . . . 575 35 tuning-fork separately while the other is removed from the
resonance box.
Setup Measuring example
The experimental setup is illustrated in Fig. 2.
– Set the switch (a) of the multi-purpose microphone to ~ .
– Lower the frequency of one tuning-fork by means of the
clamping screw (b).
– Put the tuning-forks on the resonance boxes, and direct the
openings of the boxes towards the microphone.
– Connect the multi-purpose microphone to the oscilloscope
via the BNC/4 mm adapter:
Zero line: middle
Coupling: AC
Scan: 20 mV/DÍV.
Trigger: Auto Fig. 3 Experimental setup for displaying acoustic beats on the
oscilloscope
Time base: 20 ms/DIV.
Table 1: The individual periods T1 and T2 and the beat
period Tb:
T1 T2 TS
ms ms ms
2.20 2.25 112
Fig. 2 Display of an acoustic beat on the oscilloscope
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LD Physics Leaflets P1.7.1.2
Evaluation
Table 2: The individual frequencies f1 and f2 and the beat
frequency fb
f f fS f1 − f2
Hz Hz Hz Hz
455 444 9 11
Eq. (VI) is confirmed by comparing the measured beat
frequency fb with the difference f1 – f2.
Results
When two acoustic oscillations with a slight difference in
frequency are superposed, beats that are clearly audible are
generated. These beats can be displayed on an oscilloscope.
Supplementary information
More precise investigations are possible if the beats are re-
corded with a storage oscilloscope or with the computer-as-
sisted data logging system CASSY.
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