17.13
Beats
Die Studie der Musik bietet viele zahlreichte Beispiele für die Superposition von Wellen und die konstruktive und destructive Interferenz, die auftret…
Wenn zwei Schallwellen gleicher Amplitude und leicht unterschiedlicher Frequenzen aufeinander treffen, kann die Verschiebungsgleichung für die resultierende Welle an einer bestimmten Position bestimmt werden. Seine Intensität steigt und fällt abwechselnd mit der Zeit.
Die Gesamtlänge der Zeitachse stellt eine Sekunde dar. Wenn die beiden Wellen in Phase sind, addieren sich ihre Amplituden.
Und wenn die beiden Wellen genau phasenverschoben sind, ist die Gesamtamplitude Null.
Diese periodische Variation der Klangintensität mit der Zeit bildet Schwebungen.
Die resultierende Welle sieht aus wie eine einzelne sinusförmige Welle mit unterschiedlicher Amplitude, die in einer Sekunde zwei Maxima und zwei Minima durchläuft. Die Frequenz dieser Amplitudenvariation beträgt zwei Hertz.
Die Anzahl der pro Sekunde gehörten Schläge wird als Schwebungsfrequenz bezeichnet und entspricht der Frequenzdifferenz der Schallwellen.
Das Zeitintervall zwischen zwei aufeinanderfolgenden Schlägen ist definiert als die Schwebungsperiode, die den Kehrwert der Schwebungsfrequenz darstellt.
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Q1: What happens when two sound waves of slightly different frequencies interfere with each other?
When two sound waves of equal amplitude and slightly different frequencies interfere, their displacement creates a resultant wave with varying amplitude. The intensity rises and falls alternately as the waves move in and out of phase. This periodic variation in sound intensity with time forms beats, producing a single sinusoidal wave that appears to pulse in loudness.
Q2: How is beat frequency defined and calculated?
Beat frequency is the number of beats heard per second and equals the difference between the frequencies of the two interfering sound waves. For example, if one wave has a frequency of 440 Hz and another has 442 Hz, the beat frequency is 2 Hz. The beat period, which is the time interval between successive beats, is the inverse of the beat frequency.
Q3: Why do beats occur when sound waves are in and out of phase?
Beats result from constructive and destructive interference between waves. When the two waves are in phase, their amplitudes add together, creating maximum intensity. When exactly out of phase, the total amplitude becomes zero. This alternating pattern of constructive and destructive interference produces the characteristic amplitude variation that creates the beating sound effect.
Q4: What does the resultant wave look like when beats are formed?
The resultant wave appears as a single sinusoidal wave with varying amplitude that goes through two maxima and two minima in one second when the beat frequency is 2 Hz. The wave oscillates between loud and quiet, with the amplitude envelope rising and falling periodically. This creates the characteristic pulsing or throbbing sound that listeners perceive as beats.
Q5: How do piano tuners use beats to tune instruments?
Piano tuners strike a tuning fork and play a note on the piano simultaneously. As they adjust the piano string, the beat frequency decreases as the note's frequency approaches the tuning fork's frequency. When the frequencies match, the beats disappear entirely. This method works because beats between two tones can be heard up to approximately 6 or 7 Hz, making small frequency differences audible.
Q6: Why must multi-engine aircraft propellers be synchronized?
If propeller frequencies are not synchronized, the sound waves from different engines interfere and create annoying beats heard as loud throbbing sounds. Synchronization prevents this unwanted beat phenomenon. Some aircraft use electronic systems for synchronization, while others rely on pilots adjusting engine speeds by ear, similar to the piano tuning process.
Q7: What role does constructive and destructive interference play in creating beats?
Beats are a direct result of constructive and destructive interference between two slightly different frequencies. Constructive interference occurs when waves align, amplifying the sound, while destructive interference occurs when they oppose each other, reducing the sound. The periodic cycling between these two interference states creates the amplitude modulation characteristic of beats.