Acoustic Wave Interference
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Acoustic Wave Interference
What is interference? Most people have heard this term and often associate it vaguely with waves or vibrations. However, a physicist would define it precisely as the phenomenon of overlapping waves, leading to an increase or decrease in the amplitude of the resultant wave. Interference occurs with all types of waves (electromagnetic, mechanical, de Broglie, etc.) and in all media where such waves can propagate.
This phenomenon follows the principle of superposition, which states that the resultant amplitude A at any point where partial waves overlap is given by the equation:
where: A1, A2 - amplitudes of the partial waves, and φ - the phase difference between the two waves.
As shown, the resultant wave can have a maximum amplitude of A = A1 + A2 when φ = 2k. In this case, the phases of both waves are perfectly aligned: corresponding points of the partial waves (crests and troughs) coincide in space and time. Thus, the resultant amplitude is the sum of the amplitudes of both partial waves, resulting in constructive interference.
For opposite phases (φ = 2k + 1), the resultant amplitude is minimized, described by A = A1 - A2. Here, the waves are out of phase, and their opposite elements (crests and troughs) overlap, causing destructive interference. If both partial waves have the same amplitude, the resultant amplitude is A = 0.
Generating Acoustic Waves Using Software
A simple way to generate acoustic waves with precisely controlled frequency and phase is by using a computer and appropriate software. While there are many software-based generators, we will take a slightly unconventional approach and use Audacity—a music editing tool. This advanced, multi-track audio editor is distributed under the GNU GPL license (available for free download). It is available for multiple operating systems, including Unix/Linux, Microsoft Windows, and Mac OS.
After downloading and installing Audacity, you will need the project file I have prepared. You can download it here. After extracting the archive, you will find a folder containing the file "interferencja.aup." Double-clicking it should launch Audacity with two loaded audio waveforms. It should look like this:
We see two audio waveforms: the upper one is for the left speaker, and the lower one is for the right speaker. To play the audio, press the green play button on the top toolbar. Notice that the frequencies of these sounds differ slightly: 440Hz for the left channel and 441Hz for the right channel. The difference between the frequencies is 1Hz. The signal level remains constant throughout the duration of the sound, as seen in both channels. Now, turn on your speakers and play the audio. Does the sound appear uniform? Not at all! You can clearly hear alternating moments of increased and decreased loudness. By listening carefully, you will notice that the time between two moments of decreased amplitude (or increased amplitude) is one second, indicating an amplitude modulation frequency of 1Hz. Interestingly, this frequency is exactly the difference between the two generated frequencies. Why?
Let’s analyze why we hear these variations in loudness. Each speaker receives a steady signal with no amplitude fluctuations. You can verify this by muting one channel—you’ll hear a continuous sound. This indicates that the variations are due to the interaction of both frequencies. To investigate this further, use the zoom tool (a small magnifying glass with a plus sign on the top toolbar) to zoom in until the sinusoidal waveform is clearly visible:
We see two textbook sine waves. Their frequencies differ slightly, although this difference is not easily noticeable at this scale. Initially, both waves are in phase at t = 0s (as indicated by the time scale above the waveforms). Their crests and troughs align perfectly, resulting in constructive interference that increases the resultant amplitude. Note that the lower waveform has a slightly higher frequency, causing its crests to gradually shift ahead of the upper waveform's crests. Although this shift is minor at this point, let's scroll to t = 0.5s (half the period at 1Hz):
The situation changes significantly. Now, the waveforms are in opposite phases (crests align with troughs and vice versa). Their amplitudes cancel out, causing destructive interference and a significant drop in loudness.
These effects repeat alternately every second (check it!). This results in an amplitude modulation frequency of 1Hz. As we observed, this frequency is always the difference between the frequencies of the two waveforms. The two interfering frequencies produce a resultant wave whose oscillation frequency is 1Hz. Let's prove this by summing the displacements of both sine waves from their equilibrium positions. To do this, open the track menu (on the left side of the upper waveform; click the black triangle next to the track name) and select "Make Stereo Track." Then, from the top menu, choose Tracks - Mix - Mix Stereo Down to Mono. This sums the amplitudes, as we discussed earlier. The result should look like this:
This result confirms our observations. The amplitude changes with a frequency of 1Hz, with maximum reinforcement at full seconds and cancellation in between. The principle of superposition holds true. The animation below illustrates this process in detail: the red and blue waveforms represent the partial waves (left and right channels), while the black waveform represents the resultant wave.
The phenomena we observed here also occur with all other types of waves.
Enjoy experimenting! :)
Further readings:
- Born M., Wolf E., Principles of Optics, Cambridge University Press, Cambridge, 1999
- Longhurst R.S., Geometrical and Physical Optics, Longmans, Londyn, 1968
- Resnick R., Halliday D., Fizyka t.1, wydanie IX, Państwowe Wydawnictwo Naukowe, Warszawa, 1993
Marek Ples