Two loudspeakers emit out-of-phase sound waves along the x-axis. The sound has maximum loudness when the speakers are 30 cm apart. The sound becomes quieter as the distance between the speakers is increased, reaching zero at a separation of 60 cm.
If you keep increasing the separation between the speakers, at what separation will you hear the loudest sound again?
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The two speakers produce the loudest sound when the waves of both speakers add with constructive interference. Meaning, when the speakers are 30 cm apart, they are adding the waves thus making the sound at its loudest. However when the distance is increasing, the destructive interference starts to dominate. Consequently it decreases the sound. At 60 cm apart, the two waves “cancel” each other thus no sound will be produced. Now you can notice that as you will increase the distance again, the sound will increase again. This is because in the waveform, the distance between the peak and the trough is 1/2 of the wavelength. 30cm will now be the 1/2 of the wavelength and 60cm must be the full wavelength. Adding another 30 cm will make another half of the wavelength. Therefore, at distance 90 cm (60 + another 30) you will hear the loudest sound again.
The best way to visualize this is draw two identical sine waves on two pieces of paper. Flip one over so that the peaks of one line up with the valleys of the other. The two waves would interfere and give you no sound. This represents if both speakers were at zero on the x-axis. Now slide one of the waves forward until peaks line up with peaks. This is where the sound is the loudest. The first instance this happens represents 30cm of separation.
Blue and red overlap and add together having the effect of louder sound. (Constructive interference)
Continue to move until you have peaks line up with valleys again. You should have noticed that you moved half a wavelength in doing so and ended up with double the original separation of 60cm. You can also see that the wavelength of the sound is 60cm. Moving full wavelengths from the first antinode (point of maximum intensity) will get you the subsequent antinodes.
Moving 30cm more causes the sound to destructively interfere such that no sound is heard.
Thus another 30 cm, should have the peaks line up again!
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