Two speakers driven in a phase by the same amplifier at a frequency of 640 Hz, are 7 m apart. A listener originally at the position of one of the speakers starts walking away from that speaker in a direction that is perpendicular to the line connecting the two speakers.
How far do they have to walk to find their first sound minimum or destructive interference? the speed of sound is 340m/s.
To find the distance at which the listener experiences the first sound minimum or destructive interference, we need to consider the concept of path difference.
When two sound waves interfere, they can either reinforce each other (constructive interference) or cancel each other out (destructive interference). In the case of destructive interference, the two waves are out of phase and have a path difference equal to an odd multiple of half the wavelength.
First, let's find the wavelength of the sound wave:
Speed of sound = 340 m/s
Frequency = 640 Hz
The formula to calculate wavelength is given by:
Wavelength = Speed of sound / Frequency
Wavelength = 340 m/s / 640 Hz = 0.53125 meters or 53.125 cm
Now, let's consider the situation when the listener is at one of the speakers originally. The sound waves from both speakers will reach the listener in phase since they travel the same distance. As the listener starts walking away perpendicular to the line connecting the two speakers, the distance to one of the speakers increases while the distance to the other speaker remains constant.
Let's denote the distance the listener travels as "x" meters. The path difference between the two sound waves will be the difference in the distances traveled by the sound waves from the two speakers to the listener.
For destructive interference to occur, the path difference should be an odd multiple of half the wavelength:
Path difference = (distance from one speaker to listener) - (distance from the other speaker to listener)
For the listener at one of the speakers initially, the path difference when they are at a distance x will be:
Path difference = (7 + x) - 7 = x
To find the distance at which destructive interference occurs, we set the path difference equal to an odd multiple of half the wavelength:
x = (2n + 1) * (wavelength / 2)
where n is any positive integer (1, 2, 3, ...). The smallest positive value of x for destructive interference will correspond to n = 0.
Therefore,
x = (2 * 0 + 1) * (0.53125 / 2)
x = 0.265625 meters or 26.5625 cm
So, the listener needs to walk approximately 0.265625 meters or 26.5625 cm to experience the first sound minimum or destructive interference.