Plane sound wave of frequency 500 Hz fall normally on a smooth wall.At what distance from the wall will the air particles have maximum and minimum amplitude of vibration and the velocity of sound in air may be taken as 340 meter per second.
32CM
To determine the distance from the wall at which the air particles will have maximum and minimum amplitude of vibration, we need to consider the phenomena of sound interference.
When a sound wave reflects off a smooth wall, it creates both incident and reflected waves. The interference between these waves produces areas of constructive and destructive interference.
Constructive interference occurs when the incident and reflected waves align perfectly, resulting in greater amplitude (loudness) of sound. Destructive interference occurs when the waves are perfectly out of phase, leading to a cancellation of sound (minimum amplitude).
To find the distance at which these interference patterns occur, we need to consider the relationship between the wavelengths of the sound wave and the distance traveled by the waves.
The formula to calculate wavelength is given by:
λ = v / f
where:
λ = wavelength
v = velocity of sound in air (given as 340 m/s)
f = frequency of the sound wave (given as 500 Hz)
Using the formula, we can calculate the wavelength as follows:
λ = 340 m/s / 500 Hz
λ ≈ 0.68 meters
Now let's consider the interference patterns:
1. Maximum amplitude (constructive interference):
Constructive interference occurs when the difference in path lengths traveled by the reflected and incident waves is an integer multiple of the wavelength.
To find the distance for maximum amplitude, we consider that the extra distance traveled by the reflected wave is equal to an integer multiple of the wavelength:
d_max = m * λ
where:
d_max = distance for maximum amplitude
m = integer (1, 2, 3, ...)
2. Minimum amplitude (destructive interference):
Destructive interference occurs when the difference in path lengths traveled by the reflected and incident waves is equal to a half-integer multiple of the wavelength.
To find the distance for minimum amplitude, we consider that the extra distance traveled by the reflected wave is equal to a half-integer multiple of the wavelength:
d_min = (m + 0.5) * λ
where:
d_min = distance for minimum amplitude
m = integer (0, 1, 2, 3, ...)
Now, let's calculate both the distance for maximum and minimum amplitude using the calculated wavelength:
For maximum amplitude:
d_max = m * λ
For m = 1,
d_max = 1 * 0.68 meters
d_max ≈ 0.68 meters
For minimum amplitude:
d_min = (m + 0.5) * λ
For m = 0,
d_min = (0 + 0.5) * 0.68 meters
d_min ≈ 0.34 meters
Therefore, at a distance of approximately 0.68 meters from the wall, the air particles will have maximum amplitude of vibration (constructive interference), and at a distance of approximately 0.34 meters from the wall, the air particles will have minimum amplitude of vibration (destructive interference).