Rituraj Tiwari
Last Activity: 4 Years ago
To determine the maximum wavelength for which a person receives maximum sound intensity from a source, we need to consider how sound waves behave when they travel. In this scenario, we have sound reaching the listener directly from the source and also after reflecting off a rigid boundary, creating a situation where interference can occur.
Understanding Sound Waves and Interference
When sound waves travel, they can either add together constructively (increasing intensity) or destructively (decreasing intensity) when they meet. For the person standing 6 meters away from the source, the distance to the rigid boundary plays a crucial role in how these sound waves interact.
The Role of Wavelength
The wavelength of a sound wave is the distance between successive peaks (or troughs) of the wave. For maximum sound intensity, we want to achieve constructive interference, which occurs when the waves are in phase. This means that they arrive at the listener's position at the same time and their peaks align.
Calculating for Maximum Intensity
In this case, the distance to the boundary is important. If the person is 6 meters from the source, the direct path is 6 meters. The reflected sound wave travels to the boundary and back, which adds to the distance the wave travels. Therefore, the total distance for the reflected wave is:
- Distance to the boundary = 6 m
- Distance back to the listener = 6 m
- Total distance for reflected wave = 6 m + 6 m = 12 m
For constructive interference to occur, the difference in path lengths (the direct path compared to the reflected path) must be an integer multiple of the wavelength. Thus, we can set up the equation:
Path difference = Total distance of reflected wave - Distance of direct wave = 12 m - 6 m = 6 m
Finding the Maximum Wavelength
For maximum sound intensity, the path difference should equal an integer multiple of the wavelength (λ). This can be expressed as:
Path difference = nλ
Where n is an integer (1, 2, 3...). For the first harmonic (n=1), we set:
6 m = 1λ
From this, we find:
λ = 6 m
Conclusion: The Maximum Wavelength
Thus, the maximum wavelength for which the person receives maximum sound intensity, considering the direct and reflected waves, is 6 meters. This wavelength ensures that the two sound waves constructively interfere, maximizing the intensity of the sound heard by the listener.
