Solution:

Given:
Length of the closed organ pipe, l
Maximum amplitude of the 3rd overtone, a

To find:
Amplitude at a distance of l/7 from the closed end of the pipe

Concept:
In a closed organ pipe, only odd harmonics are present. The frequency of the nth harmonic is given by:

fn = (2n-1) v/4l

Where v is the speed of sound in air.

As we know that the pipe is vibrating in the 3rd overtone, the frequency of the 3rd harmonic will be:

f3 = (2 x 3 - 1) v/4l = 5v/4l

Let us assume that the displacement amplitude of the air particles at the closed end of the pipe is a. Since the pipe is closed at one end, the displacement amplitude at the open end of the pipe will be zero.

Now, we need to find the displacement amplitude at a distance of l/7 from the closed end of the pipe.

Approach:
The displacement amplitude of the air particles in a closed organ pipe varies with distance according to the equation:

y(x) = a sin(nπx/l)

Where y(x) is the displacement amplitude at a distance x from the closed end, a is the displacement amplitude at the closed end, n is the harmonic number, and l is the length of the pipe.

To find the displacement amplitude at a distance of l/7 from the closed end of the pipe, we need to substitute the values of n, x, a, and l in the above equation.

Calculations:
Given, n = 3, a = maximum amplitude of 3rd overtone, x = l/7, l = length of the pipe

Substituting these values in the above equation, we get:

y(l/7) = a sin(3π/7)

Using the trigonometric identity sin(3π/7) = sin(4π/7 - π/7), we can write:

y(l/7) = a sin(4π/7) cos(π/7) - a cos(4π/7) sin(π/7)

Simplifying this expression using the values of sin(4π/7) = sin(3π/7), cos(4π/7) = -cos(3π/7), and sin(π/7) = sin(6π/7), we get:

y(l/7) = -a sin(3π/7) sin(6π/7) - a cos(3π/7) cos(6π/7)

y(l/7) = -a sin(3π/7) sin(3π/7) - a cos(3π/7) cos(3π/7)

y(l/7) = -a cos(3π/7)^2 - a sin(3π/7)^2

y(l/7) = -a

Therefore, the displacement amplitude at a distance of l/7 from the closed end of the pipe is equal to -a.

Conclusion:
The displacement amplitude at a distance of l/7 from the closed end of the pipe is equal to -a. This means that the air particles at this point are displaced in the opposite direction to the closed end of the pipe.