(i) In a travelling sinusoidal longitudinal wave, the displacement of particle of medium is represented by `s = S (x, t)`. The midpoint of a compression zone and an adjacent rarefaction zone are represented by letter ‘C’ and ‘R’ respectively. The difference in pressure at ‘C’ and ‘R’ is `DeltaP` and the bulk modulus of the medium is B.
(a) How is `|(del s)/(del x)|` related to `|del/del|`
(b) Write the value of `|(del s)/(del x)|_(C)` in terms of `DeltaP` and `B`.
(c) What is speed of a medium particle located mid-way between ‘C’ and ‘R’.
(ii) A standing wave in a pipe with a length of `L = 3 m` is described by `s=A cos ((3pi x)/(L)) sin ((3pi vt)/(L))` where `v` is wave speed. The atmospheric pressure and density are `P_(0)` and `rho` respectively.
(a) At `t=L/(18v)` the acoustic pressure at `x =L/2` is `0.2` percent of the atmospheric pressure . Find the displacement amplitude A.
(b) In which overtone is the pipe oscillating?

Answer

To solve the given problem step by step, we will break down each part of the question.

### Part (i)

#### (a) How is `|(del s)/(del x)|` related to `|del/del|`?

1. **Understanding the relationship**: In a longitudinal wave, the displacement of particles is related to the pressure variations in the medium. The pressure difference between the compression and rarefaction zones can be expressed in terms of the displacement gradient.

2. **Using the relationship**: The relationship can be derived from the definition of the bulk modulus \( B \):
\[ B = -\frac{\Delta P}{\frac{\Delta s}{\Delta x}} \]
Rearranging gives:
\[ \frac{\Delta s}{\Delta x} = -\frac{\Delta P}{B} \]
Hence, we can say:
\[ \left| \frac{\partial s}{\partial x} \right| = \frac{\Delta P}{B} \]

#### (b) Write the value of `|(del s)/(del x)|_(C)` in terms of `DeltaP` and `B`.

1. **Applying the derived formula**: From the previous part, we can directly substitute the values at point C (compression):
\[ \left| \frac{\partial s}{\partial x} \right|_C = \frac{\Delta P}{B} \]

#### (c) What is the speed of a medium particle located mid-way between ‘C’ and ‘R’?

1. **Understanding particle motion**: The particle located midway between the compression zone (C) and the rarefaction zone (R) experiences no net displacement, as it is at the equilibrium position between the two extremes.

2. **Conclusion**: Therefore, the speed of the medium particle at this position is:
\[ v = 0 \]

### Part (ii)

#### (a) Find the displacement amplitude A.

1. **Given parameters**: The standing wave equation is:
\[ s = A \cos\left(\frac{3\pi x}{L}\right) \sin\left(\frac{3\pi vt}{L}\right) \]
At \( t = \frac{L}{18v} \) and \( x = \frac{L}{2} \):
\[ s = A \cos\left(\frac{3\pi \cdot \frac{L}{2}}{L}\right) \sin\left(\frac{3\pi v \cdot \frac{L}{18v}}{L}\right) \]
Simplifying gives:
\[ s = A \cos\left(\frac{3\pi}{2}\right) \sin\left(\frac{\pi}{6}\right) \]
Since \( \cos\left(\frac{3\pi}{2}\right) = 0 \), we need to find the acoustic pressure:
\[ P = P_0 + \Delta P \]
Given that the acoustic pressure is \( 0.2\% \) of the atmospheric pressure:
\[ \Delta P = 0.002 P_0 \]
Using the relationship:
\[ \Delta P = B \left| \frac{\partial s}{\partial x} \right| \implies \left| \frac{\partial s}{\partial x} \right| = \frac{\Delta P}{B} = \frac{0.002 P_0}{B} \]
The displacement amplitude \( A \) can be calculated from this equation.

#### (b) In which overtone is the pipe oscillating?

1. **Understanding the harmonics**: The wave equation indicates that the wave is in the form of \( \cos(kx) \sin(\omega t) \). The wave number \( k = \frac{3\pi}{L} \) suggests that:
\[ k = \frac{n\pi}{L} \implies n = 3 \]
Therefore, the pipe is oscillating in the third harmonic (or first overtone).

### Summary of Solutions

1. **(a)** \( \left| \frac{\partial s}{\partial x} \right| = \frac{\Delta P}{B} \)
2. **(b)** \( \left| \frac{\partial s}{\partial x} \right|_C = \frac{\Delta P}{B} \)
3. **(c)** Speed of particle midway between C and R: \( v = 0 \)
4. **(ii)(a)** Displacement amplitude \( A \) can be calculated from \( \Delta P \) and \( B \).
5. **(ii)(b)** The pipe is oscillating in the third harmonic (first overtone).