Following are expressions for four plane simple harmonic waves (i) $$y_1 = A \cos 2\pi \left(n_1 t + \dfrac{x}{\lambda_1}\right)$$ (ii) $$y_2 = A \cos 2\pi \left(n_1 t + \dfrac{x}{\lambda_1} + 0.5\right)$$ (iii) $$y_3 = A \cos 2\pi \left(n_2 t + \dfrac{x}{\lambda_2}\right)$$ (iv) $$y_4 = A \cos 2\pi \left(n_2 t - \dfrac{x}{\lambda_2}\right)$$. The pairs of waves which will produce destructive interference and stationary waves respectively in a medium, are

For destructive interference, two conditions must be fulfilled:
1. Both waves must have the same amplitude, frequency and direction of propagation.
2. The phase difference between them must be $$\pi$$ (or an odd multiple of $$\pi$$), i.e. path difference $$=\frac{\lambda}{2}, \frac{3\lambda}{2}, \ldots$$

Case 1: Waves (i) and (ii)
Let $$\phi = 2\pi\!\left(n_1 t + \dfrac{x}{\lambda_1}\right)$$.
$$y_1 = A\cos\phi,\qquad y_2 = A\cos(\phi+\pi) = -A\cos\phi$$
Superposition gives $$y = y_1 + y_2 = A\cos\phi - A\cos\phi = 0$$ for every $$t$$ and $$x$$.
Thus the two waves cancel everywhere - complete destructive interference.

For a stationary (standing) wave, two identical waves must travel in opposite directions with the same amplitude and frequency. Their superposition is of the form $$y = 2A\cos(\omega t)\cos(kx)$$, exhibiting fixed nodes and antinodes.

Case 2: Waves (iii) and (iv)
Write
$$\theta = 2\pi\!\left(n_2 t + \dfrac{x}{\lambda_2}\right),\qquad \theta' = 2\pi\!\left(n_2 t - \dfrac{x}{\lambda_2}\right)$$
Both have the same $$n_2, \lambda_2, A$$ but opposite signs before $$x$$, so they travel in opposite directions.
$$y_3 = A\cos\theta,\qquad y_4 = A\cos\theta'$$
Using the identity $$\cos a + \cos b = 2\cos\!\left(\dfrac{a+b}{2}\right)\cos\!\left(\dfrac{a-b}{2}\right)$$, we get
$$y = y_3 + y_4 = 2A\cos\!\left[2\pi n_2 t\right]\cos\!\left[\dfrac{2\pi x}{\lambda_2}\right]$$
This expression has a time-dependent part $$\cos(2\pi n_2 t)$$ and a space-dependent part $$\cos\!\left(\dfrac{2\pi x}{\lambda_2}\right)$$, but no term of the form $$f(x\pm vt)$$, hence it represents a stationary wave with nodes where $$\cos\!\left(\dfrac{2\pi x}{\lambda_2}\right)=0$$.

Therefore:
• Destructive interference pair: (i) and (ii).
• Stationary-wave pair: (iii) and (iv).

Option D which is: (i, ii), (iii, iv)