The equation of two waves having the same amplitude, wavelength, and speed but propagating in opposite directions is
\(y_1 = a\sin \frac{2 \pi} \lambda (v t - x)\) and
\(y_2 = a \sin \frac{2\pi }\lambda (vt + x)\)
Where a is the amplitude, \(\lambda\) is the wave-length and v is the velocity of the wave. A stationary wave is formed due to the superposition of these two waves. The resultant displacement of a particle is given by,
\(y = y_1 + y_2\)
\(= a\sin \frac{2\pi} \lambda (vt - x) + a\sin \frac{2 \pi } \lambda (vt + x)\)
Using the relation,
\(\sin C + \sin D = 2 \sin \frac {C + D} 2 \cos \frac{C - D} 2\)
we have \(y = 2a \cos \frac{2 \pi } \lambda x . \sin \frac{2π}\lambda vt\)
\(=A \sin \frac {2\pi }\lambda vt\)
where \(A = 2a \cos \frac {2 \pi }\lambda x\) represents the amplitude of the resultant wave.
