Question

obtain an expression for the stationary wave formed by two sinusoidal waves travelling along the same path in the opposite direction and obtain the position of nodes and antinodes​

Answer 1

Answer:

For position of nodes we have

$x = (2N + 1)\frac{\lambda}{4}$

For position of antinodes we have

$\lambda = \frac{N\lambda}{2}$

Explanation:

As we know that equation of travelling wave is given as

$y = A sin(\omega t - kx)$

now another wave is coming from opposite side

$y = A sin(\omega t + kx)$

now by superposition of above two waves we have

$y = A sin(\omega t - kx) + A sin(\omega t + kx)$

so we will have

$y = A (2sin\omega t cos kx)$

$y = (2A coskx) sin\omega t$

now for position of nodes net amplitude of the point must be zero

$2Acoskx = 0$

$\frac{2\pi}{\lambda} x = (2N + 1)\frac{\pi}{2}$

$x = (2N + 1)\frac{\lambda}{4}$

for position of anitinode net amplitude must be maximum

$2Acos kx = \pm 1$

$\frac{2\pi}{\lambda} x = N\pi$

$\lambda = \frac{N\lambda}{2}$

#Learn

Topic : Standing waves

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