Question

Two waves travelling in opposite directions along $ \pm x$ the axis. Their wave equations are expressed as:
${y_1} = 10\cos \left( {\dfrac{x}{2} + 5t} \right)$ and
${y_2} = 10\cos \left( {\dfrac{x}{2} - 5t} \right)$
Where x and y are in cms and t is in seconds. After superposition, they form a standing wave. The amplitude at the nodes will be
A. 0 cm
B. 10 cm
C. 20 cm
D. 40 cm

Answer 1

Hint: We all know that stationary waves are generally said to be standing waves. These waves are usually travelling opposite to each other, and the moment they do superposition, then their energy component is added or cancelled out.

Complete step by step answer:
When we represent a wave, then we denote it by the general equation:
$y = A\cos \left( {\omega t + \varphi } \right)$
Here, y is the displacement from the mean position, A is the amplitude, $\omega $ is the angular frequency and $\phi $ is the phase angle.The phase angle is an imaginary angle which means that when a harmonic motion is mapped into a circular motion, then we can say that the angle of the particle from the initial position while travelling is phase angle. Angular frequency is the number of radians a particle is covering in a second.
We know that when a standing wave after striking from the fixed point follows the superposition on the same wave itself, then the final amplitude is the summation of the separate amplitude of two waves. We know that node is the point of least intensity and the amplitude is zero at node because the displacement of the particle is zero from the mean position. Therefore the amplitude at node is zero and the correct option is (A).

Note: We must be knowing about the travelling wave, which is produced when two stationary waves interfere with each other while moving in the same direction. Still, we should keep in mind that the oscillating wave is created when two standing waves are moving in the opposite direction.