Question

Consider the three waves \[{z_1},{z_2}\] and \[{z_3}\] as
\[{z_1} = A\sin \left( {kx - \omega t} \right)\]
\[{z_2} = A\sin \left( {kx + \omega t} \right)\]
\[{z_3} = A\sin \left( {ky + \omega t} \right)\]
Which of the following represents a standing wave?
A. \[{z_1} + {z_2}\]
B. \[{z_2} + {z_3}\]
C. \[{z_3} + {z_1}\]
D. \[{z_1} + {z_2} + {z_3}\]

Answer 1

Hint: Two waves travelling in opposite directions form standing waves. The phase from the equation of motion of the wave gives an idea about the direction of the motion of the wave.

Complete step by step solution:
The wave with the equations \[{z_1} = A\sin \left( {kx - \omega t} \right)\] is moving towards +x-axis.
The wave with the equations \[{z_2} = A\sin \left( {kx + \omega t} \right)\] is moving towards the -x-axis.
The wave with the equations \[{z_1} = A\sin \left( {ky + \omega t} \right)\] is moving towards -y -axis.

The standing wave is the combination of two waves moving in opposite directions. When two waves moving in opposite directions encounter the interference then there is addition of the intensities of the individual waves and cancellation of the intensities. The point with maximum intensity is called the antinode and the point with minimum intensity is called the node.

As the intensity is proportional to the square of the amplitude, so; at node the amplitude is zero and at antinode the amplitude is maximum. The standing wave is also called a stationary wave. The waves in a standing wave should have the same amplitude and frequency. To form the standing wave the waves should be having the same frequency.

The amplitudes of all three given waves are equal. The frequency of all the three given waves are equal. But, the wave \[{z_3}\] is moving along the -y-axis which is perpendicular to the motion of the other two waves. The waves \[{z_1}\] and \[{z_2}\] are having equal frequency as well moving towards each other. Hence, the standing wave will be formed by the waves \[{z_1}\] and \[{z_2}\].

Therefore, the correct option is A.

Note: The direction of the wave is decided using the phase. As the phase of the wave must be constant. So on differentiating the phase if the velocity comes negative then it is travelling towards left or it is moving towards right.