If the amplitudes of the two waves are 3 units and 1 unit respectively, show by the principle of superposition that the ratio of the amplitudes of the stationary wave at an antinode and node respectively 2:1
Answer
To show the ratio of the amplitudes of the stationary wave at an antinode and node, we can use the principle of superposition.
The principle of superposition states that when two or more waves meet at a point, the resultant displacement at that point is the algebraic sum of the individual displacements.
Let's consider two waves with amplitudes 3 units and 1 unit.
When these waves meet, the displacement at any point is obtained by adding the individual displacements of the waves at that point.
At an antinode, the waves are in phase and have the same displacements. Therefore, the resulting amplitude at an antinode is obtained by adding the amplitudes of the two waves.
Amplitude at antinode = 3 units + 1 unit = 4 units
At a node, the waves are out of phase and have opposite displacements. Therefore, the resulting amplitude at a node is obtained by subtracting the amplitude of one wave from the other.
Amplitude at node = 3 units - 1 unit = 2 units
So, the ratio of the amplitudes at an antinode and node is 4 units : 2 units, which simplifies to 2 : 1.
Therefore, by the principle of superposition, the ratio of the amplitudes of the stationary wave at an antinode and node is 2 : 1.