# Waves - Class 11 - Physics

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## Extra Questions - Waves | NCERT | Physics | Class 11

Consider two waves passing through the same string. The principle of superposition for displacement says that the net displacement of a particle on the string is the sum of the displacements produced by the two waves individually. Suppose we state similar principles for the net velocity of the particle and the net kinetic energy of the particle. Such a principle will be valid for:

A) Both the velocity and the kinetic energy.

B) The velocity but not for the kinetic energy.

C) The kinetic energy but not for the velocity.

D) Neither the velocity nor the kinetic energy.

The correct answer is **Option B: The velocity but not for the kinetic energy**.

According to the **principle of superposition**, the total displacement, $\vec{y}{\text{net}}$, of a particle on a string due to two waves is given by the vector sum of the displacements caused by each wave:
$$
\vec{y}{\text{net}} = \vec{y}_1 + \vec{y}_2
$$

Here, $\vec{y}_1$ and $\vec{y}2$ are the displacement vectors produced by the two individual waves. To find the net velocity, we differentiate the displacement equation with respect to time: $$ \frac{d\vec{y}{\text{net}}}{dt} = \frac{d\vec{y}_1}{dt} + \frac{d\vec{y}2}{dt} $$ $$ \vec{v}{\text{net}} = \vec{v}_1 + \vec{v}_2 $$

Hence, the velocities also add vectorially, and the principle of superposition is applicable to **net velocity**.

To determine if this principle applies to **kinetic energy**, consider the kinetic energy, which is proportional to the square of the velocity. Squaring the net velocity:
$$
v_{\text{net}}^2 = (\vec{v}_1 + \vec{v}2)^2
$$
Expanding this, we get:
$$
v{\text{net}}^2 = v_1^2 + v_2^2 + 2\vec{v}_1 \cdot \vec{v}_2
$$

Since $v_{\text{net}}^2$ is not simply $v_1^2 + v_2^2$, but includes an additional cross-term, $2\vec{v}_1 \cdot \vec{v}_2$, it is evident that the **principle of superposition does not apply to kinetic energy**. This cross-term involves a product of velocities, indicating that kinetic energies do not simply add when two waves interact. Thus, any effect like interference (constructive or destructive) could alter the combined kinetic energy differently from merely summing them.

Therefore, **the principle of superposition holds for the net velocity of the particle but not for the net kinetic energy** in the context of waves on a string.

The equation of two light waves are $y_{1} = 6 \cos \omega t$, $y_{2} = 8 \cos (\omega t + \phi)$. The ratio of maximum to minimum intensities produced by the superposition of these waves will be

A) $49: 1$

B) $1 : 49$

C) $1 : 7$

D) $7 : 1$

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The displacement due to a wave moving in the positive $x$-direction is given by $y = \frac{1}{\left(1 + x^{2}\right)}$ at time $t = 0$ and by $y = \frac{1}{1 + (x-1)^{2}}$ at $t = 2$ seconds, where $x$ and $y$ are in meters. The velocity of the wave in m/s is:

A) 0.5

B) 1

C) 2

D) 4