Assertion : In a small segment of string carrying sinusoidal wave, total energy is conserved.

Reason: Every small part moves in $\mathrm{SHM}$ and in SHM total energy is conserved.

Assertion : Two waves moving in a uniform string having uniform tension can have different velocities.
Reason : Elastic and inertial properties of string are same for all waves in same string. Moreover, speed of wave in a string depends on its elastic and inertial properties only.

Assertion :In standing wave pattern particle of medium between two consecutive nodes vibrates in same phase but with different amplitude.

Reason : In stationary wave the amplitude of vibration does not depend on the position of the particle.

Assertion : The transverse wave is travelling along a string in the positive $x$ axis as shown

Points $\left(A\&P_1\right)$ moving $\downarrow$ (downward) and points $\left(C\&P_2\right)$ moving (upward).

Reason : In a wave propagating in positive $x$ direction the points with $+ve$ slope move downward and vice versa.

Assertion : In transverse wave particle velocity is perpendicular to the direction of wave velocity.

Reason : In wave motion energy always transferred in the direction of wave propagation.

Assertion : The wave function of a pulse is given by $y=\frac{3}{(2x+3t+4)}$ is propagates in ($-\text{ve} x-$ direction)

Reason : The given wave function is of the form $y=f(kx+\omega t)$ which represent a wave travelling in negative $x$- direction.

Assertion : Standing waves do not transfered energy in the medium.

Reason : Every particle vibrates with its own energy and it does not share its energy with any other particle

Assertion : A wave can be represented by function $y=f(kx\pm \omega t).$ 

Reason : Because it satisfies the differential equation
$\frac{{\partial }^2\mathrm{y}}{\partial {\mathrm{x}}^2}=\frac{1}{{\mathrm{v}}^2}\left(\frac{{\partial }^2\mathrm{y}}{\partial {\mathrm{t}}^2}\right)\text{ where }\mathrm{v}=\frac{\omega }{\mathrm{k}}$