Question

Assertion: In a sinusoidal traveling wave on a string, the potential energy of the deformation of the string element at extreme position is maximum.
Reason: The particles in the sinusoidal traveling wave perform SHM.
A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C) Assertion is correct but Reason is incorrect
D) Assertion is incorrect but Reason is correct

Answer 1

Potential energy of deformation of a spring element can be related to the position of the string element. A sinusoidal wave function’s amplitude or frequency remains the same even if the position of the wave is changed. The sinusoidal oscillation of a mass on a spring is one of the best examples to understand simple harmonic motion.

Complete step-by-step solution:
We are provided with an assertion and a reason. We are required to find if the assertion as well as the reason are correct and if the given reason is the correct explanation of the assertion. The assertion in the question states that: In a sinusoidal traveling wave on a string, the potential energy of deformation of the string element at extreme position is maximum. The reason is that the question states that: The particles in the sinusoidal traveling wave perform SHM. Let us check if these statements are true.
A spring-mass system is one of the best examples of simple harmonic motion. Simple harmonic motion can be defined as a special type of periodic motion in which the restoring force tends to bring back a particle to its equilibrium position and has a magnitude proportional to the distance of the particle from its equilibrium position. If we release a mass from a spring of a particular spring constant, the mass executes simple harmonic motion. The mass tries to deform the spring, but a restoring force pushes it back to its equilibrium position. It is to be kept in mind that the potential energy of a spring element can be calculated at any given position of the spring, whether it be the equilibrium position or an extreme position. Oscillation of the mass can be expressed as a sinusoidal wave. The general expression for a sinusoidal wave is given by
$y(x,t)=A\sin (kx-\omega t+\phi )$
where
$y(x,t)$ is the sinusoidal wave
$A$ is the maximum amplitude
$\omega$ is the angular frequency
$x$ is the spatial coordinate
$k$ is the wavenumber
$t$ is the time at which the wave is recorded
$\phi$ is the phase constant
Now, let us change the spatial coordinate of the wave and see if the equation of wave changes or not. Suppose
$x={{x}_{0}}$
where
${{x}_{0}}$ is another spatial coordinate
The sinusoidal wave equation of oscillation for this spatial coordinate can be written as:
$y({{x}_{0}},t)=A\sin (k{{x}_{0}}-\omega t+\phi )$
where
$y({{x}_{0}},t)$ is the sinusoidal wave
$A$ is the maximum amplitude
$\omega$ is the angular frequency
${{x}_{0}}$ is the spatial coordinate
$k$ is the wavenumber
$t$ is the time at which the wave is recorded
$\phi$ is the phase constant
It can be concluded from the above expressions that the sinusoidal wave oscillates with the same amplitude and frequency, even when spatial coordinates are changed.
Since potential energy is directly proportional to the square of spatial coordinate, the amplitude, and frequency of the sinusoidal equation of potential energy to remain the same, at any given point of consideration. It needs to be noted here that phase differences can happen.
Hence, the given assertion is incorrect.
Also, since all the particles of the sinusoidal wave travel with the same amplitude and frequency all the time, it can be said that these particles execute simple harmonic motion always.
Hence, the given reason is correct.
Therefore, the correct answer is option D.

Note: The relation between potential energy and spatial coordinate is given by
$P.E=\dfrac{1}{2}k{{x}^{2}}$
Students can perform the calculation by writing potential energy in a waveform and perform necessary calculations, to see for themselves, that potential energy oscillates with the same amplitude and frequency at any point of consideration, even though there is a change in phase.