Question

The equation of the stationary wave is
$y = 2A\sin \left( {\dfrac{{2\pi ct}}{\lambda }} \right)\cos \left( {\dfrac{{2\pi x}}{\lambda }} \right)$
Which of the following statements is wrong?
(A) The unit of $c\lambda$ is the same as that of $\lambda$.
(B) The unit of $x$ is the same as that of $\lambda$.

Answer 1

Angles have no dimensions. They can be expressed as mass, length, and time raised to powers of zero.

Complete Step by Step Solution: Standing wave, also called stationary wave, combination of two waves moving in opposite directions, each having the same amplitude and frequency. The phenomenon is the result of interference; that is, when waves are superimposed, their energies are either added together or cancelled out. It is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.
Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.
Dimensionless quantities are those which do not have dimensions but have a fixed value.
-Dimensionless quantities without units: Pure numbers, $\pi$, e, $\sin \theta$, etc.
-Dimensionless quantities with units: Angular displacement – radian, Joule’s constant – joule/calorie, etc.
Since angles have no dimensions, dimensions of $\dfrac{{2\pi ct}}{\lambda } = {M^0}{L^0}{T^0}$
Thus, $ct = {M^0}{L^1}{T^0}$
$ct = \left[ {{L^1}} \right] = \lambda$
So the dimension of The unit of $ct$ is the same as that of $\lambda$.
Dimensions of $\dfrac{{2\pi x}}{\lambda } = {M^0}{L^0}{T^0}$. This is because angles have no dimensions.
$x = {M^0}{L^1}{T^0}$
$x = \left[ {{L^1}} \right] = \lambda$
Hence, the unit of $x$ is the same as that of $\lambda$.

Thus, both options A and B are correct.

Note: We already know that $2\pi$ is dimensionless, since it is a number.
Dimensions of $\dfrac{{2\pi c}}{\lambda } = \dfrac{{{L^1}{T^{ - 1}}}}{{{L^1}}}$
Dimension of $\dfrac{{2\pi x}}{{\lambda t}} = \dfrac{{{L^1}{T^{ - 1}}}}{{{L^1}}}$.
So dimensions of both are equal.
Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of the physical quantities. These dimensions are independent of the numerical multiples and constants and all the quantities in the world can be expressed as a function of the fundamental dimensions.
The expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the dimensional formula of that quantity.
The important applications of dimensional analysis are
1. To convert the value of a physical quantity from one system to another.
2. To check the correctness of a given relation.
3. To derive a relation between various physical quantities.