The displacement of a progressive wave is represented by y = A sin(wt - kx), where x is distance and t is time. Write the dimensional formula of i. w and ii. k.
Understand the Problem
The question asks us to find the dimensional formula (or dimensional analysis) of angular frequency (ω) and wave number (k) given the displacement equation of a progressive wave y = A sin(ωt - kx). This involves understanding the dimensions of displacement (y), amplitude (A), time (t), and distance (x), then using the principle of dimensional homogeneity to deduce the dimensions of ω and k.
Answer
The dimensional formula for angular frequency $\omega$ is $[T^{-1}]$, and for wave number $k$ is $[L^{-1}]$.
Answer for screen readers
The dimensional formula for angular frequency $\omega$ is $[T^{-1}]$.
The dimensional formula for wave number $k$ is $[L^{-1}]$.
Steps to Solve
- Dimensions of displacement (y) and amplitude (A)
Since $y$ represents displacement and $A$ represents amplitude, both are lengths. Therefore, their dimensions are $[L]$.
- Dimensions of time (t) and distance (x)
The dimension of time $t$ is $[T]$ and the dimension of distance $x$ is $[L]$.
- Dimensional homogeneity of the argument of the sine function
The argument of the sine function, $(\omega t - kx)$, must be dimensionless. This means that each term in the argument must also be dimensionless.
- Dimensional analysis of $\omega t$
For $\omega t$ to be dimensionless, the dimension of $\omega t$ must be 1. This can be written as:
$[\omega t] = 1$
$[\omega][t] = 1$
$[\omega][T] = 1$
Solving for $[\omega]$ gives:
$[\omega] = [T]^{-1}$ or $[T^{-1}]$
- Dimensional analysis of $kx$
Similarly, for $kx$ to be dimensionless:
$[kx] = 1$
$[k][x] = 1$
$[k][L] = 1$
Solving for $[k]$ gives:
$[k] = [L]^{-1}$ or $[L^{-1}]$
The dimensional formula for angular frequency $\omega$ is $[T^{-1}]$.
The dimensional formula for wave number $k$ is $[L^{-1}]$.
More Information
Angular frequency ($\omega$) represents the rate of change of an angle, measured in radians per second. Wave number ($k$) represents the spatial frequency of a wave, indicating the number of waves per unit distance.
Tips
A common mistake is to assign dimensions to the sine function itself. Remember that trigonometric functions are dimensionless; only their arguments must be dimensionless.
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