Question

A wave pulse is travelling on a string with a speed $u$ toward the position $X-axis$ . The shape of the string at $t=0$ is given by $g\left(x\right)=ASin\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)$, where $A$ and $a$ are constants.

(a) What are the dimensions of $A$ and $a$?

(b) Write the equation of the wave for a general time , if the wave speed is $u$.

Answer 1

Step 1. Given Data,

At $t=0$, $g\left(x\right)=ASin\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)$

Step 2. Find the dimension of $A$ and $a$

$$\begin{gathered} Dimensionofg\left(x\right)=DimensionofA \\ \left[L\right]=A \end{gathered}$$

Similarly,

$$\begin{gathered} Dimensionofg\left(x\right)=Dimensionofa \\ \left[L\right]=a \end{gathered}$$

Step 3. Write the equation of the wave for a general time , if the wave speed is $u$,

$$\begin{gathered} T=\frac{\lambda }{\nu } \\ here,\lambda =a \\ \Rightarrow T=\frac{a}{\nu } \end{gathered}$$

Therefore, wave equation for general time $t$,

$$\begin{gathered} Y=ASin\left[\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)-\left(\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)\right] \\ \Rightarrow Y=ASin\left[\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)-\left(\raisebox{1ex}{$vt$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)\right] \end{gathered}$$

Hence, the equation of the wave for a general time is $Y=ASin\left[\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)-\left(\raisebox{1ex}{$vt$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)\right]$.