Question

A wave pulse is travelling on a string with a speed $\nu$ towards the positive X-axis. The shape of the string at t = 0 is given by g(x) = A sin(x/a), 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 t, if the wave speed is $\nu$.

Answer 1

The shape of the string at t = 0 is given by g(x) = A sin(x/a), where A and a are constants.
Dimensions of A and a are governed by the dimensional homogeneity of the equation g(x) = A sin(x/a).
Now,
$$\begin{gathered} \left(a\right)\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^0\right]=\left[\mathrm{A}\right] \\ \Rightarrow \left[A\right]=\left[\mathrm{L}\right] \\ \mathrm{And}, \\ \left[a\right]=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^0\right] \\ \Rightarrow \left[a\right]=\left[\mathrm{L}\right] \\ \left(b\right)\mathrm{Wave}\mathrm{speed}=\nu \\ \therefore \mathrm{Time}\mathrm{period},T=\frac{a}{\nu } \\ \mathrm{Here}, \\ a=\mathrm{Wave}\mathrm{length}=\lambda \\ \mathrm{The}\mathrm{general}\mathrm{equation}\mathrm{of}\mathrm{wave}\mathrm{is}\mathrm{represented}\mathrm{by} \\ y=A\sin \left\{\frac{x}{a}-\frac{t}{{\displaystyle \frac{{\displaystyle a}}{v}}}\right\} \\ =A\sin \left\{\frac{x-\nu t}{a}\right\} \end{gathered}$$