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) = Asin(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,

\[(a)    \left[ M^0 L^1 T^0 \right] = \left[ A \right]\] 

\[ \Rightarrow \left[ A \right] = \left[ L \right]\] 

\[And,   \] 

\[\left[ a \right] = \left[ M^0 L^1 T^0 \right]\] 

\[ \Rightarrow \left[ a \right] = \left[ L \right]\] 

\[\] 

(b)  Wave  speed =\[ \nu\] 

\[ \therefore  \text{ Time  period, }  T = \frac{a}{\nu}\] 

Here,

a = Wave  length = \[\lambda  \] 

 The  general  equation  of  wave  is  represented  by

\[y = A\sin\left\{ \frac{x}{a} - \frac{t}{\frac{a}{v}} \right\}\] 

\[       = A\sin\left\{ \frac{x - \nu t}{a} \right\}\]