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