The mass per unit length of a uniform wire is 0.135 g cm$$^{-1}$$. A transverse wave of the form $$y = -0.21\sin(x + 30t)$$ is produced in it, where $$x$$ is in meter and $$t$$ is in second. Then, the expected value of tension in the wire is $$x \times 10^{-2}$$ N. Value of $$x$$ is ______ (Round-off to the nearest integer)

The transverse wave is given by $$y = -0.21\sin(x + 30t)$$, where $$x$$ is in meters and $$t$$ is in seconds. Comparing with the standard form $$y = A\sin(kx + \omega t)$$, we get the wave number $$k = 1$$ m$$^{-1}$$ and angular frequency $$\omega = 30$$ rad s$$^{-1}$$.

The wave speed is $$v = \frac{\omega}{k} = \frac{30}{1} = 30$$ m s$$^{-1}$$.

The mass per unit length is $$\mu = 0.135$$ g cm$$^{-1} = 0.0135$$ kg m$$^{-1}$$.

For a transverse wave on a string, the wave speed is related to tension by $$v = \sqrt{\frac{T}{\mu}}$$. Solving for tension:

$$T = \mu v^2 = 0.0135 \times (30)^2 = 0.0135 \times 900 = 12.15$$ N

Expressing as $$x \times 10^{-2}$$ N: $$12.15 = 1215 \times 10^{-2}$$, so $$x = \mathbf{1215}$$.