The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be ______ %.

The speed of a transverse wave in a stretched string is given by $$v = \sqrt{\frac{T}{\mu}}$$, where $$T$$ is the tension and $$\mu$$ is the linear mass density.

If the tension is increased by 4%, the new tension is $$T' = 1.04T$$. The new speed is $$v' = \sqrt{\frac{1.04T}{\mu}} = \sqrt{1.04} \cdot v$$.

Using the binomial approximation for small changes: $$\sqrt{1 + x} \approx 1 + \frac{x}{2}$$ for small $$x$$. So $$v' \approx \left(1 + \frac{0.04}{2}\right)v = 1.02v$$.

The percentage increase in speed is $$\frac{v' - v}{v} \times 100 = 0.02 \times 100 = 2\%$$.

Therefore, the percentage increase in speed is $$2$$.