A sonometer wire '$$A$$' of diameter '$$\mathrm{d}$$' under tension '$$T$$' having density '$$\rho_1$$' vibrates with fundamental frequency '$$n$$'. If we use another wire '$$B$$' which vibrates with same frequency under tension '$$2 \mathrm{~T}$$' and diameter '$$2 \mathrm{D}$$' then density '$$\rho_2$$' of wire '$$B$$' will be

The path difference between two waves, represented by $$\mathrm{y}_1=\mathrm{a}_1 \sin \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}\right)$$ and $$y_2=a_2 \cos \left(\omega t-\frac{2 \pi x}{\lambda}+\phi\right)$$ is

Two progressive waves are travelling towards each other with velocity $$50 \mathrm{~m} / \mathrm{s}$$ and frequency $$200 \mathrm{~Hz}$$. The distance between the two consecutive antinodes is

A string fixed at both the ends forms standing wave with node separation of $$5 \mathrm{~cm}$$. If the velocity of the wave on the string is $$2 \mathrm{~m} / \mathrm{s}$$, then the frequency of vibration of the string is