At time t = 0 the point at x = 0 has velocity v0 and displacement y0. The phase constant φ is given by tanφ =:

This question is incomplete, the complete question is;

The displacement of a string carrying a traveling sinusoidal wave is given by y(x,t)=ymsin(kx - ωt -φ) .

At time t = 0, the point at x = 0 has velocity v₀ and displacement y₀.

The phase constant φ is given by tanφ =:

A) ωv₀ /y₀    

B) ωv₀ y₀  

C) v₀ /ωy₀  

D) y₀ /ωv₀      

E) ωy₀ /v₀

E) ωy₀ /v₀

Explanation:

Given that;

displacement of a wave is; y(x,t) = ym sin (kx - ωt - φ)

we differentiate the given equation with respect to time

d/dt (y(x,t)) = d/dt(ym sin(kx - ωt - φ) )

v(0,0)) = -ym ωcos (k(0) - ω(0) - φ) )

v₀ = -ym ωcos (-φ)   lets leave thisas equ 1

At t = 0, x = 0

the displacement of the wave is

y(0,0) = ym sin (k(0) - ω(0) - φ)

y₀ = ym sin(-φ) let this be equ 2

y₀/v₀ = (ym sin(-φ)) / (-ym ωcos (-φ)) = ( -ym sin(φ)) / (-ym ωcos (φ))

(tanφ)/ω = y₀/v₀

tanφ = y₀ω/v₀

therefore the required value is y₀ω/v₀

option (E).