A wave read from two graphs

Problem statement

The equation of a transverse wave travelling along the positive direction of the $x$ axis is $y(x,t)=A\sin(\omega t-kx+\phi)$. Figure (a) shows the displacement of the different points of the medium at the instant $t=0$. Figure (b) shows the oscillation velocity of the point $x=0$ of the medium as a function of time. Determine:

a) (1 mark) The amplitude of the wave $A$ and the wave number $k$.

b) (1 mark) The phase of the wave $\phi$ and the maximum value of the oscillation velocity.

c) (0.5 marks) The propagation speed of the wave.

[Figure (a), «$t=0$»: $y(x,0)$ in cm against $x$ in m, from 0 to 12 m. The curve is a sinusoid that is 0 at $x=0$ and goes down: minimum of $-4$ cm at $x=1.5$ m, zero at 3 m, maximum of $+4$ cm at 4.5 m, zero at 6 m, and it repeats up to 12 m. Figure (b), «$x=0$»: oscillation velocity in m/s against $t$ in s, from 0 to 9 s. The curve starts at its positive maximum, a little above the 0.05 m/s line, is zero at $t=1$ s, has its minimum at 2 s, is zero at 3 s and returns to the maximum at 4 s and at 8 s.]

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Physics · Question 3.A. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.