Question

A wave has a wavelength of 3.0 m, a frequency of 25.0 Hz, and an amplitude of 14.0 cm. The wave travels in the positive x-direction and has a displacement of zero at t = 0 and x = 0. How many complete oscillations has the wave made at t = 20.0 s and x = 4.2 m?

Answer 1

y=A sin(ωt-kx) where ω is the angular frequency and k is the wavenumber.

In terms of frequency f and wavelength λ, the wave is y=Asin[2π(f t - x/λ) ].

y = 0 when x = 0 and t = 0.

Therefore, calculating how many times it happens till the given time and distance.
since   complete oscillation means that the phase 2pi*(f t - x/λ) changes by 2pi and y returns to zero
y = 0 when x = 0 and t = 0.

x=4.2 t=20 ; 2pi*(f t - x/λ) = 2pi*(25*20 -4.2/3) =2pi*498.6

Since a change of 2pi = 1complete oscillations

Therefore ,2pi*498.6 =2pi*498.6 *1/2pi =498.6 Complete oscilations