Answer to Question #316466 in Physics for Wellie

Question

A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical position of an element of the medium at t = 0 and x = 0 is also 15.0 cm, Determine the phase constant , and write a general expression for the wave function 

Answer 1

Given:

$A=15.0\:\rm cm$

$\lambda=40.0\:\rm cm$

$f=8.00\:\rm Hz$

$y(0,0)=15.0\:\rm cm$

The general expression for the wave function

$$y(x,t)=A\cos\left(\frac{2\pi}{\lambda}x-2\pi ft+\phi_0\right)$$

$$y(x,t)=15.0\cos\left(\frac{2\pi}{40.0}x-2\pi*8.00t+\phi_0\right)$$

$$y(x,t)=15.0\cos\left(0.157x-20.2t+\phi_0\right)$$

The initial condition gives

$$y(0,0)=15.0\cos\left(\phi_0\right)=15.0$$

Hence

$$\cos\left(\phi_0\right)=1,\quad \phi_0=0$$

Finally

$$y(x,t)=15.0\cos\left(0.157x-20.2t\right)$$