Question

Write the expression for y as a function of x and t for a sinusoidal wave along a rope in the negative x direction with the following characteristics: A = 8.00 cm: lambda = 80.0 cm: f= 3.00 Hz and y(0, t) = 0 at t = 0. Write the expression for y as a function of x and t for the wave in part (a) assuming y(x, 0) = 0 at x = 10.0 cm.

Answer 1

Amplitude A = 8.00 cm

wavelength $\lambda$ = 80.0 cm

frequency f = 3.00 Hz

general equation for a sinusoidal wave travelling along the negative x axis

$y = A \sin {(\frac{2\pi\,x}{\lambda}+2\pi\,f\,t+\phi)}$

where $\phi$ is the phase constant.

(a)

since y (0, t) = 0 for t = 0

$\phi$ = 0

hence wave equation is $y = (0.08 ) \sin {(\frac{2\pi\,x}{0.80}+2\pi\times\,3.00\,t)}$

where A and $\lambda$ are in meters.

(b)

since y (x, 0) = 0 for x = 10.0 cm

wave equation is $y = (0.08 ) \sin {(\frac{2\pi\times0.10}{0.80}+2\pi\times\,3.00\times\,0+\phi)}$

where A and $\lambda$ are in meters.

so $\phi=\frac{-\pi}{4}$

hence wave equation is $y = (0.08 ) \sin {(\frac{2\pi\,x}{0.80}+2\pi\times\,3.00\times\,t+\frac{-\pi}{4})}$

where A and $\lambda$ are in meters.