Question

Write an expression describing a transverse waves traveling along the string in the+ x direction wave length 11.4 cm,frequency 385hz ,and amplitude 2.13

Answer 1

Explanations & Calculations

• The relationship for the displacements of a particle on a transverse sinusoidal wave moving in +x direction is given by,

$\qquad\qquad \begin{aligned} \small y_{(x,t)}&= \small A\sin (kx-\omega t)\\ \end{aligned}$

• A = amplitude of the wave
• $\small k=\large \frac{2\pi}{\lambda}$ and $\small \omega =2\pi f=2\pi \large\frac{v}{\lambda}$
• $\small f=$ frequency and $\small v=$ speed of the wave

• Substituting the given values,
• $\small k=\large\frac{2\pi}{0.114m}=\small 55.12\,m^{-1}$
• $\small \omega = 2\pi\times 385s^{-1}=2419.03\,s^{-1}$

• These yeild,

$\qquad\qquad \begin{aligned} \small\bold{ y_{(x,t)}}&= \small \bold{2.13(55.12x-2419.03t)} \end{aligned}$

• Additionally, the speed of a transverse wave on a string is given by

$\qquad\qquad \begin{aligned} \small v&= \small \sqrt {\frac{T}{\rho}} \end{aligned}$ : $\small \rho$ is the mass per unit length/linear density

• And also waves on a string obays the wave equation,

$\qquad\qquad \begin{aligned} \small \frac{\partial^2y}{\partial x^2}&= \small \frac{1}{v^2}\frac{\partial^2y}{\partial t^2} \end{aligned}$