Question

A transverse harmonic wave on a string is described by $y(x,t)=3.0\sin (36t+0.018x+\pi /4)$ Plot the displacement $\left(y\right)$ versus $\left(t\right)$ graphs for$x=0,2$ and $4cm.$ What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency, or phase?

Answer 1

Given,
A transverse harmonic wave on a string is described
$y(x,t)=3.0\sin (36t+0.018x+\frac{\pi }{4})$

At $x=0,$ displacement can be given as

$y(x,t)=3.0\sin (36t+\frac{\pi }{4})$ …$\left(i\right)$

Time period,$T=\frac{2\pi }{\omega }=\frac{2\pi }{36}=\frac{\pi }{18}$

The displacement versus time can be plotted using equation $\left(i\right)$ at different instants. The plot will be sinusoidal with some initial displacements. Similarly, it can be plotted at positions $x=2$ and $4cm$

Initial displacement of the particle at $x=0$

$y=3.0\sin \left(\frac{\pi }{4}\right)=\frac{3}{\sqrt{2}}$

$=2.12cm$

Initial displacement of the particle at $x=2cm$

$y=3.0\sin (0.018\times 2+\frac{\pi }{4})$

$=3.0\sin (0.036+\frac{\pi }{4})$

$=2.19cm$

Initial displacement of the particle at $x=4cm$

$y=3.0\sin (0.018\times +\frac{\pi }{4})=3.0\sin (0.072+\frac{\pi }{4})$

$=2.27cm$

All oscillatory motion have same amplitude and frequency but can have different initial phase.