Question

For the wave described in Exercise 14.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. 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

All the waves have different phases.

The given transverse harmonic wave is:

\(y(x,t)=3.0\,sin(36\,t+0.018x+\frac{\pi}{4})...(i)\)

For x = 0, the equation reduces to:

\(y(0,t)=3.0\,sin(36\,t+\frac{\pi}{4})\)

Also, ω=\(\frac{2\pi}{T}=36\,rad/s^{-1}\)

\(∴T=\frac{\pi}{8}s\)

Now, plotting y vs. t graphs using the different values of t, as listed in the given table.

t(s)	0	\(\frac{7}{8}\)	\(\frac{2T}{8}\)	\(\frac{3T}{8}\)	\(\frac{4T}{8}\)	\(\frac{5T}{8}\)	\(\frac{6T}{8}\)	\(\frac{7T}{8}\)
y(cm)	\(\frac{3\sqrt2}{2}\)	3	\(\frac{3\sqrt2}{2}\)	0	\(\frac{-3\sqrt2}{2}\)	-3	\(\frac{-3\sqrt2}{2}\)	0

For x = 0, x = 2, and x = 4, the phases of the three waves will get changed. This is because amplitude and frequency are invariant for any change in x. The y-t plots of the three waves are shown in the given figure.