Question

The transverse displacement of a string (clamped at its both ends) is given by y(x,t)=0.06 sin (2π/3x) cos (120πt) where x and y are in m and tin s. the length of the string is 1.5 m and its mass is 3.0×10² kg. Does the function represent a travelling wave or a stationary wave?

Answer 1

given, wave function , y(x,t ) = 0.06sin(2π/3x)cos(120πt)
we have to resolve the equation of wave to show which type of it.

$y=0.06sin\left(\frac{2\pi}{3}x\right)cos(120\pi t)\\\\=0.03\left[2sin\left(\frac{2\pi}{3}x\right)cos(120\pi t)\right]\\\\\textbf{use formula},2sinA.cosB=sin(A+B)+sin(A-B)\\\\=0.03\left[sin\left(\frac{2\pi}{3}x+120\pi t\right)+sin\left(\frac{2\pi}{3}x-120\pi t\right)\right]\\\\=0.03\left[sin\left(\frac{2\pi}{3}x+120\pi t\right)-sin\left(120\pi t-\frac{2\pi}{3}x\right)\right]$

if we assume $y_1=0.03sin\left(\frac{2\pi}{3}x+120\pi t\right)$
$y_2=0.03sin\left(120\pi t -\frac{2\pi}{3}x\right)$
we get, given equation is the superposition of two waves. hence definitely, it is a stationary wave.