Question

A wave disturbance in a medium is described by y(x,t)=0.02 cos(50pit+(pi/2)) cos(10pix) where x and y are in meter and t in seconds. Find first node and anti node speed of interfering waves and wavelenght full step wise.

Answer 1

Dear Student
the given equation
$$\begin{gathered} y=0.02\cos \left(50\mathrm{\pi t}+\frac{\mathrm{\pi }}{2}\right)\cos \left(10\mathrm{\pi x}\right) \\ isoftheform \\ Y=2A\cos \left(\omega t+\varphi \right)\cos \left(kx\right) \\ \omega =50\mathrm{\pi } \\ \mathrm{k}=10\mathrm{\pi } \\ \mathrm{A}=0.01 \\ \mathrm{\varphi }=\frac{\mathrm{\pi }}{2} \end{gathered}$$
At node Y=0
Hence cos(10πx)=0
=10πx = nπ/2
x = n/20

at anti node Amplitude is max
= 0.02cos(10πx) should be maximum
=10πx = 0 or nπ
​= x = 0 or n/10

$$\begin{gathered} k=2\pi /\lambda =10\pi \\ \lambda =0.2m \\ \omega =2\mathrm{\pi f}=50\mathrm{\pi } \\ \mathrm{f}=25{\mathrm{s}}^{-1} \\ \mathrm{velocity}=\mathrm{f\lambda } \\ =0.2\times 25 \\ \mathrm{v}=5\mathrm{m}/\sec \end{gathered}$$