Question

The displacement of a standing wave on a string is given by
`y (x,t) = 0.4 sin (0.5x) cos (30t)`
where x and y are in centimetres.
(a) Find the frequency, amplitude and wave speed of the component waves.
(b) What is the particle velocity at `x=2.4 cm` at `t=0.8 s`?

Answer 1

Correct Answer - A::B::C
(a) The given equation can be written as the sum of two component waves as
`y(x,t) = 0.2 sin (0.5x -30t)+0.2sin (0.5x +30t)`
Hence, the two component waves are
` y_1(x,t) = 0.2 sin (0.5x - 30t)`
travelling in positive x-direction and
Now, `omega = 30"rad"//s`
and ` k=0.5 cm^-1`
Frequency, `f=omega/(2pi) = 15/pi Hz`
Amplitude, A=0.2 cm
and wave speed, `v=omega/k = 30/0.5`
=60 cm/s .
(b) Particle velocity,
`v_P(x,t)= (dely)/(delt)=-12sin(0.5x)sin(30t)`
`:.v_P(x= 2.4 cm, t=0.8s)`
`=-12sin(1.2) sin(24)`
=10.12 cm/s.