A travelling harmonic wave on a string is described by

`y(x,t) = 7.5 sin (0.0050x + 12t + pi/4)`

(a) What are the displacement and velocity of oscillation of a point at *x *= 1 cm, and *t =*1 s? Is this velocity equal to the velocity of wave propagation?

(b) Locate the points of the string which have the same transverse displacements and velocity as the *x = *1 cm point at *t = *2 s, 5 s and 11 s.

#### Solution 1

a) The given harmonic wave is

`y(x,t) = 7.5 sin (0.0050x + 12t + pi/4)`

For x = 1 cm and t = 1 s

`y = (1,1) = 7.5 sin (0.0050 + 12 + pi/4)`

`= 7.5 sin(12.0050 + pi/4)`

`= 7.5 sin theta`

Where, theta = 12.0050 + pi/4 = 12.0050 + 3.14/4 = 12.79

`= 180/3.14 xx 12.79 = 732.81^@`

`:. y = (1,1) = 7.5 sin (732.81^@`)`

`= 7.5 sin (90xx 8 + 12.81^@) = 7.5 sin 12.81^@`

`= 7.5 xx 0.2217`

`= 1.6629 ~~ 1.663 cm`

The velocity of the oscillation at a given point and time is given as:

`v = d/dt y(x,t) = d/(dt) [7.5 sin(0.0050x + 12t + pi/4)]`

`= 7.5 xx 12 cos (0.0050x + 12t + pi/4)`

At x = 1 cm and t = 1s

`v = y(1,1) = 90 cos (12.005 + pi/4)`

`= 90 cos(732.81^@) = 90 cos (90 xx 8 + 12.81^@)`

`=90cos(12.81^@)`

`=90xx0.975 = 87.75` cm/s

Now, the equation of a propagating wave is given by:

`y(x,t) = a sin(kx + wt + phi)`

where

`k = (2pi)/lambda`

`:. lambda = (2pi)/k`

And `omega = 2pi/k`

`:. v = omega/(2pi)`

`Speed, = vlambda = omega/k`

where

`omega = 12 "rad/s"`

`k = 0.0050 m^(-1)`

`:. v = 12/0.0050 = 2400 "cm/s"`

Hence, the velocity of the wave oscillation at *x* = 1 cm and *t* = 1 s is not equal to the velocity of the wave propagation.

b) Propagation constant is related to wavelength as:

`k = (2pi)/lambda`

`:. lambda = 2pi/k = (2xx 3.14)/0.0050`

= 1256 cm = 12.56 m

Therefore, all the points at distances *n**λ* `n = +-1, +-2...`, i.e. ± 12.56 m, ± 25.12 m, … and so on for *x* = 1 cm, will have the same displacement as the *x* = 1 cm points at*t* = 2 s, 5 s, and 11 s.

#### Solution 2

THe travelling harmonic wave is `y (x,t) = 7.5 sin(0.0050x + 12t + pi/4)`

At x = 1 cm and t =1 sec

`y(1,1)= 7.5 sin(0.005 xx 1 + 12 xx 1 pi/4) = 7.5 sin (12.005 + pi/4)` ...(i)

Now 'theta = (12.005 + pi/4) radian"

`= 180/pi (12.005 = pi/4) "degree" = (12.005xx180)/(22/7)+ 45 = 732.55^@`

:. From (i), `y(1,1) = 7.5 sin(732.55^@) = 7.5 sin(720 + 12.55^@)`

`= 7.5 sin 12.55^@ = 7.5 xx 0.2173 = 1.63 cm`

"Velocity of oscillation", `v = (dy)/(dt) (1,1) = d/dt[7.5 sin(0.005x + 12 + pi/4)]`

`= 7.5 xx 12 cos [0.005x + 12t + pi/4]`

At x = 1 cm, t = 1 sec

`v = 7.5 xx 12 cos (0.005 + 12 + pi/4) = 90 cos(732.35^@)`

= 90 cos (720 + 12.55)

`v = 90 cos(12.55^@) = 90 xx 0.9765 = 87.89` cm/s

Comparing the given equation with the standard form `y(x,t) = t sin[pi/4 (vt+x) + phi_0]`

We get r = 7.5 , `(2piv)/lambda = 12 ` or `2piv = 12`

`v = 6/pi`

`(2pi)/lambda = 0.005`

`:. lambda = (2pi)/0.005 = (2xx3.14)/(0.005) = 1256 cm = 12.56 m`

Velocity of wave propagation, `v = vlambda = 6/pi xx 12.56 = 24` m/s

We find that velocity at x =1 cm, t = 1 sec is not equal to velocity of wave propagation

b) Now all points which are at a distance of +- lambda +-2lambda, +-3lambda from x =1 cm will have sametransverse displacement and velocity. As lambda = 12.56 m, therefoe all points at distances +- 12.6 m +- 25.2 m , +- 37.8 m .. form x = 1 cm will have same displacement and velocity as at x = 1 point t = 2 s, 5 s and 11 s.