Question

Find the displacement w(x.t) of an elastic string subject to the following conditions: 1) (sint, 0, 0 sts 2TT w(0,t) = f(t) = otherwise lim w(x, t) 0 Wave equation: дх2 W(x,0) 3D 0 and — (х, 0) — 0 дt

Answer 1

Elaboration:

->All points on the string are at zero displacement at t=0 and have zero initial velocity at t=0

->There is a oscillation that is given to the leftmost part of the string from t=0 to t= 2 pi. It is a sinusoidal oscillation.

->c is the speed at which any disturbance travels along the string. (unit: m/s)

So, it is quite easy to picture that after a certain time say t seconds, all points on the string will still be at zero displacement except one cycle of a sine wave travelling along the string at a speed c m/s.

W One cycle of sine wave X 1

So the solution to the displacement of the string w(x,t) will be like, for all other points, other than the place where the single sine wave cycle is will be zero

so , partially, for all points not in the sine wave cycle.

u(α, t) Ο υι ".

How to determine points that are in the sine wave cycle?

Well, the front of the sine wave cycle was the first to start traveling. So, at time t , it is at a distance x= ct . The backmost part of the sine wave is thus a wavelength behind.

C c= λv=>入%3=D-

the sine wave's general form is:

Asin(wt k) = Asin (2Tvt k)

on comparing with the given sine wave, we get:

2πνt-t ν 2π

Thus

ΧΕTΕ ΕΣΤα

Thus, the back most part of the sine wave cycle is at

ct 2TC= c(t 2m) -27T

So, now we can complete the solution:

sin (a), c(t 2) x ct 0, otherwise w(x(t), t)