Fourier Series: Solutions of One Dimensional Wave Equation - Vibrating String with Initial Velocity and Initial Displacement Given

ONE DIMENSIONAL WAVE EQUATION: VIBRATING STRING WITH INITIAL VELOCITY AND INITIAL DISPLACEMENT GIVEN

Type 3. Vibrating string initial velocity and initial displacement given:

The initial and boundary conditions are

(i) y (0, t) = 0

(ii) y (l, t) = 0

(iii) ( ∂y/∂t )_{(x, 0)} = g(x)

iv) y(x, 0) = f(x)

The suitable solution is a

 y(x, t) = (C₁ cos px + C₂ sin px) (C₃ cos p at + C₄ sin p at)

   …………(1)

Apply condition (i) we get C₁ = 0

Apply condition (ii), we get p = nπ/l

The most general form is

Substitute, C_n, D_n values in (2), we get the general solution.

PROBLEM ON VIBRATING STRING WITH INITIAL VELOCITY AND INITIAL DISPLACEMENT GIVEN

Example 23: Solve the problem of the vibrating string for the following boundary conditions: (i) y (0, t) = 0, (ii) y (l, t) = 0

 (iii) ∂y/∂t (x, 0) = x(x−1), 0<x<l (iv) y (x, 0) = 

Solution: The wave equation is 

From the given problem, we get the following boundary and initial conditions

(i) y (0, t) = 0 for all t> 0

(ii) y (l, t) = 0 for all t > 0

(iii) ( ∂y/∂t )_{(x, 0)} = g(x) = x(x‒l),  0<x<l

iv) y(x, 0) = f(x) = 

Now, the suitable solution which satisfies our boundary conditions is given by

 y (x, t) = (c₁ cos px + c₂ sin px) (c₃ cos pat + c₄ sin pat)   …….(1)

Applying condition (i) in equation (1), we get

 y (0, t) =  (c₁+0) (c₃ cos p at + c₄ sin p at) = 0

Here, c₃ cos pat + c₄ sin pat ≠ 0       [. It is defined for all t]

Therefore, we get c₁ = 0

Substitute, c₁ 0 in equation (1), we get

 y (x, t) = c₂ sin px (c₃ cos pat + c₄ sin pat)

Applying condition (ii) in (2), we get

 y (l, t) = c₂ sin pl (c₃ cos pat + c₄ sin pat) = 0

Here, c₃ cos pat + c₄ sin pat ≠ 0           ['. it is defined for all t]

Therefore, either c₂ = 0 or sin pl = 0

Suppose, we take c₂ = 0 and already we have

 c₁ = 0 then we get a trivial solution.

Therefore, we consider c₂ ≠ 0 and

sin pl = 0

pl = nπ   

      [sin nπ= 0]

 p = nπ / l

      [n being a_n integer]

EXERCISE

1. A string is stretched between two fixed points at a distance of π cm and the points of the string are given initial velocities v, where

 ν= x, in 0 < x < π/2

 = π‒x, in π/2 <x<π after displacing it to the position

 y = x(π‒x) at t=0. Find the displacement of the string at any time.

2. Find the displacement of a tightly stretched string of length 7 cm vibrating between fixed end points, if initial displacement is 10 sin (3πx/7) and initial velocity is 15 sin(9πx/l).