Question 4.6.1: If the string is stretched between two fixed points with dis......
If the string is stretched between two fixed points with distance L between them, the boundary conditions are y(0, t)=y(L, t) = 0. Determine the general solution y(x,t) of the stretched string.
With reference to Equation (4.26)
Y=A\;{\mathrm{Sin}}\left({\frac{\omega}{c}}\,x\right)+B\;{\mathrm{cos}}\;\left({\frac{\omega}{c}}\,x\right), \quad \quad (4.26a) \\ U = C sin ωt + D cos ωt, \quad \quad (4.26b)and the condition that y(0,t) = 0, one requires that B = 0 so that the solution becomes
Y =Asin ( \frac{ω}{c} x) (C sinωt + D cosωt) (i)
The condition that y(L,t) = 0 leads to the equation sin ( \frac{ω}{c} L) = 0 which gives \frac{ω_{n} L}{c} = nπ, with n = 1,2,3,…
By definition, c = fλ where f is the frequency of oscillation and λ is the wave length. One can then write
\frac{ω_{n} }{c} = \frac{n π}{L} (ii)
Therefore, by making use of Equation (4.26a) and recalling that B = 0, the mode shape or normal mode or eigenvector of the vibrating string is given as
Y =Asin (nπ \frac{x}{L} ). (iii)
Note that this equation originates from the rhs of Equation (i).
In general, free vibration of a string contains many of the normal modes, and the equation for the displacement may be written as
y(x ,t) = \sum\limits_{n = 1}^{∞}{sin (nπ \frac{x}{L} ) (C_{n} sinω_{n}t + D_{n} cosω_{n} t )}\quad \quad (iv).
where the arbitrary constants C_{n} and D_{n} can be evaluated with the initial conditions of y(x, 0) and dy(x, 0)/dt. The integer n is the number of mode shapes or eigenvectors.