Chapter 16

Consider the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0

It was shown in Section \(13.2\) that the equation of a vibrating string is $$\frac{T}{\rho} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}$$ where \(T\) is the constant magnitude of the tension in the string and \(\rho\) is its mass per unit length. Suppose a string of length 60 centimeters is secured to the \(x\) -axis at its ends and is released from rest from the initial displacement $$f(x)=\left\\{\begin{array}{lr} 0.01 x, & 0 \leq x \leq 30 \\ 0.30-\frac{x-30}{100}, & 30