# Physical manifestation of mechanical waves for a non-defined value of the cotangent function

Consider a horizontal rod that is fixed on one of its ends. The other end is under sinusoidal excitation. The wave propagation in the rod can be described with a wave function which is a solution to the following wave equation with the boundary conditions:

$$c^2{\partial ^2u(t,x) \over \partial x^2}={\partial ^2u(t,x) \over \partial t^2} \tag1$$

$$u(0,x)=0\tag2$$ $${\partial u(t,x) \over \partial t}\Bigg|_{t=0}=0 \tag3$$ $$u(t, 0) = A \sin (w_ot)\tag4$$ $$u(t, L)=0 \tag5$$ $$t \geq 0 \tag6$$ $$L \geq x \geq 0 \tag7$$

where $$u(t, x)$$ is the wave function, $$c$$ is the wave speed, $$w_o$$ is circular frequency, $$t$$ is time, $$x$$ is space and $$L$$ is the rods length. The solution to this problem is derived here. The wave function is formulated as:

$$u(t,x) = {A \over 2} \Bigg[ \sin \Biggl(w_0\bigg(t+ {x\over c}\bigg)\Bigg) + \cot \bigg( w_0 {L \over c} \bigg) \cos \Bigg( w_0\bigg(t+ {x\over c}\bigg)\Bigg) \Bigg] + \Bigg[A \sin \Bigg(w_0\bigg(t - {x\over c}\bigg)\Bigg)- {A \over 2} \Bigg( \sin \Bigg(w_0 \bigg(t- {x\over c}\bigg)\Bigg) + \cot \bigg( w_0 {L \over c} \bigg) \cos \Bigg(w_0 \bigg(t- {x\over c}\bigg)\Bigg)\Bigg)\Bigg] \tag{8}$$

In theory, we can choose the material and length of the rod so the argument in the cotangent function is $$n\pi$$, where $$n$$ is a whole number. Since the cotangent function is not defined for $$n\pi$$, what would actually happen if we designed the rod so that $$w_0 {L / c} = n\pi$$? Would the rod explode?

If you choose your frequency such that $$w_0L/c=n\pi$$, you are pushing the system into resonance. The amplitude will steadily grow in this case, and at some point nonlinear effects would kick in. Also, as the time wore on, more energy would be taken from the system. Thus, the amplitude would be large, but it would not be likely to explode.