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An air bubble has formed inside a pneumatic pipe. A random vibration causes the bubbble to burst, sending an acoustic pulse through the pipe which is described by \[ \frac{\partial p}{\partial t}+K_{0} \frac{\partial u}{\partial x}=0 \] and \[ \frac{\partial u}{\partial t}+\frac{1}{ ho_{0}} \frac{\partial p}{\partial x}=0 \] wherepis the fluid pressure perturbation anduthe perturbation velocity.0is the ambient density in the fluid andK0is the bulk modulus of the air.

Question: Solve the systems of equations in characteristic form.

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