# How can I mathematically describe the reflected sound waves in an open tube?

When a sinusoidal sound wave passes through a tube open at both ends, it will get reflected when it reaches the end of the tube. The reflected wave will itself reflect when it reaches the opposite end of the tube, and so on, generating infinitely many reflection waves that interact which each other and the fundamental. This website allows you to see the reflection waves end their sum.

I know the fundamental is described as $$\cos \left( k x - \omega t \right)$$, but I can't figure out an equation for the reflected waves. How would I do that, as a function of the wavelength and the length of the tube? I want to sum each of the equations mathematically and see for which values of the wave and tube length the resulting wave is 0 everywhere.

• Like basics state in their comment, you have to use the two boundary conditions in order to reach an equation for the steady state of travelling waves (make sure to use both left and right travelling waves). Then you will be able to solve for any field variable you wish. Oct 10, 2022 at 15:34

$$p(x_b,t) = p_{atm} \qquad$$ for open ends
$$u(x_b,t) = 0 \qquad \quad \$$ for closed ends