4. Lab Question: Modes in a Tube Note: In the lab, you used a couple different tube lengths and varied the frequency. In this problem, we will change the length of the tube continuously while holding the frequency constant. A tuning fork vibrating at frequency fo is held above a long cylindrical pipe with an open top end. The pipe is partially filled with water up to a slowly rising height, and the distance from the slowly rising water level to the top of the pipe is L. (Note that L is not a constant and should be treated as an unknown.) For some values of the water level, the air column in the pipe (the air above the water) resonates to the tuning fork. Assume the speed of sound Us is known. For this problem, write your answers in terms of the given variables Us, fo, and any other known constants. If you create a variable when making a list, you must define it. (a) Draw pictures of the lowest 3 modes (standing waves) which resonate with the tuning fork in an air column of length L. (b) Write an expression for all the allowed values of the length L of the air column which resonate with the tuning fork Note: You must justify your answer if you want to receive partial credit. Hint: Use your drawings! (c) As the water height rises, a student performing the experiment notices equal differences in water height between successive resonances. Calculate the height difference between modes. (d) What is the last air column length L which resonates before the pipe completely fills with water? SALE I NIN