The physics of wind instruments is based on the concept of standing waves. When the player blows into the mouthpiece, the column of air inside the instrument vibrates, and standing waves are produced. Although the acoustics of wind instruments is complicated, a simple description in terms of open and closed tubes can help in understanding the physical phenomena related to these instruments. For example, a flute can be described as an open-open pipe because a flutist covers the mouthpiece of the flute only partially. Meanwhile, a clarinet can be described as an open-closed pipe because the mouthpiece of the clarinet is almost completely closed by the reed. (a) Consider a pipe of length L open at both ends. If the speed of sound in air is v, what is the fundamental frequency f1 of the sound wave produced when you blow into the pipe (in terms of L and v)? (b) If you drill a hole at a position half the length of the pipe, what is the new fundamental frequency f1' of the sound that can be produced in the pipe (in terms of the original length L and v)? (c) For which values of n does the nth harmonic of the original pipe of length L from part (a) also exist as a harmonic of the pipe in part (b). (d) What length of open-closed pipe would you need to achieve the same fundamental frequency f1'' as the original open-open pipe of length L from part (a)? (e) What is the frequency of the first possible harmonic after the fundamental frequency in the open-closed pipe from part (d) (in terms of L and v)?