Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s. Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, deter-mine the locations along the pipe (measured from the left end) of the displacement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.
Solution 25E Step 1 If L is the length of the pipe open at both end, then the possible wavelengths are given by m m 2L Now for fundamental mode, that is for m = 1 we have = 2L 1 L = 1 2 So, for the fundamental tone, the pipe will contain only half of the wavelength. So for the fundamental tone, the displacement node will be at the center which means 0.60 m from the left end and pressure node at the two open ends, i.e. at the 0.00 m and 1.20 m from the left end. For next overtone, we have 2 2 2L 2 L Hence the pipe will contain one full wavelength. Since the open end will have antinode, there will be three antinode and two node. So the position of the displacement node will be 0.30 m and 0.90 m. And the position of the pressure node will be 0.00 m, 0.60 m and 1.20 m. For the 3’rd overtone, we have 3 3 2L 2 3 L 3 3 So the pipe will contain2 wavelength. Hence there will be 4 antinodes and 3 nodes. So the position of the displacement nodes are 0.20 m, 0.60 m and 0.10 m from the left. And the position of the pressure nodes are at 0.00 m, 0.40 m, 0.80 m and 1.20 m.