The number of nodes, including the end points, in a standing wave that is three wavelengths long is
A)4.
B)5.
C)6.
D)7.
E)none of the above

Question

Quizplus · Accepted Answer

A standing wave that is three wavelengths long will have four nodes per wavelength (including the endpoints). Thus, for three wavelengths, the total number of nodes is 3*4 - 2 (since the two endpoints are counted in each wavelength calculation, they should be subtracted once to avoid double counting) = 12 - 2 = 10. However, this calculation was incorrect. The correct approach is to consider that a single wavelength has two nodes at its ends. For three wavelengths, we would have 3*2 = 6 nodes at the ends of the wavelengths, plus one additional node at each internal boundary between wavelengths, adding 2 more nodes, for a total of 8. However, this explanation also contains a mistake in the calculation and understanding of nodes in a standing wave. The correct explanation is simpler: In a standing wave, nodes are points of zero amplitude. For a wave that is three wavelengths long, there are 4 nodes per wavelength (including the endpoints). However, this description mistakenly counts segments instead of nodes. The correct count is simpler: each wavelength has two nodes at its ends, and since wavelengths are contiguous, the nodes at the ends of each wavelength (except the very first and last ones) are shared. Thus, for three wavelengths, there are 3 + 1 = 4 nodes, which is also incorrect. The correct calculation is based on the understanding that nodes occur at both ends of a wavelength and at every half-wavelength interval. For three wavelengths, the correct number of nodes, including endpoints, is actually 7, reflecting a misunderstanding in the initial explanation.

The Number of Nodes, Including the End Points, in a Standing