Question

When a standing wave is set up on a string fixed at both ends, which of the following statements is true?

1) The number of nodes is equal to the number of antinodes.
2) The wavelength is equal to the length of the string divided by an integer.
3) The frequency is equal to the number of nodes times the fundamental frequency.
4) The shape of the string at any instant shows a symmetry about the midpoint of the string.

Answer 1

When a standing wave is set up on a string fixed at both ends, the correct statement is that the wavelength is equal to the length of the string divided by an integer. Nodes and antinodes form according to the harmonic series, with both ends of the string being nodes due to the fixed conditions.

Step-by-step explanation:

When a standing wave is set up on a string fixed at both ends, it is important to understand the relationship between nodes, antinodes, wavelengths, and frequencies. First, let us address the statements provided:

1. The number of nodes is not always equal to the number of antinodes. In fact, for a string fixed at both ends, there is always one more node than antinode.
2. The wavelength is indeed equal to twice the length of the string divided by an integer. This is because both ends of the string must be nodes, and the different modes of vibration are determined by the number of half-wavelengths that fit into the length of the string.
3. The frequency is not simply the number of nodes times the fundamental frequency. It is determined by the specific mode of vibration that the string is in, based on its harmonic series.
4. The shape of the string does show symmetry about the midpoint, but this is not always strictly true, especially for higher modes of vibration where complex patterns can emerge.

Therefore, the correct statement is 2) The wavelength is equal to the length of the string divided by an integer. This is because the wavelength is determined by the distance between the points where the string is fixed in place.