An example of a wave with an integral multiple wavelength and frequency is a standing wave. A standing wave is formed when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other. The interference creates stationary points called nodes and regions of maximum displacement called antinodes.
In a standing wave, the wavelength and frequency are related in such a way that the wavelength is an integral multiple of the distance between adjacent nodes or antinodes. This relationship allows the wave to maintain a fixed pattern without appearing to propagate.
A commonly observed example of a standing wave is a vibrating string or a musical instrument string. When a string is plucked or struck, waves travel along the string in both directions. When the incident and reflected waves interfere, they can create a standing wave pattern on the string.
In this case, the fundamental frequency (the lowest possible frequency) of the standing wave corresponds to the first harmonic or the fundamental mode of vibration. The string vibrates in a pattern where it has nodes at both ends and an antinode in the middle. The wavelength of this standing wave is twice the length of the string.
As the frequency increases, higher harmonics or overtones can be produced. Each higher harmonic has an integral multiple of nodes and antinodes along the length of the string. For example, the second harmonic has two antinodes and one node, the third harmonic has three antinodes and two nodes, and so on. The wavelength of each harmonic is a fraction of the fundamental wavelength, corresponding to the integral multiple of the number of nodes or antinodes.
Therefore, a standing wave is an example of a wave with an integral multiple wavelength and frequency, where the wavelength is determined by the distance between nodes or antinodes, and the frequency is determined by the number of complete oscillations per unit time.