Question

A stretched string of length $1$, fixed at both ends, can sustain stationary waves of wavelength $\lambda $ correctly given by:
A) $\lambda = \dfrac{{{l^2}}}{{2p}}$
B) $\lambda = \dfrac{{{p^2}}}{{2l}}$
C) $\lambda = 2lp$
D) $\lambda = \dfrac{{2l}}{p}$

Answer 1

Hint: When a standing wave is sustained by a string there are certain forms or states in which they appear. These forms or states are called harmonics. The number of nodes and antinodes vary from one harmonic to another harmonic. Which harmonic will appear is decided from the frequency of the wave.

Complete step by step solution:
Standing waves are waves that seem to vibrate vertically without horizontal travel. Created from waves that interfere with each other with the same frequency and amplitude while travelling in opposite directions. Node is the position on a permanent wave when, due to destructive interference, the wave remains in a fixed position over time. Antinode is the standing wave position, with a wave vibrating with the maximum amplitude. Harmonic is a standing wave which is a positive integer of the basic frequency. Standing wave harmonics is a wave travelling through a rope which is mirrored at the end of the rope. The wave returns to the right side when the end of the seal is free. The wave is reversed once the end of the rope is fixed. Standing wave patterns represent wave patterns created in a medium when two waves of the same frequencies interfere so that points along the medium appear to be still. Such standing wave patterns are generated in the medium when at certain frequencies it is vibrated. A different standing wave pattern is associated with each frequency. The related frequencies and wave forms are known as harmonics.If the string is long and both ends are set, only vibrational modes are permissible, where an integer of half-wave lengths equals the length of the string.
Therefore, the fundamental wavelength of the wire can be written as,
\[\lambda = \dfrac{{2l}}{p}\]
Where, $p = 1, 2, 3$
Hence, the correct option is D.

Note: A diligent analysis of standing wave patterns shows that the longitude of the wave generating the motive and the duration of the medium in which the design is presented is simple and mathematical. In addition, a predictability about this mathematical equation enables a conclusion about this relationship to be generalised and assumed.