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We know that stationary waves can only form when the wavelength of the wave and the length of the string satisfies certain conditions.

My question is, how will the string behave when this condition is not met? Because when we flip the string, the waves reflected from each end (same amplitude, same speed and wavelength and travels at opposite directions) still overlap with each other and the resultant wave function is still a function of a stationary wave. If there is no wave on the string because the boundary condition is not met, how does the wave we create disappear?

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The waves in a string (or waves in general) shoudl satisfied the boundary conditions of the problem. If the ends of the string are fixed, the only waves can exist in the string are such that the ends of the strings do not move.

The harmonics corresponding to the condition that the integer number of wave-lengths fit into the string length form a complete basis in the Hilbert space, and any other solution can be represented in terms of these harmonics.

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  • $\begingroup$ I think my confusion lies in how to describe it mathematically. Because I can flip the string and create a sin wave, and it still superposes with the reflected wave to create a resultant wave which has the function of a standing wave. If there is no wave on the string because the boundary condition is not met, how does the wave I create disappear? $\endgroup$
    – Winniebear
    Dec 2, 2021 at 10:57
  • $\begingroup$ @Winniebear not sure what you mean by "flipping the string" - doesn't it simply change the phase of the oscillations by $\pi$? $\endgroup$ Dec 2, 2021 at 11:00
  • $\begingroup$ 'flipping the string' means I touch the string and create two waves of the same amplitude, frequency and travels in opposite directions. Those two waves will be reflected at each end. My confusion lies in if the boundary condition is not met, how will the resultant wave disappear? $\endgroup$
    – Winniebear
    Dec 2, 2021 at 11:02
  • $\begingroup$ @Winniebear I think the problem might be in the direction of your reasoning (the cause and the consequence): the boundary condition is always met, and the only waves one can excite are those satisfying this condition. However you pull the string, its ends remain always attached. This determines how the wave is reflected. $\endgroup$ Dec 2, 2021 at 11:07
  • $\begingroup$ Do you mean that if the length of the string does not satisfy the condition to form a stationary wave, no wave will form when I pull the string? If so, where did the energy go? $\endgroup$
    – Winniebear
    Dec 2, 2021 at 11:55

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