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yes, Its possible to have a longitudinal wave on a stretched string by stroking it along its length by a rosined cloth. However, in the wave motion of a stretched string ( tant string with fixed ends) wave (disturbance) produced at one fixed end travels along the length of the string and get reflected back at the other end. Since the original wave and the reflected wave have the same frequency and amplitude, they superimpose to produce stationary transverse disturbance.

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